1 Introduction What Is Numeracy? What Makes a Person Numerate?

Section 1 Introduction 1

1 Introduction

What is numeracy? What makes a person numerate?

In broadest terms, numeracy (or numeric literacy or quantitative literacy) can be viewed as a combination of specific knowledge and skills, which are needed (but do not suffice) to function in the modern world. What other skills, apart from numeracy, do we need to posses? For instance, social intelligence, cross-cultural competency, and new-media literacy. Numeracy involves reasoning from, and about, numeric information (data), which can be presented in a variety of ways (such as numeric, graphic, narrative, visual, and dynamic forms). Inspired in part by questions we routinely ask in mathematics (What is this? Why is this true? How do we know?), we expand numeracy to include critical, evidence-supported thinking, common sense, and logical reasoning in situations and/or contexts that do not explicitly nor implicitly involve numbers or quantitative information. (Examples follow.) In our conceptualization of numeracy, and for the purpose of this course, a numerate person is assumed to be a university student, rather than a general member of our society. (One reason lies in the fact that we must make certain assumptions about the background, mathematical and otherwise.) To make this concept of numeracy more transparent, we now illustrate its aspects in several examples (which are in no way an exhaustive). In the article After 7,500% rally, cryptocurrency founder sells his coins published in the Globe and Mail, on 20 December 2017, we read “Litecoin dropped about 4 per cent to $ 319 at 1:02 p.m. in New York, according to prices on Bloomberg. The coin is still up about 75-fold since the end of 2016, according to http://coinmarketcap.com prices. The market value was $ 17.5-billion.” A numerate person is able to understand and work with the percent information (for instance, they know that the value of the Litecoin, before it dropped 4% to $ 319, could not have been $ 500); they understand that 75-fold increase means 75 times, i.e., that the Litecoin value at the end of 2016 is multiplied by 75; they can visualize $ 17.5 billion, by comparing, or relating to other quantities, such as saying that “$ 17.5 billion is the salary of 175 thousand top paid high school teachers in Ontario.” A numerate person can compute their body mass index (from a formula they recovered from Wikipedia, for instance), lookupitsvalueinthecharttodetermine whether they are deemed overweight or not. They can interpret results of their blood test by relating numbers: if their cholesterol reading is 5.78, and the desired level is less than 5.20, they realize that their cholesterol level is outside normal limits. A numerate person can reason logically: they understand that while nausea, headache, fever, and vomiting are symptoms of bacterial meningitis, they do not cause meningitis; as well, someone showing these symptoms does not necessarily have meningitis. (Thus, one has to be careful how to interpret answers from online symptom checkers.) A numerate person understands the difference between causation and correlation. A numerate person has basic understanding of probability and risk. For instance, if something is known to occur twice a week, then 4 occurrences in a certain week might not constitute an epidemic, but rather reflect a probabilistic fact that unlikely events nevertheless do occur. As well, a numerate person will recognize that statistics derived from a small sample is very likely worthless, and often not in any way representative of the population from which the sample has been drawn. 2 NUMERACY

A numerate person is aware of their digital footprint in social media and electronic marketplaces, and is concerned about privacy and confidentiality. Mathe- matical algorithms can spy on the interactions within social media, in order manip- ulate the users’ behaviour. For instance, read about the case of Cambridge Analyt- ica at http://www.cbc.ca/news/opinion/cambridge-analytica-opinion-1.4588857). A numerate person is familiar with basics and biases of search algorithms and targeted marketing. A numerate person is able to use graphs to understand, illustrate and illumi- nate concepts, presented in both static and dynamic forms. In the article This chart shows how bread prices soared during the price-fixing scheme published in MacLeans, 22 December 2017, https://goo.gl/U5CV7A we find the following figure (Source: Statistics Canada, CANSIM Table 326-0021):

We read: “A glance at Statistics Canadas data on food prices and the consumer price index, which compares the cost of a ﬁxed set of goods and services over time, seems to show the price ﬁxing in action.” A numerate person is able to look up the meaning of consumer price index (CPI), and by learning that CPI was reset to 100 in 2002, note that the graphs are accurate, as they all cross the CPI of 100 line in 2002. As well, they would realize that indeed the CPI for bread, rolls and buns has been increasing at a fast pace since 2002, and grew substantially larger than other quantities represented in the graph. For instance, they could determine that it is about 50 units larger than the CPI for all items in 2015. A numerate person is a critical person, and asks what “ecotourism,” or the label “green” on a laundry detergent actually mean. They do not pay for an “all inclusive” vacation before insisting to know what is not included. A numerate person reads the small print and, for instance, understands how the annual percentage rate (APR) is used to compute the monthly interest on their credit card. As well, a numerate person holds certain beliefs and values. They understand and appreciate the importance of mathematical and logical reasoning for living Section 1 Introduction 3

and making decisions; they accept the fact that numeracy takes time and practice to achieve and is an important part of life-long learning; they are willing to adopt attitudes, beliefs and work habits in order to overcome potential learning and other (personal) barriers that might exist; they are inquisitive, motivated to learn on their own, and are willing to engage with challenging topics and ideas. Why numeracy? A survey conducted by the Conference Board of Canada in 2012-2014 [1] claims that about 55% of Canadian adults have inadequate numeracy skills, and that it is a “signiﬁcant increase from a decade ago.” Based on the ﬁrst results of the Programme for the International Assessment of Adult Competencies, Statistics Canada reports that “Canada ranks below the OECD average in numeracy, and the proportion of Canadians at the lower level is greater than the OECD average.” [2] (References appear at the end of this chapter.)

First steps ...

Numeracy is about looking critically at information (often involving numbers), and making sense of it. But above all, is it about asking questions, actually, asking good questions, so that we can understand, and based on our understanding, make good decisions. To illustrate this, we look at several examples. (1)Whatdoesnatural in Natural spring water mean? In 100% natural spring water?Whatdoespure in Pure Life premium drinking water mean? In pure drinking water? (All of these were taken from labels of bottled water.) When we look at food labels on bottled water, we find ingredients such as magnesium sulfate, potassium chloride, salt, calcium chloride, magnesium chloride, and potassium bicarbonate, together with - of course - purified water (i.e, filtered tap water). Routinely, bottled water companies purify water, and then add some of the ingredients listed above (or some others) back into it. Why? Usually, they claim, to give taste to water. So natural is not really natural, and neither is pure pure, 100 percent water. (2) The table below shows a sodium content for four beverages, together with the percent daily value of sodium.

Drink Sodium content % daily value Monster Rehab Energy Iced Tea 110 mg/240 mL 5 Coca Cola 30 mg/250 mL 1 Starbucks Doubleshot Fortiﬁed Coﬀee Drink 160 mg/444 mL 7 Gatorade Perform Orange Thirst Quencher 250 mg/591 mL 11

So what is the daily value of sodium? Based on Coca Cola, if 30 mg is 1 percent, then the daily value is 3000 mg. If 110 mg represents 5% (Monster), then the daily value is (110/5)*100 = 2200 mg. If 160 mg represents 7% (Starbucks), then the daily value is (160/7)*100 = 2285 mg. If 250 mg represents 11% (Gatorade), then the daily value is (250/11)*100 = 2272 mg. According to Health Canada (https://goo.gl/Q4qqjm), adequate intake of sodium for teens and adults is 1500 mg/day, with upper limit of 2200-2300mg/day. Comments? (3) Shinerama is Canada’s largest post-secondary fundraiser in support of Cystic Fibrosis Canada (McMaster students routinely participate). What is there to ask about, isn’t this a good cause? 4 NUMERACY

Yes it definitely is, but – it costs to raise money! Let’s check how Cystic Fibrosis Canada does it. The web page 2017 Charity 100: Grades at https://goo.gl/ED1pA4 gives a ranking of charitable institutions, by evaluating them on important parameters. The screenshot from the web page shows that, in terms of charity efficiency, Cystic Fibrosis Canada is given the grade of D. What does it mean? Comparing with other organizations, we see that that’s a fairly low ranking (actually among the worst two in its category Fundraising Organizations). Probing further, we look at two important parameters: charity efficiency (defined as the percentage of total funds collected that actually go toward the cause), and fundraising efficiency (the amount of money needed to raise $ 100). (These definitions can be found on https://goo.gl/ED1pA4.)

The figure shows that the charity efficiency of Cystic Fibrosis Canada is 56.0%, and its fundraising efficiency is $ 32.00. As comparison (data taken from the same web page): the charity efficiency of Terry Fox Foundation is 84.0%, and its fundraising efficiency is $ 16.00. United Way of Calgary and Area does it even better: its charity efficiency is 89.0%, and the fundraising efficiency is $ 6.00. What is Really Important about the Fundraising case Study? The fact that Cystic Fibrosis Canada has the charity efficiency of 56.0%, on its own, means very little. Is it good, bad, average? It only made sense when compared with other organizations. Hence the message - numbers by themselves are not as informative, or powerful. We need to place them into context, such as comparing to a reference frame obtained by looking at other organizations. Closer to home example. You got your Math 2UU3 test back, and your grade is 28/40=70%. That, by itself does not as much as knowing what other people got on the same test. For instance, if the class average was 55%, then you know that you did very well, but if the class average was 85%, then it’s quite the opposite. Can you think of other examples where establishing a reference frame provides a more valuable information? Question (always ask questions!): is the site we used in this case study reliable? Who created it and who has been and maintaining it? Section 1 Introduction 5

(4) Consumers looking at the following label in a British supermarket chain Tesco

might conclude that the meat they are buying is from a farm, and (looking at the name) believe that it is a local farm. However, investigations show that although Woodside Farm exists, they do not supply the meat to the supermarket in question. As a matter of fact, Woodside Farms is a brand name, and not a farm. In Guardian, 13 December 2017, Tesco faces legal threat over marketing its food with ’fake farm’ names we read “In March 2016 Tesco, the UKs largest retailer, sparked controversy after launching a budget range of seven own-label farm brands including Woodside Farms and Boswell Farms for fruit and veg as well as meat based on British-sounding but ﬁctitious names. Some foods were imported from overseas and given British names to make them sound local.” (5) Many more examples coming, starting with the next section.

Not everything is quantitative, nor quantitative is useful

There are situations when knowing the temperature does not say the whole story. If it is a few degrees (Celsius) below zero, we might feel cold, very cold, or freezing (or not at all cold). The size of a plot of land might not be as useful–for instance for farmers–as the quality of the grass on it (hence the cow index,usedin Ireland or mother cow index used in parts of the US; both indices “measure” the ability of the plot of land to sustain certain number of cattle). To many people the phrase very low risk of side effects seems to be lot more useful than 0.3% chance of side effects. We can find many more examples where measuring, using quantitative scales, metres, grams, seconds, etc. do not tell the whole story, or tell it in ways which are not useful nor meaningful. The article How we measure without maths published by BBC News (27 July 2018, https://bbc.in/2vxBEne) talks about qualitative scales and the need, or preference for narratives over quantitative data. We read “These are yardsticks that measure observable, but not necessarily numerical, properties and we use them all the time. Qualitative scales are sometimes humorous and often downright bizarre, but they are just as valuable as quantitative scales for imagining relationships between properties and standardis- ing ideas.” These qualitative assessments/ scales “range from chili pepper heat to mineral hardness to ocean breezes [...] Qualitative scales allow us to label variables with little or no quantitative information. These unusual units of measurement are often colloquial: guesstimations and as-the-crow-flies rules of thumb that allow for quick assessments and comparisons.” 6 NUMERACY

And yet, qualitative descriptions “prove their usefulness time and again. With- out them, we would struggle to conceptualise ideas of pain (a doctor might ask a patient to rank his symptoms) or grade the severity of weather conditions (like the Beaufort Scale does).” Suggestion: heck Wikipedia for Beaufort Scale; in particular, look at the chart, and note the verbal descriptions. A part of the chart is reproduced here:

(Source: Wikipedia https://bit.ly/1jxYJ0y)

Section references [1] The Conference Board of Canada (n.d.; the page claims that the data is accurate as of June 2014). Adults With Inadequate Numeracy Skills. https://goo.gl/jx6Jny [2] Statistics Canada. Skills in Canada: First Results from the Programme for the International Assessment of Adult Competencies (PIAAC) Section 2 Mathematical Reasoning and Numeracy 7

2 Mathematical Reasoning and Numeracy

In this chapter we learn about building blocks of mathematics (deﬁnition, theorems, and logical reasoning), and what they represent in numeracy, i.e., how they relate to real-life situations.

Deﬁnitions

In mathematics, a definition introduces a new object, a new property of objects, or a new relation between objects, based on previously defined or established objects, properties and/or relations. By “established” we mean through accepted mathematical routines, such as by providing acceptable evidence (“proof”). Mathematical definition is succinct, clear and precise, and does not leave space for ambiguity. It answers the question “What?” In mathematics, it is not possible to work with anything that is not defined. (there is a caveat to this; read on). Consider the following example of a mathematical definition, which introduces a new concept, that of a prime number: A prime number is a positive integer which has exactly two distinct divisors: the number 1 and itself. By reading (and re-reading) this definition carefully we can figure out what a primen number is. First of all, only positive integers (e.g., 1, 2, 3, 4, 5, etc.) could be prime numbers. Since the number 1 has only one divisor (namely, itself), it is not a prime number (as the definition requires two distinct divisors). The number 2 is divisible by 1 and by 2, and thus, having exactly two distinct divisors, it is aprimenumber.Thenumber6isnotaprimenumber,asithasmorethantwo distinct divisors: 1, 2, 3, and 6. There are alternative ways to define a prime number (try to find one!), but they all have the same meaning. No matter what definition is used, 1 and 6 are not prime numbers, and 2 and 17 are prime numbers. Once established, mathematical definitions do not change (for instance, the definition of a prime number that we wrote is more than 2300 years old). As well, they do not get adjusted based on culture, language, geography, nor for political pressures, nor for any other reason. Outside of mathematics, in other disciplines, definitions can rarely be qualified in the same terms. One can argue that, in real life, there is nothing that would qualify as a mathematical definition. Real life definitions may depend on authority, history, context, geography, time, and so on. Let us explore definitions in real life. First of all, we do not routinely use the word definition in everyday language. Instead of asking “Can you define this for me?” we might say “Can you explain?” “What does it mean?” “What is our agreement on this?” “What is our understanding about this?” “What is the convention?” and so on. As our case studies will show, it is a challenge (and often not possible) to create a statement that would share the properties of a mathematical definition when dealing with non-mathematical contexts. In some cases (can you suggest some?), it is even desirable to keep things unclear and vague. In mathematics, there is no ambiguity about the meanings of the words used (not just in definitions, but everywhere), as all of them are precisely defined. In contrast, in real life, our understandings are often implicit. A “square of a chocolate” is not a square, but a three-dimensional shape. As a two-dimensional figure, a square has no thickness, so we cannot bite into it (actually nothing on our planet is two-dimensional). Red wine is not red, and white wine is not white. 8 NUMERACY

A “red herring” does not have to be red, nor a herring. So, when it’s not a dried smoked herring, turned red by smoke, what is a “red herring”?

Case study What is a Planet? After the revision of the definition of a planet at the meeting of the Interna- tional Astronomical Union in August 2006, our Solar system “lost” its most distant planet, Pluto. The new definition (see Why is Pluto no longer a planet? BBC News, 13 July 2015 https://goo.gl/vxpMUw) states that a planet is a celestial body that (a) is in orbit around the Sun (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and (c) has cleared the neighbourhood around its orbit.

In the BBC article, we read: “Pluto met the ﬁrst two of the these criteria, but the last one proved pivotal. ‘Clearing the neighbourhood’ means that the planet has either ‘vacuumed up’ or ejected other large objects in its vicinity of space. In other words, it has achieved gravitational dominance.”

Case study What is Climate Change? Everyone seems to be talking about it – but what is a good definition (convention, understanding) of the term “climate change”? David Suzuki Foundation is a Canadian non-profit organization with head- quarters in Vancouver (and offices in Toronto and Montreal), working on protect- ing the natural resources and environment, and on creating a sustainable living for all Canadians. On the web page https://goo.gl/T3Jnno), under What is Climate Change? we read: “In a nutshell, climate change occurs when long-term weather patterns are altered – for example, through human activity. Global warming is one measure of climate change, [as] is a rise in the average global temperature.” NASA (National Aeronautics and Space Administration) is a U. S. agency in charge of space programs, and aeronautics and aerospace research. In its answer to What is Climate Change?,athttps://goo.gl/T3Jnno), we find NASA’s understanding: “Climate change is a change in the usual weather found in a place. This could be a change in how much rain a place usually gets in a year. Or it could be a change in a place’s usual temperature for a month or season. Climate change is also a change in Earth’s climate. This could be a change in Earth’s usual temperature. Or it could be a change in where rain and snow usually fall on Earth. Weather can change in just a few hours. Climate takes hundreds or even millions of years to change.” Australian Academy of Science https://goo.gl/cfthcz defines climate change as “Climate change is a change in the pattern of weather, and related changes in oceans, land surfaces and ice sheets, occurring over time scales of decades or longer.” Section 2 Mathematical Reasoning and Numeracy 9

As we can see, there are similarities, but also diﬀerences in how these organizations conceptualize climate change. Clarity and sharpness of a mathematical deﬁnition are no longer there.

Case study ADHD and Social Anxiety Disorders Unlike mathematical definitions, the definition of ADHD (Attention-Deficit/ Hyperactivity Disorder) has undergone numerous revisions, and the most recent version appears in 2013 in the fifth edition of Diagnostic and Statistical Manual of Mental Disorders [1]. It is not just revisions and updates; sometimes, people (experts, researchers) do not agree on a definition. As soon as the Diagnostic and Statistical Manual (Third Edition), in 1980, introduced the term “social phobia”, there has been confusion as to whether or not shyness is a mental disorder (later, “social phobia” was replaced by a new category “social anxiety disorder” or SAD). In the article When does benign shyness become social anxiety, a treatable disorder? [2] which aimed at clarifying the issue, we read: “While many people with social anxiety disorder are shy, shyness is not a pre-requisite for social anxiety disorder.” Note: the sentence “While many people with social anxiety disorder are shy, shyness is not a pre-requisite for social anxiety disorder” can be represent visually (these kinds of representations will became very useful), as in this figure.

shy people

people with SAD

Shy people are represented by the blue oval, and SAD people by a yellow oval. The overlap represents the fact that “many people with social anxiety disorder are shy.” The part of the yellow oval which is outside the blue oval in the diagram represents the fact that one can suffer from SAD without being shy. The adjustments to the definitions of SAD have had serious consequences. In their online article Shyness ... Or Social Anxiety Disorder? Social Anxiety Insti- tute https://goo.gl/F8UFpf states: “The definition of ‘social anxiety disorder’ has shifted over the past thirty years as the seriousness of the situation became clearer, and government epidemiological data consistently showed a larger percentage of the general population suffering from social anxiety symptoms.” Not everyone agrees. “In Shyness: How Normal Behavior Became a Sickness (Yale University Press, October 2007, check https://goo.gl/WkkSML), Northwest- ern’s Christopher Lane chronicles the ‘highly unscientific and often arbitrary way’ in which widespread revisions were made to ‘The Diagnostic and Statistical Man- ual of Mental Disorders’ (DSM), a publication known as the bible of psychiatry that is consulted daily by insurance companies, courts, prisons and schools as well as by physicians and mental health workers.” Of course, there is potential agenda to this. The article continues: “By labeling shyness and other human traits as dysfunctions with a biological cause, the doors 10 NUMERACY

were opened wide to a pharmaceutical industry ready to provide a pill for every alleged chemical imbalance or biological problem, he adds.” In Summer 2018, we found ads in Hamilton (HSR) buses that solicited vol- unteers to participate in a McMaster University “research study evaluating an investigational medication in Social Anxiety Disorder.”

Case study How to Define the Colour Red? In physics, the coulour red is defined as “the color at the end of the visible spectrum of light, next to orange and opposite violet. It has a dominant wavelength of approximately 625–740 nanometres.” (Wikipedia, https://goo.gl/9Uyzh6;a nanometre is one-billionth of a metre). For numerous tasks that computers do, for the use in software, or on internet (say, background of a web page), the colour red is defined using the string FF0000 (which represents the mix of red, green and blue pigments); alternatively, this string can be represented as the RGB (red, green, blue) triple (255, 0, 0).

Left: colour picker as found, for instance, in Adobe Illustrator or Photoshop. Right: picking a red colour (Adobe Dreamweaver interface, and HTML code below). One can make a deﬁnition precise, but then its use might be impractical, or might not make much sense in routine real-life interactions. We will not be buying FF0000 car any time soon, nor order online a t-shirt in the 625–740 nanometres spectrum. In real-life we migt describe red as colour of ﬁre or colour of blood. Section 2 Mathematical Reasoning and Numeracy 11

Investigate Further Examples of Definitions in Real Life; Hints and Leads (1) Within sciences, many terms are misunderstood, and/or used in a number of ways, such as: uncertainty, energy, genetic, trauma, minerals, and so on. Read about it at https://bit.ly/2N2TYhr. (2) In biology, the basic term “species” is not defined. What constitutes acceptable evidence (“proof”) is unclear. (3) After a long and challenging process, in January 2017, McMaster University produced (updated in late 2019) and made public a policy document addressing sexual assault, as mandated by the Liberal Government. Read about it at https://goo.gl/yGNanX. In your view, is it easy to produce such a document? What can you say about definitions? (4) Gender and Olympic games. Read https://goo.gl/60TgMZ) (5) What does “driving under influence” (DUI) mean in Ontario? Is it the same in all Canadian provinces? How does it compare to U.S., or elsewhere in the world? What is the difference between DWI (driving while intoxicated) and DUI? (6) What does it mean that a yoghurt is 88% fat free? What is the precise meaning of the term “fat free”? What is the definition of a green product? (7) What is “ecotourism”? (Is this an example of a red herring?) (8) Are we truly accepting of diverse opinions if we label some opinions as not desired, and block people from expressing them? If we claim to be tolerant, then should we accept those who are not tolerant? (9)Whatismilk?Readaboutwhyitmatters:https://goo.gl/9d8WUi. (10) Apex car rental in New Zealand sells insurance that “has you fully covered.” However, on closer inspection, you realize that you are not covered at all if you drive intoxicated, or if you drive on the wrong side of the road and get into an accident. Moreover, if your pour diesel instead of gasoline in the tank and damage the engine, you have to pay for the repairs. If you drive on unsealed roads and damage the undercarriage, or a flying stone hits a windshield (windscreen in New Zealand) and cracks it, you pay for that. Any damage to the roof of the car is not covered by insurance. So much for full coverage. (11) Pick an ad for an “all inclusive” vacation and try to figure out what it does not include. (12) Definition of overweight depends on geography (which is affected by other things). According to North American and European Union standards (check, for instance, Centres for Disease and Prevention https://goo.gl/z8KGwE)apersonin called obese is their body mass index (BMI) is larger than 30. In Japan, a person whose BMI is larger than 25 is declared to be obese (source: New Criteria for ’Obesity Disease’ in Japan, https://goo.gl/YqbXKf). Recall that BMI is computed by dividing one’s mass (measured in kg) by the square of their height (height measured in metres); thus, the units of BMI are kg/m2. (13) What does “no sodium” mean? No dandruff? [class slides]

Remember Question We Must Always Ask What? That’s the question! Important information about a product we are buying, or on ﬁnancial documents (such as mortgages, loans, or investments), legal and other contracts, software download agreements, etc. is contained in “small print,” and often there are pages and pages of it. Routinely, we skip it (and the other side knows that!), and do not think much about it. How many of us actually check all information about privacy as we update software on our laptop, or upload an app to our phone? Do we know exactly what 12 NUMERACY

information about us, and how is going to be widely shared and used? How many of us check details of a car rental agreement when renting a car? We should never sign a car loan, a mortgage, or any document, paper or online, without knowing what exactly we are signing, and what the ramifications are. The following examples serve as warning of things that could happen if we don’t do it. (1) In the online article Major pet insurer says dog injuries from ’jumping, running, slipping, tripping or playing’ not covered published on 3 October 2016 at https://goo.gl/uWqivx, we read about a pet owner (Ms Richardson) who bought dog insurance policy from the company called Petsecure. One day, while running, her dog injured its hind leg. The owner claimed that is a common injury, as dog’s paws, while running, can get caught in a rabbit or fox hole, and the resulting momentum forces the knee to twist, causing injuries to ligaments or muscles. However, when the owner filed a claim to get reimboursed for the hospitaliza- tion of her dog, the company denied her claim. According to the article, “Petsecure pointed to a clause in her policy denying coverage if a dog is injured while ‘jumping, running, slipping, tripping or playing.’ ” If Ms Richardson, the dog owner, had carefully read the policy in the first place, she would have realized that the policy was completely worthless (and we add highly unethical; what is dog supposed to do, if not to jump, run, slip, trip or play?). She should not have bought the policy in the first place. (The good news it that, in the end, under the pressure from media, Petsecure did honour her claim and reimboursed 80% of her expenses). In the same article, we read: “A Vancouver lawyer who specializes in animal rights law says Richardson’s policy has ‘one of the craziest clauses she’s ever come across. Basically, what that policy says is, the dog can’t be a dog.’ ” The lawyer says that she “gets a lot of complaints from pet owners about insurance policies not covering things they expected would be covered” and ends with important message “Regardless of what that glossy brochure says ... always, always read the fine print. And not just the front page, or the first page, but the entire policy to make sure that what you think you’re getting, is what you’re actually getting.” (2) On 1 September 2017, CBC News (https://goo.gl/9iri68) published the article Calgary man warns Cuba travellers about fine print after paying 10X cost of damaged TV. Long story short: a couple goes on an all-inclusive vacation to Varadero, Cuba. One day the husband, accidentally, broke the TV set in their hotel room. The couple fully admitted fault for the damage, but were shocked to realize that they were to pay about $ 5000, i.e., 10 times the value of the broken TV set. Was that a scam? Not really - Sunwing, the company they booked the vacation through, states on their webpage that the “Rule of 10 will be in place, established by local authorities. In the case of damaged items, customers will be charged the value of the item multiplied by 10.” Although they tried to help the couple, Sunwing representatives were not successful. We read: “the property [the hotel] deferred to their published policy which reads ‘when damages caused by a break or loss of property, whether classified as fixed or useful assets, are the result of an intentional act of the clients or are linked to vandalism, the responsible person will be charged ten (10) times the value of the purchase price of the asset broken or lost.’ ” In the end, the hotel management reserved their right to apply the full penalty charge as per the stated policy. Section 2 Mathematical Reasoning and Numeracy 13

Case study What is Radio Frequency Exposure? Initially, when cell phones came out, their makers strongly denied any effects of cell phone radiation (also known as radio frequency (RF) exposure) could possibly have on human health. The article Mother who kept her phone in her bra every day for 10 years is convinced it caused her terminal breast cancer in U.K.’s Daily Mail (9 September 2015; https://goo.gl/qBLWm5) reports about the case of a 51-year old woman who was diagnosed with an aggressive form of breast cancer. The woman believed that her cancer was caused by the radiation from her phone. What is cell phone radiation? We can find some information about RF exposure burried deep in our phones. On iPhone 5, following the sequence Settings > General > About > Legal > RF Exposure, we find the following:

This carefully crafted narrative (written, no doubt, by lawyers) seem to be implying that it’s a good idea to keep our phone away from our body (and there is definitely no need to keep it close to us when we sleep). In Measurements of Radiofrequency Radiation with a Body-Borne Exposimeter in Swedish Schools with Wi-Fi published in “Frontiers of Public Health” [3], the authors report on measuring the intenisty of RF radiation in school classrooms in Sweden. Introducing the topic, they write: “Lately, it has been discussed if radiofrequency (RF) radiation can have long-term adverse effects on children’s health” and later: “Exposure to RF radiation was classified as a possible human carcinogen, Group 2B, by the International Agency for Research on Cancer (IARC) at WHO in 2011. The decision was mainly based on case-control human studies on use of wireless phones by the Hardell group from Sweden and the IARC Interphone study, which showed increased risk for brain tumors, i.e., glioma and acoustic neuroma” (see references in [3]). To read more about RF exposure, and about some attempts at determining whether or not the RF radiation is harmful, follow the link (American Cancer Society) https://goo.gl/HLfVBp In the article The inconvenient truth about cancer and mobile phones published by The Guardian (14 July 2018, https://bit.ly/2La9RD4) we read “We (average 14 NUMERACY

citizens, op.a.) dismiss claims about mobiles being bad for our health but is that because studies showing a link to cancer have been cast into doubt by the industry?” However, the paper claims that “On 28 March this year, the scientiﬁc peer review of a landmark United States government study concluded that there is clear evidence that radiation from mobile phones causes cancer, speciﬁcally, a heart tissue cancer in rats that is too rare to be explained as random occurrence.” Note: Relate the danger of RF exposure to the Internet of Things, which will multiply the number fo deviced connected (wirelessly) to internet.

Investigate What Did the Artist Want to Say? In this painting, Belgian surrealist artist Ren´e Magritte drew a pipe. The text below the pipe says “This is not a pipe.”

Why is this not a pipe? If it is not a pipe, what is it?

Theorems and Causation

A theorem in mathematics is a statement that establishes a new relationship between previously defined mathematical objects, relations or properties. In order to be accepted, a theorem needs to be supported by (mathematically) acceptable evidence, i.e., a proof. A theorem, applied to real life, suggests that we ask how and why (‘how’ is sometimes answered by an algorithm). The key word is evidence, and this is where mathematics differs from any other discipline, or any other real-life situation. To be accepted, mathematical proof must be based on previously established facts and results, and must follow rigid rules of logical reasoning. In principle, a proof can be traced all the way back to the beginning of a mathematics discipline, whose foundations consist of facts and statements that must be taken for granted (these statements are called axioms). Are there axioms (i.e., statements whose validity we take at face value) in real life? Perhaps religious beliefs (often qualified as ‘truth’), can be taken as axioms? For instance What is acceptable evidence outside of mathematics? In most sciences, repeated experiments (with – as much as possible – identical starting/initial conditions) that produce identical, or very similar outcomes, are accepted as evidence. How many experiments? In general, the more, the better (however, easier said than done). Section 2 Mathematical Reasoning and Numeracy 15

Evidence-based medicine (EBM) is an approach to evaluating medical prac- tices by requiring solid, strong evidence, based on well-conducted and well-designed research. (Although medicine is a very old discipline, EBM is a relatively new phenomenon, as it emerged in the 1960s). EBM will support a strong recommendation for a certain practice if it comes from a randomized control trial (there are other possible designs, such as meta- analysis; read more about EBM on Wikipedia, https://goo.gl/TpAFkK.) Ran- domized control trial involves a statistical analysis of the (possible) diﬀerences in the behaviour or characteristics between the treatment and the control groups. Meta-analysis combines results of multiple studies; systematic review is a complete, exhaustive summary of current evidence, based on literature search and secondary data.

A mathematician would not accept repeated experiments as evidence, unless they cover absolutely all possible cases, and all yield identical results. Goldbach Conjecture states that every even number greater than 2 can be written as a sum of exactly two prime numbers. For instance, 4 = 2 + 2, 6=3+3, 18 = 13 + 5, 24 = 11 + 13, and so on. Although the experiment of checking for this property has been successfully performed for the first 2 billion billion even numbers (i.e., repeated 2 · 1018 times) all with identical outcomes (i.e., even number = sum of exactly two primes), this is not accepted as proof that the conjecture is true for all even numbers. Formulated in 1742, Goldbach Conjecture has become the oldest and the most famous unsolved (as of today) mathematical problem. Math theory cannot be built on conjectures, hypotheses or other statements which have not been rigorously proven. This is why the development of some math theories “waits” until certain results are proven (for example, certain subareas of number theory need the Riemann hypothesis, which has not been proved true (or false) yet). Needless to say, in real life (physics, chemistry, etc.) this kind of evidence (billions of repeated experiments with identical outcomes) would definitely be accepted, and, if appropriate, would be viewed as a solid foundation for a theory. Here is an anecdote that illustrates the difference in the way we might think of acceptable evidence. Persons A, B, and C are on the train in Ireland, and see acowinthefield.

Person A says: “All cows in Ireland are black.” Person B says: “There is a black cow in Ireland.” Person C says: “There is a cow in Ireland one of whose sides is black.” Which one is a mathematician?

Repeated experiments are not always possible. For instance, cosmology (study of our universe) is based on one experiment, i.e., our Universe, as we do not know of (i.e., are not aware of) any other universes. When exactly repeated experiments yielding similar outcomes are accepted as signiﬁcant is formalized by a statistical approach which we will discuss later. 16 NUMERACY

Case study How Much Water? We have all heard about how drinking 6 to 8 glasses of water daily benefits our health and well-being. What evidence supports this claim? The article 8 glasses of water a day ‘an urban myth’ published by CBC News (10 June 2012, https://bit.ly/1AI1j9I) re-states the view that drinking water is beneficial. However, it seems that this particular quantity of 6-8 glasses per day has no scientific evidence. The article provides a more appropriate, common-sense advice - listen to your body! When you’re thirsty, drink some water, when you are not - don’t! The article What drove us to drink 2 litres of water a day? published in Australian and New Zealand Journal of Public Health (https://bit.ly/2PEXwGj) gives a revealing historic background on the 6-8 glasses of water suggestion, going back to 19th century advice for healthy living. As well, it emphasizes that there is a difference between thirst and dehydration. (Note: know the definitions! Thirst and dehydration are two different things; so are sadness and depression.) We read “In today’s western society there is an accepted popular view that the moment one feels thirsty, one is dehydrated. Consequently, the only way to avoid this highrisk situation is to consume copious amounts of water. Supporters of this view believe consuming beverages other than water will only lead to further dehydration. ” The article concludes with “Water is important for health; however, the recommendation of 8 glasses of pure water per day appears an overestimation of requirements. All fluids are important in meeting requirements and water should not be singled out. We should be educating the general public that beverages like tea and coffee, despite their caffeine content, do not lead to dehydration and will contribute to a person’s fluid needs, something worth considering when discussing fluid requirements.” Sinister forces? In The truth about sports drinks (New Zealand Herald, 16 November 2016, https://bit.ly/2f4iJGa) we read “One of industry’s greatest successes was to pass off the idea that the body’s natural thirst system is not a perfect mechanism for detecting and responding to dehydration [Note: mixing thirst with dehydration!] These include claims that: ‘The human thirst mechanism is an in- accurate short-term indicator of fluid needs ... Unfortunately, there is no clear physiological signal that dehydration is occurring.’ ” Consequently, “healthcare organisations routinely give advice to ignore your natural thirst mechanism. Dia- betes UK, for example, advises: ‘Drink small amounts frequently, even if you are not thirsty – approximately 150 ml of fluid every 15 minutes – because dehydration dramatically affects performance.’ ” People tend to exaggerate: one can have serious consequences by consuming too much water. We learn of one such case in Hiker Fatality From Se- vere Hyponatremia in Grand Canyon National Park publish in Wilderness and Environmental Medicine (September 2015, Volume 26, Issue 3, Pages 371-374, https://bit.ly/2wmZv9u) The authors, T. M. Myers and M. D. Hoffman, write: “We present the case of a hiker who died of severe hyponatremia at Grand Canyon National Park. The woman collapsed on the rim shortly after finishing a 5-hour hike into the Canyon during which she was reported to have consumed large quantities of water.” Furthermore, in Drinking too much water can be fatal to athletes published in Science Daily (14 Sept 2014, https://bit.ly/2Lu8Txw) we read “The recent deaths of two high school football players illustrate the dangers of drinking too much water and sports drinks, according to Loyola University Medical Center sports medicine physician Dr. James Winger.” In particular, “Over-hydration by athletes is called exercise-associated hyponatremia. It occurs when athletes drink even when they are not thirsty. Drinking too much during exercise can overwhelm the body’s ability to remove water. The sodium content of blood is diluted to abnormally low levels. Cells absorb excess water, which can cause swelling – most dangerously Section 2 Mathematical Reasoning and Numeracy 17

in the brain.” This is a serious condition, as hyponatremia “can cause muscle cramps, nausea, vomiting, seizures, unconsciousness, and, in rare cases, death.” The article continues “Georgia football player Zyrees Oliver reportedly drank 2 gallons of water and 2 gallons of a sports drink. He collapsed at home after football practice, and died later at a hospital. In Mississippi, Walker Wilbank was taken to the hospital during the second half of a game after vomiting and complaining of a leg cramp. He had a seizure in the emergency room and later died. A doctor conﬁrmed he had exercise-associated hyponatremia.”

Case study Sports drinks What kind of evidence is there? In Sports drinks are selling consumers a myth that is slowing them down (Busi- ness Insider, 1 November 2016, https://read.bi/2MwelVW) we read “From eight glasses of water a day to protein shakes, we’re bombarded with messages about [what] we should drink and when, especially during exercise. But these drinking dogmas are relatively new. For example, in the 1970s, marathon runners were dis- couraged from drinking fluids for fear that it would slow them down. Now we’re obsessed with staying hydrated when we exercise, not just with water but with specialist drinks that claim to do a better job of preventing dehydration and even improve athletic performance.” The authors claim that the reason “behind this huge rise in sports drinks lies in the coupling of science with creative marketing. An investigation by the British Medical Journal has found that drinks companies started sponsoring scientists to carry out research on hydration, which spawned a whole new area of science.” As well, “These same scientists advise influential sports medicine organisations, developing guidelines that have filtered down to health advice from bodies such as the European Food Safety Authority and the International Olympic Com- mittee. Such advice has helped spread fear about the dangers of dehydration.” Punchline: “One of industry’s greatest successes was to pass off the idea that the body’s natural thirst system is not a perfect mechanism for detecting and responding to dehydration.” Is there evidence? We read “Many of the claims about sports drinks are often repeated without reference to any evidence. A British Medical Journal review screened 1,035 web pages on sports drinks and identified 431 claims they enhanced athletic performance for a total of 104 different products. More than half the sites did not provide any references - and of the references that were given, they were unable to systematically identify strengths and weaknesses. Of the remaining half, 84% referred to studies judged to be at high risk of bias, only three were judged high quality and none referred to systematic reviews, which give the strongest form of evidence.” However, “the current evidence is not good enough to inform the public about the benefits and harms of sports products. What we can be almost sure about is that sports drinks are not helping turn casual runners into Olympic athletes. In fact, if they avoided these sugar-laden drinks they would be probably be slimmer and so faster.”

Investigate Evidence and “Evidence” (1) (Coincidence; Ear infections in children) The article The new ear infection rules published at https://bit.ly/2dZlgG7 discusses the eﬀectiveness of antibiotics in the treatment of ear infections in children. Evidence that antibiotics work is 18 NUMERACY

sometimes based on coincidence–untreated, the infection would disappear on its own. New ear infection rule in many cases: give no antibiotics, just wait! So, to provide good evidence we must: show that intervention (antibiotic) indeed has an effect (ear infection gone) AND show that when there is no intervention (no antibiotic given), there is no effect (ear infection persists). (2) (Personal bias) The article Europeans greatly overestimate Muslim population, poll shows (The Guardian, 13 December 2016, https://bit.ly/2hKOmGy)shows how personal bias affects one’s understanding of data. So, when we poll people on all kinds of things, are we collecting good, accurate evidence? (3) (Sounds good) Just because it sounds good, it does not have to be good. In Self- help ’makes you feel worse’ (BBC News, 3 July 2009, https://bbc.in/2BY4GCJ) we read “Canadian researchers found those with low self-esteem actually felt worse after repeating positive statements about themselves. They said phrases such as ’I am a lovable person’ only helped people with high self-esteem.” (4) (Authority, celebrity) Evidence coming from authority, celebrity, popular culture, media influencers. The case of health and wellness strategy called GOOP: Dont blame Gwyneth Paltrow (MacLean’s 30 June 2017, https://bit.ly/2oocS5v). There is more: Gwyneth Paltrow’s Goop expanding to Canada - and some medical experts aren’t happy (Global News, 24 August 2018, https://bit.ly/2NxPr4H). (5) (Evidence exists, yet not trusted) Evidence against using BMI (body mass index): The risks of a quick fix: a case against mandatory body mass index reporting laws. (Eat Disord. 2008 Jan-Feb; 16(1):2-13. doi: 10.1080/10640260701771664. https://bit.ly/2LsLvAr) From the abstract, we learn “As the United States addresses obesity, a number of state legislatures are considering laws that require schools to track and report students’ body mass index (BMI), a measurement of body weight (weight/height squared). This article describes the state level activity on mandatory BMI reporting, offers numerous arguments against this practice, and suggests an alternative approach to promoting health in youth. Mandatory BMI reporting laws place a new and inappropriate responsibility on the schools. Proponents of such laws imply that BMI reporting will have positive outcomes, yet there is virtually no independent research to support this assumption. The authors argue that these laws could do significant harm, including an increased risk for children to develop eating disorder symptoms.” Related: School: 66-pound girl is ’overweight’ report says “A mom is outraged when her 9-year-old daughter is classified as overweight by a school health program in New York” (CNN News, https://cnn.it/TNfjO7). (6) The Guardian, 3 January 2018, published the article Watchdog bans advert’s claim eHarmony is ‘scientifically proven’ in which they report that “The advertising regulator has banned the online dating service eHarmony from claiming it has a ‘scientifically proven matching system’. Upholding a complaint about a billboard ad on London Underground, the Advertising Standards Authority (ASA) said the claim was misleading because eHarmony could not prove its service provided a greater chance of finding lasting love.” Check https://bit.ly/2M6do5i.

Causation and Implication

The logical structure of a theorem is given by implication (outside of math, more often referred to as causation). In narrative form (symbols A and B represent two statements), an implication can be expressed in many diﬀerent forms, such as: “if A then B” “assume A,thenB (follows)” Section 2 Mathematical Reasoning and Numeracy 19

“A causes B” “B is caused by A” “B follows from A” “B is a result of A” “B is a consequence of A” “in case A,thenB” “A is/are B” “if assumption(s) A then conclusion(s) B” For instance, in the statement “A cat is a mammal”, ‘cat’ is the assumption (or the cause) and ‘mammal’ is the conclusion (or the eﬀect). In a formal math language, we write “if cat then mammal,” or (using A ⇒ B to symbolize the implication “if A then B”), we write cat ⇒ mammal. Another example: the statement “Shortness of breath is a symptom of an allergy” establishes allergy as the cause of the shortness of breath. Thus, we write “if allergy then shortness of breath,” or allergy ⇒ shortness of breath. Remember that “medical condition (disease) implies symptoms”, i.e., medical condition (disease) ⇒ symptoms.

An implication “if A then B” can be visualized using so-called Venn diagrams:

A cats

B mammals

We think of these as boxes: whatever is in A (smallerbox)mustbeinB (larger box), since it contains the smaller box. The implication cat ⇒ mammal is illustrated in the picture on the right: a cat are in the smaller box (where all cats are), which is contained in a larger box, where all mammals are. This diagram suggests that not all mammals are cats, as there are animals in the larger box (dogs, rats, elephants, etc.) which are not in the smaller box. Thus, we conclude that the implication mammal ⇒ cat is not true. (More about this soon.) Likewise, other interpretations of the implication (causation) A ⇒ B can be visualized as well:

A cause disease

B effect symptoms

How do we use, i.e., correctly interpret an implication (causation) A ⇒ B? We need to check that all assumptions hold (i.e., check that everything listed under A holds). If so, then the conclusion (or conclusions) hold(s). If one or more assumptions do not hold, then the implication cannot be used. However, what is claimed (conclusion/eﬀect) may or may not be true. Going back to the implication cat ⇒ mammal – we hold an animal in our hands, and verify that it is a cat. Thus, we conclude (from the given implication) that it is a mammal. If we hold a raccoon in our hands, we cannot apply the implication (a raccoon is not a cat, so the assumption is not true). However, the conclusion (raccoon is a mammal) is true. Now imagine that we hold a spider in our hands. The assumption is not true (a spider is not a cat), so we cannot use 20 NUMERACY

the implication. In this case, the conclusion does not hold either (a spider is not a mammal). Important to remember: cause implies eﬀect, but the absence of the cause does not say anything about the eﬀect. Another illustration: strong earthquake causes destruction. However, in the absence of a strong earthquake there could be no destruction, or destruction could occur due to other causes (such as tsunami or war or a medium-strength earthquake).

From an implication A ⇒ B,wecanderiveaconverse statement, B ⇒ A (in words, we swap cause and eﬀect). There is no relation between the truthfullness of a statement and its converse. For instance, we know that prolonged UV exposure causes sunburn and skin damage (written more formally, prolonged UV exposure ⇒ sunburn and skin damage). However, sunburn and skin damage to not cause prolonged UV exposure. Or: the converse of “If the last digit of a number is 6, then it is even” is the statement “If a number is even then its last digit is 6” which is not true (as an even number could end in 0, 2, 4, or 8 as well). “A cat is a mammal” is true, but its converse “A mammals is a cat” is not. It could happen that the converse of a true statement is also true. For instance, for a student at McMaster University, both “If GPA is 12.0, then A+ in all courses” and its converse “If A+ in all courses, then GPA is 12.0” are true. If both A ⇒ B and B ⇒ A are true, or both A ⇒ B and B ⇒ A are false, we say that A and B are equivalent,andwriteA ⇔ B.

Example 2.1 Equivalent Statements In the opinion piece Smash the Wellness Industry published in New York Times (8 June 2019, at https://nyti.ms/2F1MYgZ) U.S. novelist Jessica Knoll writes “This was before I could recognize wellness culture for what it was - a dangerous con that seduces smart women with pseudoscientiﬁc claims of increasing energy, reducing inﬂammation, lowering the risk of cancer and healing skin, gut and fertility problems. But at its core, ‘wellness’ is about weight loss. It demonizes calorically dense and delicious foods, preserving a vicious fallacy: Thin is healthy and healthy is thin.” We recognize the vicious fallacy at the end of the paragraph as the equivalence healthy ⇔ thin.

Can you ﬁnd more examples of equivalent statements? (Note: not at all easy in real life; it’s easier if you think of abstract concepts, such as in math.) From any statement A we can form its negative notA (often denoted by ¬A). If A=“cat,” then notA=“not a cat,” i.e., anything but a cat (could be a mouse, cruse ship, happiness, math test, universe, etc.) Another useful modiﬁcation of an implication A ⇒ B is a contrapositive statement notB ⇒ notA. A statement and its contrapositive have the same truth value, i.e., they are either both true or both false. For instance, the contrapositive of “If cat then mammal” is “If not mammal then not cat.” Or, the contrapositive of “If you drive, then do not drink” is “If you drink, then do not drive.” Makes sense! Using Venn diagrams, we can visualise contrapositive statements: Section 2 Mathematical Reasoning and Numeracy 21

A A

B B

implication contrapositive

The diagram on the left represents A ⇒ B: the brown dot is in box A, and as box A is in the larger box (box B), the brown dot is also in box B. The diagram on the right: the brown dot is not in the larger box (so notB), and as the larger box contains the smaller box, the brown dot cannot be in the smaller box (thus notA). In other words, notB ⇒ notA.

In real life, there are very few (if any?) “mathematically strong” implications (meaning that they indeed apply to all cases). Mathematically speaking, the implication meningitis (disease) ⇒ nausea, vomiting, and headaches (symptoms) means that everyone who has meningitis experiences these symptoms. In reality, however, it could be that someone has meningitis but experiences none of the three symptoms (i.e., likely has other symptoms, or no symptoms at all). The implication “Magnitude 5 earthquake causes destruction of property and/ or life” is true in most cases, but not always. UV exposure causes skin damage in most people, but likely there is a person who is not negatively affected by UV exposure. Researchers have figured out that reversing the habit of eating breakfast (either from having breakfast to not having it, or from not having breakfast to having it) leads to initial weight loss (but again, not in every person). Thus, in real life, an implication (causation) often comes with a qualifier, such as “very likely,” “probably,” “quite often,” and so on; i.e., instead of A implies B, we find statements such as “A might imply B,”“A probably implies B,”“A implies B with 75% chance,” etc. Nevertheless, the reasoning that comes with our understanding of how implications work is very useful.

A group of friends decides that “If the weather is nice, then we go for a hike”. However, the weather turned not to be nice, so they should not go for a hike. Is this a logically sound conclusion?

Example 2.2 Math Implication vs Real Life The message on this street sign could be interpreted as the implication “If it is between 8am and 6pm, then I am not allowed to park.”

What is our common understanding of the remaining time interval, from 6pm to 8am? How does it diﬀer from a mathematician’s interpretation? 22 NUMERACY

Correlation

Cause and eﬀect is only one possible relationship between two things. By ‘things’ we mean objects, statements, events, variables, measurements, etc., for instance whatever A and B in A ⇒ B represent. A correlation is a mutual connection, or a relationship between two or more things. (Later, we will see that there is a way to quantify the strength of that relationship). The easiest way to understand what correlation is is to look at examples. In the article How the smartphone aﬀected an entire generation of kids published in MacLeans (22 August 2017, at https://bit.ly/2okfP7i Jean M. Twenge, Professor of Psychology at San Diego State University, writes “I wondered if these trends - changes in how teens were spending their free time and their deteriorating mental health - might be connected. Sure enough, I found that teens who spend more time on screens are less happy and more depressed, and those who spend more time with friends in person are happier and less depressed.” Note that Prof. Twenge did not use the word ‘cause’, but instead ‘connection/ connected’. So, there is a relationship between the way teens spend their free time and their mental health, but there is no evidence of causality. And she is careful to point that out: “Of course, correlation doesnt prove causation: Maybe unhappy people use screen devices more.” This last sentence is important, so we repeat it – Correlation is weaker than causation, and in many cases correlation does not imply causation.

Case Study 2.3 Screen Time, Smart Phones and Mental Health In Limiting children’s screen time linked to better cognition (27 September 2018, at https://bbc.in/2OXS6p7), BBC News Health Reporter Alex Therrien writes “Children aged eight to 11 who used screens for fun for less than two hours a day performed better in tests of mental ability, a study found. Combining this with nine to 11 hours of sleep a night was found to be best for performance. Researchers said more work was now needed to better understand the effects of different types of screen use. However, they acknowledge that their observational study shows only an association between screen time and cognition and cannot prove a causal link. And it did not look at how children were using their screen time, be it to watch television, play videogames or use social media.” Note how the author used the word ‘linked’, and points out that more research is needed to figure out the effects (thus, to establish a causation) of different types of screen use. In Social media, but not video games, linked to depression in teens, according to Montreal study, Kate McKenna, a CBC News reporter (15 July 2019, at https://bit.ly/2JynMkE), opens by saying that “Study [to be reported on] exam- ined mental-health implications of high levels of screen time.” Read the article to determine if ‘implication’ is meant in a mathematical sense, i.e., if the study refers to causation or not. The first three paragraphs are: “Screen time - and social media in particular -islinkedtoanincreaseindepressivesymptoms in teenagers, according to a new study by researchers at Montreal’s Sainte-Justine Hospital. The research team, led by Patricia Conrod, investigated the relationship between depression and exposure to different forms of screen time in adolescents over a four-year period. Section 2 Mathematical Reasoning and Numeracy 23

‘What we found over and over was that the eﬀects of social media were much larger than any of the other eﬀects for the other types of digital screen time,’ said Conrod, a professor of psychiatry at the University of Montreal.” Identify all words in these paragraphs that refer to a link between depression and exposure to screens. Do these words suggest correlation or causation? In an opinion piece by a psychology professor Dr. Dennis-Tiwary Taking Away the Phones Wont Solve Our Teenagers Problems published in New York Times (14 July 2018, at https://nyti.ms/2KUP47c) we read that “Although some research does show that excessive and compulsive smartphone use is correlated with anxiety and depression, there is a lack of direct evidence that devices actually cause mental health problems.” Does this is any way contradict the conclusions of the previous two articles? What can you conclude after reading therse three pieces?

Example 2.4 Causation or Correlation? In each case, figure out whether the paragraphs quoted are about causation(s) or correlation(s), and identify them (i.e., state which variables are related). Then read the entire article to find (if any) further causations or correlations. (1) In the article Bad trip from smoking pot? It could be a sign of mental illness published in Hamilton Spectator (21 July 2018, at https://bit.ly/2LYJ7qs), we read: “In particular, research in Denmark has discovered heavy cannabis users are substantially associated with the development of schizophrenia and bipolarism. In fact, of those who were hospitalized with a pot-related mental condition, almost 50 per cent were diagnosed with schizophrenia or bipolarism later on in life. The risks increase the younger a person starts using. Experts have not yet determined whether cannabis causes schizophrenia or bi-polar disorder, or whether it simply triggers a first psychotic episode.” (2) WebMD article Child Leukemia Again Linked to Power Lines (2 June 2005, https://wb.md/2MEetmv) claims that “Living near high-voltage power lines raises children’s risk of leukemia by 69%, a British study shows. That doesn’t prove that power lines cause the deadly blood cancer, the study’s authors are quick to point out. Despite 30 years of research, scientists still can’t come up with a plausible reason why the weak magnetic fields near power lines might cause leukemia.” (3) The Conversation (Academic rigour, journalistic flair) of June 17, 2019 published an analysis of teenage behaviours titled Teenage sexting linked to increased sexual behaviour, drug use and poor mental health https://bit.ly/2Yetlga We read: “While sexting is linked to sexual behaviour and mental health factors, correla- tional studies do not provide evidence to suggest that sexting is in any way the cause of risky behaviour or poor mental health.”

Example 2.5 Does Smoking Cause Cancer? Lung cancer information (Mayo Clinic, https://mayocl.in/2C1nfBl) states that “Smoking causes the majority of lung cancers - both in smokers and in people exposed to secondhand smoke. But lung cancer also occurs in people who never smoked and in those who never had prolonged exposure to secondhand smoke. In these cases, there may be no clear cause of lung cancer.” Think about what kind of experiment would convince people that smoking actually causes cancer. Would such an experiment be feasible, allowed? If not, then why do we believe that smoking causes cancer, isn’t it just a correlation? 24 NUMERACY

In many situations it might make sense to describe a correlation more precisely. If an increase in A is followed by an increase in B, or a decrease in A is followed by a decrease in B, then A and B are said to be positively correlated.For instance, exposure to screens (phone, laptop, TV) has been found to be positively correlated with depression. If A increases as B decreases and A decreases as B increases, then A and B are said to be negatively correlated. Example: the number of trees on a hill is negatively correlated to the chance of a landslide. In the case the variables are presented as graphs, we could easily spot the sign of a correlation:

positively correlated negatively correlated

Note that the blue graph follows the trend of the brown graph: it increases roughly where the brown graph increases, and decreases where the brown graph decreases. The green graph does the opposite: it decreases roughly where the brown graph increases, and increases where the brown graph decreases. Examine the three pairs of variables (A and B, A and C, and B abd C)given in this table for the sign (positive or negative) of their correlation:

Example 2.6 Positive or Negative Correlation? Identify each case as a positive or negative correlation. (a) Car speed and travel time to a destination (b) Air temperature and ice cream sales (c) Amount of food eaten and hunger (d) Salary and spending (e) Cigarette smoking and years of life remaining (f) Snowfall and the number of people driving (g) Hair length and shampoo use.

How do correlations occur? For instance, A and B can be correlated because: A implies B or B implies A (so causation implies correlation) A is equivalent to B AthirdthingC causes both A and B A implies C and C implies B Coincidence (no connection between A and B) In the following exercise we explore examples of these situations. Section 2 Mathematical Reasoning and Numeracy 25

Example 2.7 Correlations Looking at the above list, identify how each of these are correlated: (a) Children with larger feet spell better than children with smaller feet (b) The more gasoline we put in our car, the more the car loses in its resale value (c) As a student’s study time increases, they are happier (d) On a sunny day, UV index increases as the temperature increases (e) As we climb higher and higher, the air temperature decreases (f) The number of ﬁre engines at a site and the extent of the damage to the property (g) The more margarine people consume, the higher the divirce rate in the U.S. state of Maine:

(Source: Spurious correlations, http://www.tylervigen.com). (h) As the consumption of ice cream increases, so does the number of drowning deaths (i) As the demand for a product increases, so does its price.

Axioms

If a fact (formula, statement, property) in mathematics depends on previously established facts, and these facts depend on previously established facts, where does it all start? It all starts with axioms – every theory in mathematics is built from a set of statements whose validity is taken for granted, i.e., they cannot be proven to be true. For instance, the fact that x + y = y + x for all real numbers is an axiom. Of course, we can convince ourselves that this statement is true by examining examples: 3 + 4 = 7, and since 4 + 3 = 7 we conclude that 3 + 4 = 4 + 3; etc. But the axiom states that x + y = y + x is true for all (infinitely many) choices for x and y. An axiom tells us that multiplication by 1 does not change the number (x·1=x); nor does a number change if we add zero to it (x+0 = x). In Euclidean geometry (i.e., the geometry that we learn at school and which applies to out daily lives), an axiom states that a line is determined by two points. Check Wikipedia (under Historic Development) for a complete list of axioms for Euclidean geometry. As well, we need to have a “starting” object in math, as math does not allow for circular definitions (i.e., we cannot define A in terms of B, and then B in terms of A). Usually that “starting” object is a set, i.e., we do not define what a set is. We determine a set by listing its elements (for instance, we can consider the set of all people who live in Canada today). 26 NUMERACY

In real life, a good way to characterize axioms is to say that they represent a set of beliefs (and that philosophy and religion are well-positioned to determine what these axioms could be). One way or another, axioms (even in mathematics) are statements (facts) that we decide to believe in (thus, every mathematician and every scientist must be a believer!). Other disciplines have axioms, as well. For instance in astronomy and cosmology, we assume (i.e., take for granted, believe) that the same laws of physics apply at all locations in universe (except at singularities, such as black holes). As well, we assume that the Universe is homogeneous, i.e., that it looks about the same in all directions. Axioms change with new discoveries: in physics, Newton’s axioms were replaced by the axioms of Einstein’s special relativity, and then those were replaced by the axioms of general relativity. Are there axioms in biology? What would, or what do they look like? What about axioms in social sciences? Philosophy? What is the (original) meaning of the word “dogma”?

Algorithms and Formulas

Algorithms and formulas give us means of calculating quantities that we are interested in. For instance to compute someone’s body mass index (BMI), we measure their height h (inmetres)andmassm (in kg) and then use the formula mass m BMI = = height2 h2 For instance, a person of mass m = 68 kg and height h =1.7 m has the body mass index of m 68 BMI = = ≈ 23.53 h2 1.72 The symbol ≈ denotes the fact that the value of the fraction in not exactly 23.53; instead the value was rounded oﬀ to two decimal places. (How we round oﬀ a number depends on the context; more about it later). The formula a + a + a + ···+ a a¯ = 1 2 3 n n calculates the average value (the mean) of the numbers a1,a2,a3,...,an. Thus, if four houses in our neighbourhood were sold for $ 376 000, $ 412 000, $ 398 000, and $ 564 000, then the average price (denote if by P¯) of a house sold is 376 000 + 412 000 + 398 000 + 564 000 P¯ = = $ 437 500 4 We will meet many formulas in this course. An example of an algorithm is provided in your course outline and on the 2UU3 web page! It tells you how your course grade is calculated.

Case study Data Sets and Algorithms One of a major strengths of algorithms these days is their ability to extract patterns from large data sets. Often referred to as AI (artiﬁcial intelligence), algorithms are aggressively present in our lives, sifting through huge, deeply con- cerning amounts of data collected, legally or illegally (through our use of certain web pages, smart phones, etc.) about every aspect of our lives. On 16 February 2012, Forbes magazine (https://goo.gl/46Lf2X) published the article How Target Figured Out A Teen Girl Was Pregnant Before Her Father Did. We read “Every time you go shopping, you share intimate details about your consumption patterns with retailers. And many of those retailers are studying Section 2 Mathematical Reasoning and Numeracy 27

those details to figure out what you like, what you need, and which coupons are most likely to make you happy.” Target’s statistician Andrew Pole discusses how Target uses customer data to learn about them, their needs, spending patterns, and so on. For instance Pole and his analysis looked at historic buying data of women who signed up for Target baby registries and “before long some useful patterns emerged. Lotions, for example. Lots of people buy lotion, but one of Pole’s colleagues noticed that women on the baby registry were buying larger quantities of unscented lotion around the beginning of their second trimester. Another analyst noted that sometime in the first 20 weeks, pregnant women loaded up on supplements like calcium, magnesium and zinc. Many shoppers purchase soap and cotton balls, but when someone suddenly starts buying lots of scent-free soap and extra-big bags of cotton balls, in addition to hand sanitizers and washcloths, it signals they could be getting close to their delivery date.” And now the scary part: “As Pole’s computers crawled through the data, he was able to identify about 25 products that, when analyzed together, allowed him to assign each shopper a ‘pregnancy prediction’ score. More important, he could also estimate her due date to within a small window, so Target could send coupons timed to very specific stages of her pregnancy.”

Note: the words algebra and algorithm come from the same person: Abu Abd Allah Muhammad Ibn Musa al-Khwarizmi (ca 780- 850 CE), Persian mathematician, astronomer, astrologer, geographer and author, who worked in Baghdad. Al-Khwarizmi wrote the famous Book of Completion and Balancing in which he deﬁnes algebra (‘al-jabr’ means subtracting a quantity from one side of the equation and adding it to another). The word algorithm is derived from the latinization of Al-Khwarizmi name, Algoritmi.

Investigate Formulas What is wind chill, and what measurements are used to calculate its values? What about humidex? What is UV index? In the article This Is How Many Calories You Burn While Walking (source: https://bit.ly/2EhUG3i, 15 March 2016) we read “Once they had identiﬁed the components needed for an accurate prediction, the researchers created a new equation that accounted for both height and body mass. To put it to use, measure velocity (distance divided by time) in meters per second, and height in meters:”

Kcal means kilocalories; so 1 Kcal = 1000 calories, but not when food is concerned: in that case, 1 Kcal = 1 (food) calorie. Be aware when travelling:

. 28 NUMERACY

When we write the formula using math symbols, we need to make sure to identify all quantities that are used, and their units (for instance, if we use B for body mass, then we say that B represents a person’s body mass in kg, etc.) The formula below gives the energy expenditure when we run (of course, it is an approximation) BV 2 Calories per minute = 0.035B +0.029 H where B is the body mass in kilograms, V represents the velocity in metres per second and H is the height in metres. The above article states: “Height essentially economizes walking, so the taller you are, the less energy it takes per pound to walk a mile, or the slower the rate you burn calories while walking at the same speed compared to someone who is shorter.” How is this reﬂected in the formula?

Note: Reasoning about fractions A B if A increases and B does not change, then the fraction increases A B if A decreases and B does not change, then the fraction decreases A B if A does not change and B decreases, then the fraction increases A B if A does not change and B increases, then the fraction decreases

Investigate Algorithms (1) The article When technology discriminates: How algorithmic bias can make an impact published by CBC News on 10 August 2017, https://bit.ly/2vWUbvz, states that “Research has shown that algorithms can actually perpetuate–even accentuate–social inequality.” Explain what is the main concern, related to how algorithms work, presented in this article. What was the scandal involving Facebook and Cambridge Analytica in 2018 about? You can start your investigation by reading https://bit.ly/2pCFpEJ (2) ‘It’s ridiculous. It’s Picasso’: Facebook reviewing anti-nudity policy after block- ing Montreal museum ad (CBC News, 2 August 2018, https://bit.ly/2OFhwrS Montreal’s ﬁne arts museum complained after Facebook blocked ads featuring cu- bist nude paintings. Here is the oﬀending painting:

In Here’s Why Facebook Removing That Vietnam War Photo Is So Important (fortune.com, 9 September 2016 https://for.tn/2c5mu2A) we read: “In the latest controversy involving the giant social networks news judgement, Facebook (fb, +0.40%) removed an iconic photo from the Vietnam War: A picture of a young Section 2 Mathematical Reasoning and Numeracy 29

Kim Phuc running naked down a road after her village was hit by napalm.” As well: “When a Norwegian newspaper editor–who posted the photo as part of a series on war photography–tried to re-post it, along with a response from Phuc herself, his account was suspended.” In New York Times article of 9 September 2016 (https://nyti.ms/2AzgXNg) Facebook Restores Iconic Vietnam War Photo It Censored for Nudity we read “The image is iconic: A naked, 9-year-old girl ﬂeeing napalm bombs during the Vietnam War, tears streaming down her face. The picture from 1972, which went on to win the Pulitzer Prize for spot news photography, has since been used countless times to illustrate the horrors of modern warfare.”

We read: “But for Facebook, the image of the girl, Phan Thi Kim Phuc, was one that violated its standards about nudity on the social network. So after a Norwegian author posted images about the terror of war with the photo to Face- book, the company removed it.” And finally, “The move triggered a backlash over how Facebook was censoring images. When a Norwegian newspaper, Aftenposten, cried foul over the takedown of the picture, thousands of people globally responded on Friday with an act of virtual civil disobedience by posting the image of Ms. Phuc on their Facebook pages and, in some cases, daring the company to act. Hours after the pushback, Facebook reinstated the photo across its site.” (3) In Heat health risk prompts change to Environment Canada warning system (CBC News, Jul 27, 2018, https://bit.ly/2JZxWrF) we learn that the heat warn- ings algorithm need to be adjusted by regions (the article also mentions humidex, which is another formula/algorithm). How to define a heat wave is important, as people could (and do) die from it. (4) Algorithms we know from math: how to solve an equation (“move terms from one side to another”) finding extreme values (or peaks, tipping points) of a function, and so on.

To learn more about algorithms and how they work (or don’t), read the article When algorithms go bad: Online failures show humans are still needed published by CBC News (1 October 2017, at https://bit.ly/2M3jFyF). Highlights: “We’ve grown ‘dependent on algorithms to deliver relevant search results, the ability to intuit news stories or entertainment we might like,’ says Michael Geist, a professor at University of Ottawa and Canada Research Chair in internet and e-commerce law. 30 NUMERACY

These formulas, or automated rule sets, have also become essential in manag- ing the sheer quantity of posts, content and users, as platforms like Facebook and Amazon have grown to mammoth global scales. In the case of Amazon, which has over 300 million product pages on its U.S. site alone, algorithms are necessary to monitor and update recommendations eﬀectively, because it’s just too much content for humans to process, and stay on top of, on a daily basis. But as Geist notes, the lack of transparency associated with these algorithms can lead to the problematic scenarios we’re witnessing.”

Math Language and Real Life Language

Sometimes it becomes necessary to distinguish the language used in mathematics (we will refer to it as ‘math language’) and the language that we use in everyday situations (‘English language’ or ‘everyday language’). Since we want everyone who reads or listens to the same piece of math to receive the same message, we have to make sure that math language is clear, precise and unambiguous. As well, the context — which could be explicitly stated, or suggested but not explicitly stated — needs to be made clear. The sentence “The sum of two odd numbers is an even number” is an example of a math statement. Previous to this statement, concepts of the sum, as well as even and odd numbers were deﬁned. So, the statement above establishes one possible relationship between known concepts and/or known objects. From the development of the material, it is clear (but not speciﬁed in the sentence above) that we are talking about integers. In everyday language, we use words whose meaning is often vague, unclear or depends on our experience. For instance, the word “warm” in “It is warm outside” does not have the same meaning for everyone. Thus, “It is warm outside” is not a math statement. A few more examples contrast the two languages:

Math Language English (everyday) Language 14 is larger than 2. The roof is kind of red. ... is equal to 80 About 80 people showed up This is a square. This car is cheap. ... the chance is 45% It might rain today ... the tax is 13% ... somewhat larger

We are not saying that everyday language is not informative, or makes no sense. As a matter of fact, within a certain context, most people might agree with a statement (such as “This car is cheap”). As well, we know that “about 80 people” probably means 78 or 83 people, but not 47 or 150. In our discussion of probability we will see that “the chance is 45%” can be given a precise mathematical meaning, although it involves randomness. In certain situations, we use math terms but not in their precise mathematical sense. For instance, “a square of chocolate” is not a square. To say “Earth is a sphere” is not correct: Earth is a three-dimensional solid, whereas a sphere is a two-dimensional surface. When we say “area of a circle” (many school textbooks use this phrase) we mean the area of the region (i.e., the disk) enclosed by the circle. The circle is a curve and so has length, but not area. Note: We need to be careful when using the word “between”. How many whole numbers (integers) are there between 2 and 8? If we subtract, we get 8 − 2=6 which may or may not make sense. Section 2 Mathematical Reasoning and Numeracy 31

If “between” 2 and 8 means “between and including” 2 and 8, then the answer is 7. However, If “between” 2 and 8 means “between but not including” 2 and 8, then the answer is 5.

Quantiﬁers and Rules of Formal Logic

When we want to specify the quantity of things or objects which possess certain property, we use quantifiers. Two commonly used quantifiers are the universal quantifier, meaning “for all,” or “any,” or “for every,” and the existential quantifier, meaning “for some,” or “there are,” or “there exists.” In real life, we often use variations (relaxed versions) of these, such as “almost all,” “many,” “a few,” “barely any,” and so on. However, it is not really possible to develop formal logic with these quantifiers. (Actually there have been attempts at it, but that’s not relevant for what we’re trying to do here.) Note: in working with logical statements, we focus on their structure - we will not worry much about whether or not we are writing good (correct style, grammar, etc.) English sentences. To state that every single cat in Hamilton is black, we use the universal quantifier, and say “All cats in Hamilton are black” or “Every cat in Hamilton is black.” Math symbol for the universal quantifier is ∀. Denoting cats by x and the property of being black by A, we write ∀xA (might look weird, but will be useful!). The sentences “There is a black cat in Hamilton” or “There exists a black cat in Hamilton” use the existential quantifier, and mean that there is at least one black cat in Hamilton (of course, there could be two, or 10 or 295 black cats in Hamilton). Math symbol for the existential quantifier is ∃. Using the notation just introduced, we write ∃xA (where, as before x represents cat(s) and A is the property of being black). How do we work with statements that involve universal and existential quantifiers? Consider the statement involving the universal quantifier “All cats in Hamilton are black.” If we wish to prove that this statement is true, then finding one cat, or two or twenty two cats in Hamilton, and verifying they are all black will not do. We have to locate every single cat in Hamilton and verify that it is black. However, if we wish to prove that the claim “All cats in Hamilton are black” is false, all we need is to find one cat in Hamilton that is not black (it does not matter what colour it is, as long as it is not black). That cat is called a counterexample. By this reasoning, we reached an important conclusion: In order to prove that a statement involving a universal quantifier is true, we have to prove that it holds for each and every object that is implicated. To show that a statement involving a universal quantifier is false, we must find a counterexample, i.e., one case for which the statement does not hold. This latter case can be described by a math formula: ¬ (∀xA)=∃x (¬A) We read this formula as: negative of (i.e., we are proving that the universal statement is false) all objects have some property A is there exists an object which does not have the property A. Remember that we cannot use an example to prove that a statement involving a universal quantifier is true. However, to disprove such statement, we can use an example (that’s what we call a counterexample). Now, consider the following statement which expresses an existential property “There is a black cat in Hamilton.” Finding one black cat in Hamilton will prove that this statement is true. However, if we wish to prove that the statement is false, we have to show that not 32 NUMERACY

asinglecatin Hamilton is black. So, we have to find all cats in Hamilton, and check that none is black. Thus, we conclude that in order to prove that a statement involving an existential quantifier is true, we have to show that it holds for at least one object that is implicated. To disprove a statement involving a universal quantifier (i.e., to show it is false), we must show that it does not hold for all of the objects implicated. In terms of symbols, ¬ (∃xA)=∀x (¬A)

Example 2.8 Working with Quantifiers In each case, identify x (i.e., the objects involved) and A (the property that relates to the objects). Then say what you would have to do to prove that the statement is true. Finally, say what is needed to prove that the statement is false (i.e., that its negative is true). (a) “Every child with asthma has allergies” (b) “Some humanities students take Math 2UU3” (c) “Every investment generates profit” (d “There is a bird which can fly as high as 500 metres above ground” (e) “All McMaster students have to buy a Hamilton bus pass” (f) “There is a bird living on Earth that cannot fly.” Solution for (f): the object involved is bird(s), so that’s x. The property A is ‘cannot fly.’ Using the symbols we introduced, we write the given statement as ∃xA. To prove that “There is a bird living on Earth that cannot fly” is true, we need to find at least one such bird. Perhaps the most famous example is a kiwi bird, a nocturnal animal that lives in New Zealand. (Given that kiwi birds do not fly and New Zealand is an island, it’s not at all clear how they got there.) Another bird which cannot fly is takahe, which is native to Australia and New Zealand:

To disprove the given statement, we interpret the right side in the formula ¬ (∃xA)=∀x (¬A) Thus, we have to show that all birds (∀x) on Earth have the property ¬A = negative of cannot fly = can fly. In short, we have to show that all birds on Earth can fly.

Given two properties A and B,we can combine them to create new properties, A and B (denoted by A &B and formally called conjunction)andA or B (denoted by A∨B and formally called disjunction). The ‘or’ used is the inclusive or. Thus, “apples or oranges” includes three possibilities: apples only, oranges only, and apples and oranges. To prove a statement that involves ‘and’ we need to prove that both properties are satisﬁed. Thus, to prove “There is a cat in Hamilton which is black and hungry” we have to identify a cat in Hamilton which is both black and is hungry. Section 2 Mathematical Reasoning and Numeracy 33

To prove a statement that involves ‘or’ we need to prove that at least one property is satisﬁed. Thus, to prove “There is a cat in Hamilton which is black or hungry” we have to identify a cat in Hamilton which is either black (but not hungry), or a cat which is hungry (but not black), or a cat which is both black and hungry. How about disproving such statements? How can a cat not be black and hungry? For instance, if it’s a brown cat (so not black) but hungry, or if it’s a black cat with full stomach (so not hungry), or if it’s a brown cat (not black) with a full stomach (not hungry). Using the symbols we introduced, we write ¬ (A &B)=¬A ∨¬B

Likewise, ¬ (A ∨B)=¬A &¬B (Justify this formula with an example.) Note that we are doing math with sentences!

Example 2.9 Putting it All Together In each case, explain how you would prove and disprove each statement. (a) “Everyone who has flu has fever and runny nose” (b) “There is a person who smokes both traditional cigarettes and e-cigarettes” (c) “All women in Ireland have red hair and green eyes.” (d) “Some women in Ireland have red hair and green eyes.” (e) “Some women in Ireland have red hair or green eyes.” (f) “All women in Ireland have red hair or green eyes.” Solution for (c): To prove this statement, we would have to show that every single woman in Ireland has red hair and green eyes. To disprove this statement, first write it in terms of symbols: ∀x (A &B) where x represents a woman in Ireland, A represents red hair and B represents green eyes (check that it makes sense). Now use the math that we learned to compute ¬ (∀x (A &B)) First, the negative of the universal quantifier is the existential quantifier ¬ (∀x (A &B)) = ∃x (¬ (A &B)) Now use the formula for the negative of ‘and’ ¬ (A &B)=¬A ∨¬B and put it together: ¬ (∀x (A &B)) = ∃x (¬ (A &B)) = ∃x (¬A ∨¬B) Interpret the expression on the right: there exists a woman in Ireland who does not have red hair or does not have green eyes. Done! Solution for (e): To prove this statement, we would have to show find one woman in Ireland who has red hair or green eyes (meaning she has red hair (and eyes which are not green), or green eyes (and not red hair), or has both red hair and green eyes). To disprove, we proceed as above, by combining the formulas we discussed: with the same meanings for x, A, and B,, the given statement can be written as ∃x (A ∨B) 34 NUMERACY

Its negative is: ¬ (∃x (A ∨B)) = ∀x (¬ (A ∨B)) = ∀x (¬A &¬B) Thus, to disprove the given statement, we have to show that all women in Ireland do not have red hair and do not have green eyes.

Now that we have seen how formulas work, try reasoning without writing them down. Keep in mind that: (i) the negative of the existential quantifier is the universal quantifier, and vice versa: the negative of the universal quantifier is the existential quantifier (ii) the negative of ‘and’ is ‘or’ between the negatives of the statements (iii) the negative of ‘or’ is ‘and’ between the negatives of the statements One more thing and we are done. Consider the implication A ⇒ B, i.e., cause ⇒ effect. As mentioned before, this means that if the cause is present, the effect will happen. But how do we prove that an implication is not true? Consider the statement “If you use your phone continuously for 8 hours, its battery will die.” To disprove it, we have to use our phone continuously for more than 8 hours and show that its battery is not dead (there is charge left). Thus, in order to prove that cause ⇒ effect is not true, we need to demonstrate that the cause is present but the effect is absent. In symbols, ¬ (A ⇒ B)=A &¬B

Example 2.10 Proving Things Wrong Say what you would have to do to disprove each statement. (a) Singing opera causes hair loss (b) If Ann takes two Tylenol tablets, then her headache will be gone in two hours (c) Solving all homework questions guarantees the grade of A+ in this course (d) People who sleep 6 hours or less at night experience fatigue and periodic loss of concentration during the day. Solution for (c): To disprove this statement, we have to identify one student who indeed solved all homework questions, yet they did not end up with A+ in the course.

What does it mean that A does not imply B? Consider the statement “It has been demonstrated that drinking 6-8 glasses of water every day does not cause kidney problems.” Look at the diagrams below. Te diagram on the left represents the statement A implies B (the box prepresenting A is contained in the box representing B). So the arrangement of the two boxes where A is not contained in B (diagrams in the middle and right) represents A does not imply B.

A A B B B

A implies BA does not imply BA does not imply B

Thus, A does not imply B could mean two things: in some cases A implies Section 2 Mathematical Reasoning and Numeracy 35

B but in some cases it does not (middle diagram) of A does not imply B at all (diagram on the right). So the correct way to interpret “It has been demonstrated that drinking 6-8 glasses of water every day does not cause kidney problems” is to say that drinking 6-8 glasses of water every day might, or might not cause kidney problems. However, usually this statement is interpreted as “If you drink 6-8 glasses of water every day, you will not have kidney problems.” Note the subtle diﬀerence between disproving the statement A implies B and the statement A does not imply B.

Section references: [1] Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition. 2013. American Psychiatric Association. [2] Dalrymple, K. L., and Zimmerman, M. (2013, November). When does benign shyness become social anxiety, a treatable disorder? Current Psychiatry, 12(11), 21-38. [3] Lena K. Hedendahl, Michael Carlberg, Tarmo Koppel, and Lennart Hardell Measurements of Radiofrequency Radiation with a Body-Borne Exposimeter in Swedish Schools with Wi-Fi Frontiers in Public Health. 2017; 5: 279. Published online 2017 Nov 20. doi: 10.3389/fpubh.2017.00279 36 NUMERACY

3 Numbers: Quantitative in Quantitative Reasoning

Number is a concept (often labeled as mathematical concept) that we use to count, measure, label, calculate with, and so on. Every known society or civilisation that has existed on Earth has, in one way or another, developed and used number systems. One of the ﬁrst things a child learns is to count (moreover, there is evidence that children as old as a few months can distinguish between one, two and three; and so can dogs, cows and some other animals). In this chapter we discuss concepts related to numbers, as they are needed in real life situations.

Numbers

Natural numbers are numbers used to counts: one, two, three, four, five, and so on. The symbols that represent numbers, such as 1, 2, 3, 4, 5, and so on are called numerals. The system most people use today is based on the late 14th century Hindu-Arabic numeral system. It is a positional number system (i.e., the value of a numeral depends on its location within a number) based on the powers of 10 (and thus called a decimal system). For instance, the first (leftmost) occurrence of 7 in 17074 contributes 7000 to the value of the number, whereas the second 7 contributes 70 to its value. Expanding natural numbers by adding zero and negative (counting) numbers we obtain integers. By forming ratios of integres (fractions) we obtain rational numbers. By division, rational numbers can be converted to decimal numbers.These decimal numbers have either a finite number of decimals (such as 1/8=0.125), or an infinite number of decimals, which become periodic (such as 1/3=0.3333... (3 repeated), 3/11 = 0.27272727... (27 repeated), 8/21 = 0.380952380952380952... (380952 repeated)), and 1/28 = 0.03571428571428... (571428 repeated). The numbers whose decimal representation has an infinite number of decimals which do not become periodic are called irrational numbers. The most famous irrational number is π =3.1415926535.... Rational and irrational numbers, put together, form real numbers. Although called real numbers, there is not much real about them. In our everyday lives, we most often use natural numbers, integers and rational numbers, together with their decimal representations. When we measure something, we attach units to numbers. Of the amazing array of the ways people measure things, we will stick to those which are most commonly used. Many belong to the SI system (or the metric system) of units, such as the metre, the second and the kilogram. However, there ane numerous exceptions: instead of the SI unit Kelvin (for temperature) we use degrees Celsius (or Fahrenheit). As well, we use pounds, ounces, pints, as well as feet and inches (for instance, building standards in North America use these units).

Exact Values and Approximations

Some things are exact in the sense that we know their precise, unique value. For instance, the convention we accept about the time states that 1 hour = 60 minutes and 1 minute = 60 seconds (thus, there are exactly 24 · 60 · 60 = 86, 400 seconds in a day), or (again, by convention) 100 degrees Celsius = 212 degrees Fahrenheit. Section 3 Numbers: Quantitative in Quantitative Reasoning 37

Money and counting (smaller quantities) often result in exact values: one can buy 6 eggs, 4 bananas and two jars of peanut butter and pay $ 15.45. Larger counts (such as a number of people at an open air concert, population of Quebec, or the number of students attending colleges) are estimates. As well, larger amounts of money (say, government dept) are approximate, rounded-oﬀ amounts. For instance, Canadian National debt, i.e., the amount of money Canadian government owes to holders of Canadian Treasury security, such as treasury bills and bonds, is about 700 billion dollars (check the debt clock http://www.debtclock.ca/). It is not possible to exactly measure anything – it is inevitable that instru- ment(s) used and the people who use them will generate diﬀerent readings for Thus, every measurement comes with a measurement error, and can be expressed as measured value ± measurement error as in 3.15 ± 0.05. This means that the true value is believed to lie in the range from 3.10 to 3.20; using interval notation, we write [3.10, 3.20].

NASA’s Global Change Climate keeps track of the rising levels of sea water at their site https://climate.nasa.gov/vital-signs/sea-level/

The measurement 91(±4) mm is the increase in the sea level since 1993. Thus, the sea level increased between 87 mm and 95 mm. The causes of sea level increase are related to climate change: the added water comes from melting ice (glaciers), and from the thermal expansion of seawater. The graph below tracks sea level changes, with the baseline (Sea Height Variation = 0) taken to be the sea level on 1 January 1993. 38 NUMERACY

Below is a manufacturing blueprint for some metal piece, specifying its dimensions (called d1,d2,d3, and h). Note that instead of exact values, in each case a range of values is given, expressed using the minimum and the maximum allowed values. For instance d1 has the minimum of 1.7mm, and the maximum of 1.84mm, which can be written as an interval (range) [1.7, 1.84]. (Source: https://bit.ly/2JJ26T9)

A measurement error can be expressed as percent, as in “[...] the value is 12.7, with the measurement error of 1%.” This means that the interval (range) where the true measurement lies is [12.7 − 0.127, 12.7+0.127].

Quite often, even though there are errors involved, a measurement is presented as a single number. For instance, the dimensions of the table from Ikea online catalogue (https://bit.ly/2SvqjPj) are given as exact values (in inches, with metric units in parentheses). The dimensions are not presented as ranges, but as a single number. Is the length of the table exactly 220cm? It is not, however it might be confusing for an average shopper to read that, for instance, the length of the table is in the range [219.85,220.15] centimetres. But more important – do we really care if the table we’re contemplating to buy is 1.5mm longer or shorter than 220cm? Note that the measurements in inches are given as fractions (which is common practice in things related to construction, building standards, furniture, ap- pliances, etc., where fractions with denominators 2, 4, 8, 16, 32 (powers of 2) are used). Similarly, tech speciﬁcations for the size and the weigth of a MacBook Pro laptop (Apple Store, at https://apple.co/1hLVobV) use ‘exact’ values, rather than incorporating a measurement error

(Note that this time, measurements in the U.S. system of units are written as decimal numbers.) Section 3 Numbers: Quantitative in Quantitative Reasoning 39

Notes and comments: (1) The result of measuring something might depend on the device we use to measure. For an example, read How long is the coast of Great Britain? It depends how you measure it at https://bit.ly/2Y5yFTX (2) Heisenberg’s uncertainty principle (also known as the uncertainty principle) gives a limit to the precision with which certain quantities can be measured. In 1927, Heisenberg proved that the more precisely we measure the position of some (atomic) particle, the less precise our measurement of its momentum is (and vice versa). (3)Inmathwemakeadistinctionbetween exact values, when we use the equals sign = and approximate values, when we use the approximately equal sign ≈. However, often we get sloppy and write π =3.14 when we should say π ≈ 3.14. In real life, the symbols = and ≈ are not often used, but rather narrated, using words such as ‘is/are’ (as in “A table is 220 cm long”), ‘roughly’ (as in “That plot of land is roughly 15 metres wide”), ‘approximately,’ ‘about,’ ‘close to,’ and so on. (4) A common context where approximations appear is in recipes for preparing medications: instead of using interval notation such as [3.3, 3.5] milligrams, in these situations, we see ‘3.4mg±0.1 mg,’ or ‘minimum 3.3 mg and maximum 3.5 mg’.

Case Study 3.1 Approximations Everywhere (1) In the news article (NBC News, 18 June 2019, at https://nbcnews.to/2IqhX8v) More than 16 tons of cocaine worth up to $ 1B [1 billion] seized in massive bust in Philadelphia both numbers in the title are approximations. (By the way, the reporter is inconsistent when it comes to the actual amount of cocaine seized.) (2) On 20 July 2019, Iowa State Daily reports that “A press release from the city said a ‘transmission line fault’ caused the entire Ames electric service territory to lose power at 12:35 pm Saturday, causing ‘approximately’ 26,000 of their customers to go without power.” (https://bit.ly/2Y8jPXQ) (3) CBC News report How tiny homes could provide a path to security for new Canadians (21 July 2019, at https://bit.ly/30MR9W6) states that “In St. John’s, approximately 12,100 households live in unaﬀordable housing, of which nearly 65 per cent rent.” (4) In the report Woman arrested for stealing approximately $ 1,500 of items from acquaintance Indiana Daily Student (13 June 2019, at https://bit.ly/2LBVFDs) reported that “[...] suspect took a black RCA tablet, a 32-inch television, a watch, a speaker, Bluetooth headphones and clothing items valued at approximately $ 1,500.” (5) On 20 July 2019 in Halifax Pride parade 2019 takes over downtown Halifax CBC News reports that “Parade oﬃcials said there were approximately 150 parade entrants in the 2019 event.” (https://bit.ly/2YZvhq9)

Rounding Numbers

7.2·2.4 We start this section with a simple calculation – what is 0.8 ? If we round off all numbers to the nearest integer, we obtain 7.2 · 2.4 7 · 2 ≈ =14 0.8 1 If we calculate this expression without rounding off we obtain (cancel 2.4 and 0.8) 7.2 · 2.4 =7.2 · 3=21.6 0.8 which, rounded off to the nearest integer, gives 21.6 ≈ 22. Big difference! 40 NUMERACY

This illustrates a well known fact in numeric mathematics, namely that the round off errors (i.,e., replacing a true value of a number with some approximation) accumulate, rather than cancel each other. Hence, an important rule rule: We perform all calculations at maximum precision (which depends on what we use: a hand calculator, software, or an online calculator) and then round off to a desired precision once at the end. How do we round off a number? The most common rule, and the one we will use in this course, is to base the decision on the first digit (from the left) that will be dropped. ♦ Ifthefirstdigittobedroppedis0,1,2,3,or4,thenweround down, i.e., we drop that digit and also all digits to the right of it. For instance, 5.2374299 rounded off to one decimal place is 5.2, and rounded off to three decimal places is 5.237 (so rounding down refers to the fact that the rounded off value is smaller than the original value). ♦ Ifthefirstdigittobedroppedis5,6,7,8,or9,thenweround up, i.e., we drop that digit and also all digits to the right of it, and add 1 to the number that’s left. For instance, 5.2875299 rounded off to one decimal place is 5.3, rounded off to three decimal places is 5.288, and rounded off to six decimal places is 5.287530 (so rounding up refers to the fact that the rounded off value is larger than the original value). There is a variation of this rule (discussed below), as well as other rules, depending on the context where rounding off is used. Check, for instance, Cash rounding on Wikipedia (https://bit.ly/20v1Wib) and read about Swedish rounding. Let’s go back to the rule we adopted for this course (digits 0-4 are rounded down, digits 5-9 are rounded up). Rounding off changes the value of a number, except if the digit to be rounded off is zero. Thus, of the 9 cases when the number changes when rounded off, in 4 cases it is rounded down, and in 5 cases it is rounded up. This means that, in a large data set (say, 900,000 numbers) about 400,000 numbers would be rounded down, and about 500,000 numbers would be rounded up. Thus, the average of all numbers after rounding off would be larger than before rounding (and in many cases that’s not acceptable). To fix the situation, we adjust the rule for rounding off when digit 5 is involved: ♦ [same as before] If the first digit to be dropped is 0, 1, 2, 3, or 4, then we round down, i.e., we drop that digit and also all digits to the right of it. ♦ [same as before, except for the digit 5] If the first digit to be dropped is 6, 7, 8, or 9, then we round up, i.e., we drop that digit and also all digits to the right of it, and add 1 to the number that’s left. ♦ [new rule] If the first digit to be dropped is 5, then we round it so that the digit to be rounded off is even. Example for the last rule: round off 3.257 to one decimal place. If we round up, we get 3.3, and if we round down we get 3.2. The first decimal (the digit to be rounded) needs to be even, so it’s 3.2. Now round off 3.357 to one decimal place. If we round up, we get 3.4, and if we round down we get 3.3. The first decimal (the digit to be rounded) needs to be even, so it’s 3.4. This rule is good to know, as it is useful in some situations.

Notes and Comments: (1) In his work on numeric mathematics, John Von Neumann proved that round oﬀ errors propagate (i.e., do not cancel each other, no matter how we round oﬀ) and their accumulation (in numeric procedures involving a large number of calculations) can lead to serious problems. Section 3 Numbers: Quantitative in Quantitative Reasoning 41

(2) Measurements and calculations in engineering, often require a high level of precission (and numbers are routinely rounded oﬀ to six, or nine, or ten decimal places).

Example 3.2 Rounding oﬀ Round oﬀ each number to the requested number of decimal places, as shown.

number one two three four 3.72658 3.7 3.73 3.727 3.7266 0.246809 1.55555 44.050607 0.111111 0.999999

Rounding to nearest integer values is done analogously, and here we can visualize it–the question is what number is it closer to? Rounding $ 1.67 to the nearest dollar gives $ 2.00 (since 1.67 is closer to 2 than it is to 1). As well, 547 rounded to the nearest hundred is 500 (since 547 is closer to 500 than to 600). 1,500 rounded off to the nearest thousand is 2,000 (recall that when the digit in question is 5, we round up). The number 6, 946.58 rounded to one decimal place: 6, 946.6 rounded to the nearest integer: 6, 947 rounded to the nearest ten: 6, 950 rounded to the nearest hundred: 6, 900 rounded to the nearest thousand: 7, 000. Note that certain decimal numbers are always rounded off, such as periodic infinite numbers 0.333... (decimal 3 repeated) 0.405405... (group 405 repeated). The good news is we can express them as fractions (1/3and15/37 respectively) and thus record their exact value. However, as irrational numbers cannot be expressed as fractions, their values are always rounded off, and thus approximated. For instance, 3.14, 3.14159, and 3.14159266, are approximations of π.

There are cases when the context suggests that we deviate from the rules for rounding oﬀ, or they way we round oﬀ is not relevant. For instance, in a mathematical modelling of deer population in Algonquin National Park, an estimate of 1346.4 deer was obtained. For people working with deer, it’s irrelevant whether it is 1346 or 1347 deer. As well, they might be happy with an estimate of ‘about 1350,’ or even more rough, ‘about 13 hundred.’ You are in charge of a school trip for 342 students and their teachers, and need to organize bus transportation. As this is a longer trip noone is allowed to stand, i.e., every person must have a seat. Given that a bus has 55 seats, how many buses do you need to order? Dividing, you get 342/55 = 6.21, rounded to the nearest integer gives 6. But of course, you will not order 6, but 7 buses. 42 NUMERACY

Estimation

An estimation is a calculation or a judgement of the value of a quantity, usually done when relevant pieces of information are not available, or might not be known, or are not stable (i.e., change all the time) or when their values are not known precisely (so that an exact calculation is impossible). For instance, it is almost impossible to find out exactly how many paper coffee cups are used (and discarded) on McMaster campus in one day. We do not know (but can estimate) how much plastic ends up in Lake Ontario every day. Whenever we work on an estimate, we need to keep track of all data used, and assumptions made. For instance, to estimate a number of human heartbeats in a day, we need to know the heartbeat rate. This information is unstable (as it could change from person to person; as well, it depends on the type of activity: sleeping, sitting, walking, running, etc.). Consulting the American Heart Association web page https://bit.ly/2LNnjy1 we find “For adults 18 and older, a normal resting heart rate is between 60 and 100 beats per minute (bpm), depending on the person’s physical condition and age.” So, assuming that the heartbeat rate is 80 beats per minute (the average of the two extremes given), we obtain 80 · 60 · 24 = 115, 200 heartbeats in a day. Of course, with a different assumption, we would get a different estimate (thus, when we estimate, there are no unique answers). For instance: 60 heartbeats per minute yields 60 · 60 · 24 = 86, 400 heartbeats in a day 72 heartbeats per minute yields 72 · 60 · 24 = 103, 680 heartbeats in a day 84 heartbeats per minute yields 84 · 60 · 24 = 120, 960 heartbeats in a day 100 heartbeats per minute yields 100 · 60 · 24 = 144, 000 heartbeats in a day Note that we made a further (unreasonable!) assumption that one’s heartbeat rate remains constant throughout a 24-hour period (and equal to the resting rate). In reality, noone knows exactly how many times a human heart beats in one day (or even in one hour, or in one minute – unless we monitor and count). However, from the above estimates we can assert that very likely a human heart beats between 86,400 and 144,000 times in a day.

Note: Remember to state all assumptions you made when calculating an estimate.

Example 3.3 How Many Seconds are There is a Year? One thing is easy – we have already ﬁgured out that there are exactly 60 · 60 · 24 = 86, 400 seconds in one day. Now a more diﬃcult question: how many days are there is a year? Could be 365 or 366, depending on whether it’s a leap year or not. Computing the average (365 + 366)/2 = 365.5 is not appropriate. Why? Given that there is 1 leap year in 4 years, the average length of a year is 365 + 365 + 365 + 366 1461 = = 365.25 4 4 days. Thus, assuming that a year has 365.25 days, the number of seconds in a year is 60 · 60 · 24 · 365.25 = 31, 557, 600 Section 3 Numbers: Quantitative in Quantitative Reasoning 43

The fact that there are 365.25 days in a year was used to define the Julian Calendar, which was in effect until 16th century (in Europe; other parts of the world used different calendars, or used the Julian calendar longer) In the 16th century, a reform of the Julian calendar (as it was shown not to be precise; see the next example) created the Gregorian calendar, according to which certain leap years were eliminated (those divisible by 100 but not by 400; thus, 1700, 1800 and 1900 were not leap years, but 2000 was a leap year). In other words, in a span of 400 years, there are 97 leap years (366 days) and 303 non-leap years (365 days). (According to the Julian calendar, there were 100 leap years in a span of 400 years.) This makes the average length of a year 365 · 303 + 366 · 97 = 365.2425 400 days. With this number in mind, the number of seconds in a year is computed to be 60 · 60 · 24 · 365.2425 = 31, 556, 952

Of course, scientists being scientists, they did not stop there. Further estimates have been obtained, such as the one callled mean tropical year, whose length is 365.242189 ir 365.24217 days (one day being equal to 86,400 seconds). Read more about it at https://bit.ly/2JKj4Ak.

Example 3.4 Calendar Trouble, or Why Precision Matters Let us compare the Julian length of the year (365.25 days) with the more accurate (and shorter) Gregorian length of the year (365.2425 days), based on astronomic observations of the length of the solar year. The diﬀerence between the two lengths is 365.25 − 365.2425 = 0.0075 days, or 0.0075 · 86, 400 = 648 seconds, or 10.8 minutes. Thus, the ﬁrst year according to the Julian calendar was 10.8 minutes longer than it should have been. difference = 10.8 minutes end of year 1 according to the solar year

start of the Julian calendar end of year 1 according to the Julian calendar

Not a big deal. However, over a hundred years, that discrepancy grew to 10.8 · 100 = 1080 minutes, or 18 hours! In other words, people celebrating the arrival of a new year 100 years after the adoption of teh Julian calendar would be celebrating it 18 hours into the new year! difference = 18 hours end of year 100 according to the solar year

start of the Julian calendar end of year 100 according to the Julian calendar

The Julian calendar was introduced in 45 BCE. For every 100 years of its use, it was oﬀ by another 18 hours. Thus, in the year 1555 CE (so 1600 years after its adoption) the Julian calendar was ahead of the solar year by 16 · 18/24 = 12 days! At the time, people did not need these calculations–they realized that the summer solstice (which can easily be observed) did not agree with 21 June. In 1555, the day of the summer solstice was 9 June). To be in tune with the solar year, two things needed to be done: ﬁx the length of the year, and removed a certain number of days from the present (Julian) calendar. Thus, the Gregorian calendar was born (with length of a year = 365.2425 days). In 1582 it was adopted by several countries in Europe. However, instead of 12, 10 days were dropped from the calendar, in month of October; the calendar looked like 44 NUMERACY

It took about 300 years for most of the world to adopt the new Gregorian calendar, and as there were doing it they needed to remove more days to synchro- nize with those who already modified their calendars. For instance, Canada, U.S., and U.K. adopted the Gregorian calendar in 1752, and removed 11 days; Japan adopted it in 1872/73, and removed 12 days; Turkey adopted it in 1926/27, and removed 13 days. So different countries followed different calendars for quite some time, and consequently used different rules for the leap years (i.e., it was all a big mess). Currently, the Julian calendar is 13 days behind the Gregorian calendar. Thus, the start of the Fall term, 3 September 2019, is 21 August 2019. Note: There is an amazing variety in the ways people have been keeping track of time; some of these do not care about leap years, nor the precise length of a year. For instance, Mayan (Maya = Mesoamerican culture) long count calendar keeps track of the number of days from the creation of universe (corresponds to 11 August 3114 BCE). According to this calendar, 3 September 2019 is the day 1,874,447.

Note: Calendar correction is just one situation where small differences in numbers matter. Over time, small numbers could (and do) accumulate to something large. Perhaps the best example are banking fees: we might feel that individual fees are not large (say, paying $ 2 for some transaction). However, given the huge number of transactions, it is known that these ‘small’ fees contribute billions of dollars to the profits of banks and other financial institutions.

Case Study 3.5 Raw Count (Enumeration) vs Rough Estimate: Number of Sex Partners How do we arrive at estimates in everyday life? In New York Times article The Myth, the Math, the Sex (12 August 2007, at https://nyti.ms/2GDyHpe) we read “One survey, recently reported by the federal government, concluded that men had a median of seven female sex partners. Women had a median of four male sex partners. Another study, by British researchers, stated that men had 12.7 heterosexual partners in their lifetimes and women had 6.5.” These findings echo an earlier report Why Men Report More Sex Partners than Women published in LiveScience (February 17, 2006 at https://bit.ly/2Jmmmrt) “Psychologist Norman R. Brown at the University of Michigan has done several studies on the apparent flaw in these surveys. The latest was a web-based survey of 2,065 heterosexual non-virgins with a median age in their late 40s. The women reported on average 8.6 lifetime sexual partners. The men claimed 31.9.” Can these reports be true? Let’s try to figure it out. These reports are about heterosexual sex, so we assume that there are six women and six men (representing the fact that the population is roughly fifty-fifty; these numbers do not matter, as you can check yourself by picking, say 10 women and 10 men). Each segment in the diagram represents one sexual relation between a male and a female, and the number of sexual partners for every person is shown. Section 3 Numbers: Quantitative in Quantitative Reasoning 45

(2) W M (4) (3) W M (0) (0) W M (3) (3) W M (1) (3) W M (3) (2) W M (2)

Now we compute the averages: for women it is 2+3+0+3+3+2 13 = 6 6 and for men 4+0+3+1+3+2 13 = 6 6 Exactly the same, as it should be! Of course, this is an ideal situation (for instance, it could be that a woman from this group had a sexual relation with a man not included here). But this alone does not account for the discrepancy from the reports. In the LiveScience report, we ﬁnd a more plausible explanation: “ Women rely on a raw count, a method Brown says is known to result in underestimation. They tend to say, ‘I just know,’ and if you ask them to explain how they know, they say, ‘Well, there was John, Tom, etc.’ Men also rely on a ﬂawed strategy: [they] are twice as likely to use rough approximation to answer the question. And rough approximation is a strategy known to produce over-estimation.” For further statistics, read Women’s Health magazine article The Average Number Of Sexual Partners For Men And Women Revealed (30 September 2015, at https://bit.ly/2GV5J7J)

Small, Human Scale, and Large Numbers

Numbers come in all sizes, from very small to very large. How do we write them, compare them and reason with them? How do we internalize,‘get a feel for,’ numbers – for instance, how do we make ourselves understand that one billion is thousand times larger than a million, or that it takes one billion nanometres to form one metre? Here is an example, illustrating why this is important.

Example 3.6 Coffee, Anyone? The article British university ‘sorry’ after wrongly giving students 300 coffee cups worth of caffeine published on SI News web page https://bit.ly/2HgRoAx on 26 January 2017 tells a story about a medical experiment at the University of Northumbria in England that went terribly wrong. In March 2015, two sports students volunteered to participate in an experiment about effects of caffeine on exercise. They were supposed to be given a cup of coffee with 300 mg = 0.3 grams of caffeine and start an exercise routine. How much is 300 mg of caffeine? The article claims that a shot of espresso has about 65 mg of caffeine. Starbucks’ single espresso has 75 mg of caffeine (https://bit.ly/2q9P1I8) whereas Tim Hortons’ small espresso has 45 mg of caffeine (https://bit.ly/2H1yXBZ). Thus, ignoring Tim Hortons’ 300 mg of caffeine is equivalent to roughly four shots of espresso. Or, consider large drinks: Starbucks’ Grande Pike Place Brewed Coffee contains 310 mg of caffeine; Tim Hortons’ Large Coffee has 270 mg, and XLarge Coffee 46 NUMERACY

has 330 mg of caffeine (see above links). Thus, 300 mg of caffeine is contained in about one large cup of coffee. By mistake, the two students were given 30 grams (30,000 milligrams) of caffeine (in powder form), which is one hundred times larger than what they were supposed to be given – hence the equivalent of having one hundred coffees. (Note: in British media it was reported as 300 cups of coffee, assuming about 100 mg of caffeine per coffee cup–although that is not at all clear from the article.) University explained that the researchers involved “had used a mobile phone to calculate the caffeine dosage, resulting in the decimal point being in the wrong place.” The good news is that both students recovered, one after a temporary short- term memory loss. However, the question remains – how did it happen that the researchers failed to distinguish between a dose and the one which is hundred times larger?

Soon, we will learn that instead of saying “one hundred times larger” we could say “two orders of magnitude larger.”

‘Human scale’ numbers we can easily work with, relate to our experiences, and make meaningful comparisons. For instance, we know that by adding three hundred and two thousand we cannot get hundred thousand. Withough multiplying, we know that the product of 16 and 81 is larger than one hundred and smaller than ten thousand. We can easily visualize ten (e.g., ten books), or ﬁfty (e.g., the length of an olympic-size pool is 50 metres), or one million (as this is how much money (or more) we need to buy an average house in Toronto), or one tenth (1 millimetre is one tenth of a centimetre). If we have 100 dollars, we know that we can buy ten things which cost 10 dollars each. We know that we cannot buy a new car for 10,000 dollars, but we can pay a yearly tuition to study at McMaster with this money.

Example 3.7 Small(er) Numbers (1) As part of the analysis The price of making a plastic bottle The Economist (15 November 2014, https://econ.st/2fxm1Eb) published this diagram showing how cheap it is to produce a plastic bottle–and hence the reason why there are so many of them ﬁlling our recycling bins, landﬁlls, and so many large cargo ships trying to dispose of them in some poor country.

(Note: the above link is not available without subscription to The Economist, but you can ﬁnd other sources conﬁrming this.) Section 3 Numbers: Quantitative in Quantitative Reasoning 47

(2) How cheap (or expensive) is drinking water? Depends on who is buying. In Why Nestl’s Aberfoyle well matters so much to Guelph, Ont., residents (CBC News, 26 September 2016 https://bit.ly/2ONa65s) we read: “A showdown between water advocates and politicians is expected Mon- day night in Guelph, Ont., over the divisive issue of Nestl´e Waters Canada seeking renewal of a water-taking permit for its bottling plant in nearby Aberfoyle. The permit has become a ﬂashpoint in a battle between environmentalists, community leaders and corporate interests, after it was revealed that the province only charges bottled water companies $ 3.71 per million litres of water.” $ 3.71 per million litres of water amounts to 371/1, 000, 000 = 0.000371 cents per litre. How much do we–citizens of Hamilton, Ontario pay for water? As a recent bill shows, we are charged at the rate of 78 cents per cubic metre of water (1 cubic metre =1000 litres), which amounts to 0.078 cents per litre.

We can compare these numbers roughly: in scientific notation, 0.000371 = 3.71 · 10−4, and 0.078 = 7.8·10−2. Thus, we pay 2 orders of magnitude more than Nestlé Waters Canada. A precise comparison 0.078/0.000371 = 210.24 tells us that we pay 210 times more for water than Nestlé Waters Canada! To add insult to injury, we then buy bottled water at very high prices. Put (1) and (2) together to calculate how much does it actually cost to produce a bottle of water.

(Sold at Rogers Centre during Toronto Blue Jays game. Smart? Really?)

There are numbers that are beyond human scale and defy our intuition and imagination. For example: (1) A new iMac computer from Apple comes with 1 TB (= one terabite) hard drive. One terabite is one thousand billion bites. (2) Nanotechnology involves working with matter whose size is about 1 to 100 nanometers. One nanometre is one billionth of a metre (imagine cutting a stick one metre long into one billion pieces). 48 NUMERACY

(3) According to the World Bank, in 2018, the gross domestic product (GDP) of Canada was 1,709.3 billion US dollars (which is 1.7093 trillion US dollars). The diagram below shows the changes in the GDP of Canada from 2009 to 2018 (source: Trading Economics, https://tradingeconomics.com/canada/gdp).

(4) Human brain contains about one hundred billion neurons. Written out, that number is 100,000,000,000. (5) The average volume of a human red blood cell is about 100 micrometres cubed. A micrometre is one millionth of a metre. (6) One light year (the distance that light travels in vacuum in one year) is about 94,607,304,725,808,000 metres. (7) The mass of one grain of salt is about 0.00000005 kilograms. Thus, one kilogram of salt contains about 1/0.00000005, which is about 20 million grains of salt. (8) The total volume of water contained in Earth’s oceans is estimated to be about 1,350,000,000,000,000,000,000 litres.

(9) The diameter of a proton is approximately 0.000000000001 millimetres. If we could manage to align 1/0.000000000001 = 1, 000, 000, 000, 000 (one thousand billion) protons next to each other on a line, they would cover the distance of 1 millimetre. (10) At one point the rate of inﬂation in Zimbabwe was so high (called hyper- inﬂation) that the currency was quickly becoming worthless. As a result, ban- knote of astronomically high value were printed, such as the one from 2019 (from Wikipedia): Section 3 Numbers: Quantitative in Quantitative Reasoning 49

Zimbabwe has since introduced a new currency; see Why does Zimbabwe have a new currency? (BBC News, 22 July 2019, https://bbc.in/2JV2k8p). We read: “The economy is no longer in its extreme inflationary spiral, but the country has continued to suffer from severe shortages of food, medicine and fuel. Last month, the Zimbabwean authorities reintroduced the Zimbabwean dollar as the country’s sole legal tender.” On 20 May 2018, a blog EconomicPolicyJournal.com under Venezuela is Ex- periencing Hyperinflation,Turkey May Be Next shows a picture of a banana and the amount of money needed to buy it. (https://bit.ly/2YkcBA4)

(11) Pornhub is “Canadian pornographic video sharing and pornography site on the Internet.It was launched in Montreal, providing professional and amateur pornography since 2007” (Wikipedia). In its 2018 Year in Review, published on 11 December 2018 Pornhub shares major statistics about its site (https://bit.ly/2zUYKHr). In particular, they claim that they generated 4,403 petabytes of traﬃc (What is a petabyte? Rean on).

Example 3.8 When Youtube Counter Broke The article Gangnam Style music video ‘broke’ YouTube view limit (BBC News, 4 December 2014 https://bbc.in/2LKS5aj reported on the fact that the counter for the video reached its limit (and thus would not increase with new views). We read: “YouTube said the video – its most watched ever – has been viewed more than 2,147,483,647 times. It has now changed the maximum view limit to 9,223,372,036,854,775,808, or more than nine quintillion.” The screenshot below was taken in July 2019, with the counter showing over 3.3 billion views. 50 NUMERACY

The article continues: “How do you say 9,223,372,036,854,775,808? Nine quintillion, two hundred and twenty-three quadrillion, three hundred and seventy-two trillion, thirty-six billion, eight hundred and ﬁfty-four million, seven hundred and seventy-ﬁve thousand, eight hundred and eight.” Note: for technical details about how it happened, read The Economist How Gangnam Style broke YouTube’s counter https://econ.st/310OmZG

How do we make sense of these, and other (very) small and large numbers? To start, we need a productive way of writing them down–and that is accomplished by using powers of 10. Recall that 100 =1 101 =10 102 =10· 10 = 100 103 =10· 10 · 10 = 1000 and, in general, 10n =10 · 10 · 10 ·····10 = 100 ...00, where 1 is followed by n zeroes . n Using division, we deﬁne negative powers: 1 10−1 = =0.1 10 1 1 10−2 = = =0.01 102 100 1 1 10−3 = = =0.001 103 1000 and, in general, 1 10−n = =0.00 ...01 where the total number of decimals is n (thus, there 10n are n − 1 zeroes before the 1 at the end).

Example 3.9 Working With Powers of 10 Recall the multiplication formula 10m · 10n =10m+n (i.e., 10m has m zeroes, 10n has n zeroes; when multiplied, we get a number with m + n zeroes, that is, 10m+n). For example, 102 · 1011 =1013, 10 · 107 =108, etc. The division formula for the powers is 10m =10m−n 10n Section 3 Numbers: Quantitative in Quantitative Reasoning 51

The number on the top has m zeros, and the one on the bottom has n zeroes. Consider examples: 108 100, 000, 000 = = 100, 000 = 105 103 1, 000 (thus 108/103 =108−3 =105)and 105 100, 000 1 1 = = = =10−6 1011 100, 000, 000, 000 1, 000, 000 106 (in short, 105/1011 =105−11 =10−6).

Example 3.10 Multiplying Decimal Numbers and Powers of 10 Recall the rule: to multiply a decimal number by 10n, where n is positive, we move the decimal point to the right (thus making the number larger). To multiply a decimal number by 10n, where n is negative (thus, dividing), we move the decimal point to the left (which makes the number smaller). For instance: 0.003509 · 10 = 0.03509 0.003509 · 102 =0.3509 0.003509 · 103 =3.509 0.003509 · 105 = 350.9 0.003509 · 107 =35, 090 0.003509 · 1010 =35, 090, 000 As well, 26.193 · 10−1 =2.6193 26.193 · 10−2 =0.26193 26.193 · 10−3 =0.026193 26.193 · 10−5 =0.00026193

Anumberinscientific notation is written as x = D.dd...d· 10A where D is a single non-zero digit, dd...d are decimals and A is an integer (positive, negative, or zero). The number A is called the order of magnitude the number x. As we will see, these are useful notions–the order of magnitude is used for rough comparisons between numbers. In order to tell the order of magnitude of a number we have to write it in scientific notation. Examples: 5.56789 = 5.56789 · 100 (order of magnitude: 0) 55.6789 = 5.56789 · 101 (order of magnitude: 1) 556.789 = 5.56789 · 102 (order of magnitude: 2) 5567.89 = 5.56789 · 103 (order of magnitude: 3) 556789 = 5.56789 · 105 (order of magnitude: 5) We can think of one order of magnitude as one power of ten (i.e., ten-fold). For a positive integer, the order of magnitude is the number of digits minus 1. (Later, we will characterize it in a different way.) Thus, the order of magnitude of 15,000 is 4, and the order of magnitude of 3 million is 6. As well, 0.556789 = 5.56789 · 10−1 (order of magnitude: −1) 0.0556789 = 5.56789 · 10−2 (order of magnitude: −2) 0.00556789 = 5.56789 · 10−3 (order of magnitude: −3) 0.0000556789 = 5.56789 · 10−5 (order of magnitude: −5) 52 NUMERACY

Example 3.11 Order of Magnitude In Electric vehicles powered by fuel-cells get a second look (The Economist, 25 Sept 2017, at https://econ.st/2ZkmsaG) we read: “Since then, the cost of fuel-cells has come down by at least an order of magnitude, as researchers have learned how, among other things, to use less platinum in the catalyst.” Roughly, by how much has the cost of fuel-cells come down?

Example 3.12 Comparing Quantities Using Order of Magnitude Since the order of magnitude of 7,491 is 3, and the order of magnitude of 12 million is 7, we say that ‘12 million is 4 orders of magnitude larger than 7,491’ (or that 7,491 is 4 orders of magnitude smaller than 12 million). Assume that the mass of a house mouse is about 0.02 kg, the mass of a dog about 20 kilograms, the mass of a cow about 500 kg, the mass of an elephant is about 4500 kg the mass of a blue whale is about 150,000 kg. (Figure out the orders of magnitude of these numbers.) We can say that a blue whale is three orders of magnitude heavier than a cow, four orders of magnitude heavier than a dog and seven orders of magnitude heavier than a house mouse. A house mouse is three orders of magnitude lighter than a dog. A cow is one order of magnitude heavier than a dog and ne order of magnitude lighter than an elephant.

Images are not to scale!

Example 3.13 Free University Tuition for Everyone Suppose that our government decides that education in Canada should be free, and thus will to pay tuition for all university students. How much would that cost the government? Since we do not know the exact number of university students in Canada, nor the amount of tuition they pay, we can only make an estimate. How many students are there in Canadian universities? We pick a number, and increase it by the order of magnitude, and pick the one that looks right. For instance: 15, 150, 1,500, 15,000, 150,000, 1,500,000, 15,000,000. What is the most likely number of university students in canada? 150,000 is too small, when we think of the fact that just two universities nearby, McMaster (about 27,000 students) and University of Toronto (over 60,000 students) combined, yield more than half od 150,000. As well, Canada cannot have 15 million students, as it would be too many (given the tiotal population of 37 million). Thus, of all numbers listed, 1,500,000 is the most likely. Section 3 Numbers: Quantitative in Quantitative Reasoning 53

What is an average yearly tuition? Apply the same principle, starting with, say, 40: 40, 400, 4,000, 40,000, 400,000, etc. By elimination (400 is too low, 40,000 is too high) we conclude that it is 4,000. Multiplying the number of students (1,500,000) by the average tuition ($ 4,000) we obtain $ 6,000,000,000, i.e., 6 billion dollars. What if we pick diﬀerent numbers? Assume that there are 1 million university students in Canada, and the average yearly tuition is about $ 5,000. In this case, we obtain 1 million · 5, 000 = 5, 000, 000, 000 = 5 billion dollars Not the same amount, but the same order of magnitude! We can try other numbers, and we’re sure that we will either obtain a number of the same order of magnitude, or perhaps one order of magnitude larger or smaller – but deﬁnitely not three orders of magnitude lager or smaller. Let us compare our estimate with the Universities Canada data (check the web page https://bit.ly/2nSwoYf)

The average tuition for Canadian students enrolled in canadian universities is about $ 6,800 (read the article How Much Does it Cost to Study in Canada? posted at https://bit.ly/2m6A9Hs). With these numbers, we obtain 1.7 million · 6, 800 = 11, 560, 000, 000 = 11.56 billion dollars which is one order of magnitude higher than our previous estimates.

The kind of calculation done in the previous example is sometimes referred to as back of an envelope calculation.

Certain powers of 10 are given names and preﬁxes. Large numbers:

Name/preﬁx Symbol Numeric English Order of magnitude kilo K 103 thousand 3 mega M 106 million 6 giga G 109 billion 9 tera T 1012 trillion 12 peta P 1015 quadrillion 15 exa E 1018 quintillion 18 54 NUMERACY

Small numbers:

Name/preﬁx Symbol Numeric English Order of magnitude deci d 10−1 tenth -1 centi c 10−2 hundredth -2 milli m 10−3 thousandth -3 micro μ 10−6 millionth -6 nano n 10−9 billionth -9 pico p 10−12 trillionth -12 femto f 10−15 quadrillionth -15

Example 3.14 Nanoparticles and Tattoos On 13 Sept 2017 New Zealand Hedrald published the piece Why tattoos could give you cancer https://bit.ly/2Mt62ZI reporting on a study pointing at potential dangers of having tattoos. The author of the study said that they: “[...] already knew that pigments from tattoos would travel to the lymph nodes because of visual evidence: the lymph nodes become tinted with the colour of the tattoo. It is the response of the body to clean the site of entrance of the tattoo. What we didn’t know is that they do it in a nano form, which implies that they may not have the same behaviour as the particles at a micro level. And that is the problem: we don’t know how nanoparticles react.”

Note: There are two very large numbers which have special names. A googol is written as 1 followed by 100 zeroes, i.e., 10100. A googolplex is deﬁned as 10googol which is 1 followed by googol zeroes. Neither number has a meaning in math or in physics. For further names and symbols consult Wikipedia, under Ordder of Magnitude.

Going back to the list of large/small numbers (pages 47-49): In (2), 1 nanometre is 0.000000001 = 10−9 metres. One hundred million neurons (4) is 100, 000, 000 = 108 neorons. One micrometre (5) μm is 10−6 metres. One light year (6) is 9,460,730,472,580,800 metres, which is about 9.461 · 1015 metres. The mass in (7) can be written as 0.00000005 = 5.0 · 10−8, and thus its order of magnitude is −8. The number in (8) is 1, 350, 000, 000, 000, 000, 000, 000 = 1.35 · 1021 litres. In words, the order of magnitude of the volume of water in our oceans, in litres, is 21. Petabyte in (11) means 1015 bytes, i.e., 1015=1, 000, 000, 000, 000, 000 bytes or, in words, a million billion bytes.

Example 3.15 Kilobytes, Megabytes, Gigabytes, ... How large are data ﬁles? The smallest unit of that carries information is called a bit, and has only two values, which are often denoted by 0 and 1. A byte isagroupof8bits,andisa smallest unit that can be addressed (accessed) on a device that stores data. For instance, uppercase ‘A’ is represented as 01000001, lowercase ‘a’ is 01100001, and the number 7 is 00110111. One kilobyte (KB) is not exactly 1000 bytes, but 210 = 1024 bytes (this diﬀerence is often neglected, and, for practical purposes, we say that 1 KB is equal to 1,000 bytes. 1024 kilobytes form a megabyte MB, 1024 megabytes form a gigabyte GB, and 1024 gigabytes form a terabyte TB. Section 3 Numbers: Quantitative in Quantitative Reasoning 55

Based on this (and thinking of 1024 as 1000) we say that 1TB has 1000 GB, or 1,000,000 MB, one GB has one billion bytes, and so on. The size of an average text-only email (so no embedded pictures or media, and no attachments) is about 10-20 kilobytes (KB). A single photograph, taken with a phone, could require anywhere between 1-10 megabytes of space (could be more, for instance if we use a professional camera; for the argument below, assume it’s 6 MB). A full-length movie, depending on the resolution (standard definition vs high definition), could have the size of 2-6 GB (gigabytes). (Recall that 1 MB = 1 million (106) bytes and 1 GB = 1 billion (109)bytes.) For instance, this screenshot from iTunes shows that the movie Grizzlies requires abot 4 GB ih high definition (HD) and 1.6 GB instandard definition (SD).

Thus the storage required for a movie is three orders of magnitude larger than that for for a photograph, and 5 orders of magnitude larger than the size of a text-only email.

Example 3.16 Roaming Charges Why do we have to know about megabytes and gigabytes? One reason is roaming charges that companies chanrge when we are outside of our calling zones. These charges can be substantial. For instance, if we do not have a roaming (data plan) with Fido, roaming charges if we use our phone in the U.S. are $ 7.99 fpr 50 MB of data.

Suppose that we watch The Grizzlies from our hotel room in Los Angeles. In standard deﬁnition (there is no reason why we should use high deﬁnion on a 56 NUMERACY

laptop) the movie is 1.6 GB = 1,600 MB. So, the roaming charge will be 1, 600 · 7.99 = 32 · 7.99 = 255.68 50 With 13% HST, it’s about $ 289 !!! If we were watching the same movie from Morocco

we would accumulate roaming charges of 1, 600 · 9.99 = 5, 328 3 dollars, which amounts to about $ 6,020 after taxes. This is not an academic discussion only; these things happen when people are not careful. On March 2013 in the article Dad gets $ 22,000 data roaming “shock” from Fido (https://bit.ly/2JUz9DT) CBC News reports on an 11-year old who streamed YouTube videos in Mexico. We read “A B.C. dad is accusing Rogers of price gouging, after his 11-year-old son mistakenly racked up $ 22,000 worth of data charges on his fathers phone, during a family trip to Mexico. [...] his son got a sunburn and was allowed to spend time playing video games in the family’s hotel room over the course of three days. He had games installed on the phone, but also streamed several hours of video. The company later told him his son had used $ 22,000 worth, approximately 700 megabytes. According to Rogers website, thats about 12 hours of YouTube video streaming.”

Multiplication Principle

One way to generate large numbers (which can be quite useful) is to combine objects, each of which has several (or many) variations. Multiplication principle states that if X happens in m ways and Y happens in n ways, then both X and Y happen in m · n ways. This principle applies to any number of things: if X1 happens in m1 ways, X2 happens in m2 ways, X3 happens in m3 ways, ..., and Xk happens in mk ways, then X1,X2,X3,...,Xk happen in m1 · m2 · m3 · ...· mk ways.

Example 3.17 How Many Coffees? You are in a coffee shop and are faced with options: you need to pick the size of your drink (small, medium, large, extra large), choose a flavour (plain, vanilla, cinnamon), and make a decision about dairy (milk, cream, no dairy). How many different coffees can you order? To get a feel, start by writing down options in an organized way: small, plain, milk small, plain, cream Section 3 Numbers: Quantitative in Quantitative Reasoning 57

small, plain, no dairy small, vanilla, milk small, vanilla, cream small, vanilla, no dairy small, cinnamon, milk small, cinnamon, cream small, cinnamon, no dairy There are no other options which involve a small size coffee. Note that there are 9 options in this list, where 9 is the product of 3 (options for flavour) and 3 (options for dairy). We can now copy this list and replace ‘small’ by ‘medium’, to get 9 new options. Then we replace ‘small’ by ‘large’ for 9 more options and replace ‘small’ by ‘extra large’ for the final 9 options. Thus, the answer is 9 + 9 + 9 + 9 = 36. Using the multiplication principle, we see that there are 4 · 3 · 3=36different options for our drink.

Example 3.18 Credit Card Numbers Credit card numbers vary in size from 13 to 19 digits, with the most commonly used cards having 15 or 16 digits. The first six digits from the left are not random: the first digit (or the first two digits) identify a major card network that the card belongs to: Visa (4), Mastercard (5), American Express (34, 37), Diners Club (36, 38), and so on. The remaining digits in this initial group of 6 digits identify the bank which issued the card, and its type (cash back, tied to a loyalty program, travel rewards, no annual fee, low interest rates, and so on). For instance, TD Aeroplan Visa Credit Card numbers start with 4520 88, Bank of America Gold Visa cards start with 4800 11, and Citibank Platinum mastercard cards start with 5424 18.

The last digit is not random either – it is called a check digit, and is calculated from the remaining digits. Assuming that the card has 16 digits (as all Visa and Mastercard cards do) we see that only 9 digits are left for the actual credit card number. check digit 452 0 8 8

Into each of the nine empty places we can put any digit 0, 1, 2, ..., 9, i.e., we have 10 choices for each place. Using the mulitplication principle, we see that we can generate 10 · 10 · ...· 10 = 109 (one billion) numbers. In reality, not all numbers are valid credit card numbers (the same digit ap- pearing in all nine places is not acceptable, for instance). As well, Luhn’s algorithm that we mentioned earlier blocks additional candidates – if we transpose two neigh- bouring digits, we will not get a valid credit card number. As well, if we replace a single digit with another digit, we obtain a non-valid card number. 58 NUMERACY

Example 3.19 Short URLs You have noticed that many links in this book look like https://bit.ly/2JUz9DT. They are shorter versions (and thus more convenient to type and use) of the original URLs, produced by the free URL shortener https://bitly.com/.ThisURL shortener (as any other URL shortener) maintains a large dictionary where a short URL is tied to the original URL, which is how this works. For instance, the URL of Internet Live Stats web page (which provides data on the size of the Internet) is https://www.internetlivestats.com/total-number-of- websites/ and its shortened version is https://bit.ly/2KW7dC9. How many diﬀerent web pages can this shortener organize? A shortened URL is uniquely identiﬁed by the 8-letter string following https://bit.ly/.Asimple experiment convinces us that this string is case-sensitive. Thus, in each of the 8 places in the string we can put a lowercase letter (26 options), or an uppercase letter (26 options), or a digit 0-9 (10 options)–thus a total of 62 options. The multiplication principle tells us that the total number of web pages that can have bit.ly shortened URL is equal to 62 multiplied by itself 8 times, which is 628 = 218, 340, 105, 584, 896 approximately 2.18 · 1014 (or 218 million billion). This is several orders of magnitude larger than all indexed (accessible) web pages on Internet (see Internet Live Stats).

Example 3.20 Licence Plates, IP Addresses and Postal Codes (1) What is the largest number of car licence plates for vehicles registered in Ontario, assuming that the format is LLLL DDD (i.e., four letters followed by three digits). Assume that the leftmost letter is either A, B, or C. State a reason why the actual number of plates is smaller than the number you obtained. (2) An IP address is a numeric code which is assigned to every device (computer, phone, router, smart fridge, car, etc.) connected to the Internet (IP stands for Internet Protocol). Until a few years ago, the standard was Internet Protocol 4 (IPv4), introduced in 1982, with addresses of the form DDD.DDD.DDD.DDD where D represents a single digit (with leading zeros removed). For instance, the author of this text was writing it in a coﬀee shop, and his laptop was assigned the address

In theory, what is the maximum number of IPv4 addresses? In practice, Internet run out of IPv4 addresses, so since 2010s a new addressing mechanism, called IPv6, has been introduced. The format is HHHH:HHHH:HHHH:HHHH:HHHH:HHHH:HHHH:HHHH (8 groups of 4 characters) where H could be a digit 0-9 or a letter a, b, c, d, e or f. The adoption is still in progress, and it will take time until all devices switch to IPv6 addressing. Checking for IPv6 address for his laptop connected through the caﬀe’s wiﬁ, the author reached to the site https://test-ipv6.com/ and obtained the following reply Section 3 Numbers: Quantitative in Quantitative Reasoning 59

Anyway, here is an example (the address is not case sensitive): FE80:0A90:100B:2300:0202:B3FF:FB1E:8146 In theory, what is the maximum number of IPv6 addresses? (3) Postal Codes in Canada consis of 6 alphanumeric characters, written in the form letter-digit-letter space digit-letter-digit. L8S 4K1

forward sortation local delivery area unit

The leftmost letter is the postal district, which, outside of Ontario and Quebec, represents an entire province of territory (A for Newfoundland and Labrador, B for Nova Scotia, etc.) Two largest metropolitan areas have their own letter (M for Toronto (GTA) and H for the Montreal region). The digit in the second place is used to indicate the type of habitation: 0 is for rural, and 1-9 designate urban regions. The local delivery unit “speciﬁc single address or range of addresses, which can correspond to an entire small town, a signiﬁcant part of a medium-sized town, a single side of a city block in larger cities, a single large building or a portion of a very large one, a single (large) institution such as a university or a hospital, or a business that receives large volumes of mail on a regular basis” (Wikipedia). Postal codes do not include the letters D, F, I, O, Q or U. What is the theoretical maximum of postal codes in Toronto? In Alberta? (The actual maximum is smaller, since Canada Post reserves some codes for special purposes, including H0H 0H0 for Santa Claus).

Note: To think of really really large numbers, consider the number of possible chess games. It is known that after three moves of both players, there are about 121 million possible board setups/games. In the article FYI: How Many Different Ways Can a Chess Game Unfold? (Popular Science, https://bit.ly/2JVWHYM) we read “According to Jonathan Schaeffer, a computer scientist at the University of Alberta who demonstrates A.I. using games, ‘The possible number of chess games is so huge that no one will invest the effort to calculate the exact number.’ Some have estimated it at around 10100,000. Out of those, 10120 games are typical: about 40 moves long with an average of 30 choices per move.” Compared to chess, the standard (3×3×3) Rubik cube has a much smaller number of configurations: 43,252,003,274,489,856,000, i.e., about 4.32 · 1019. If interested, visit https://www.therubikzone.com/number-of-combinations/. 60 NUMERACY

Absolute and Relative Numbers

Numeric information can be presented in absolute form, i.e., as a number, with no reference to anything else (any other number) (such as “I bought bananas and paid $ 1.38”). When some kind of reference is included, we call such numeric information relative (as in “I bought 2 pounds of bananas and paid $ 1.38”). So, price on its own is an absolute number, whereas unit price ($ 1.38/2=$0.69 per pound) is a relative number.

Example 3.21 Absolute and Relative Numbers Canadian Cancer Society publishes breast cancer statistics (incidence, i.e., the number of new cases, and mortality rates) on it web page https://bit.ly/1rSTusm

Classify each number in the table as providing either relative or absolute information.

On the Government of Canada information page about Invasive meningococ- cal disease (IMD) https://bit.ly/2T8xEb8 we read: “Between 2006 and 2011, an average of 196 cases of IMD was reported annually in Canada, with an average incidence of 0.58 cases per 100,000 population.” The phrase “an average of 196 cases” gives an absolute number, whereas “average incidence of 0.58 cases per 100,000 population” is a relative information.

Examples 3.22 Absolute and Relative (1) You are given a job offer in Charlottetown, P.E.I., and the letter of offer states that your starting salary will be $ 45, 000. You are not happy with this information (why?) What kind of information would you prefer to have? (2) Starbucks’ Grande (16 fl oz ≈ 473 millilitres, which is about 473 grams) Blonde roast has 360 mg of caffeine. (Source: Caffewineinformer, https://bit.ly/2q9P1I8.) The calculation 0.360/473 = 0.00076 shows that about 0.076 % of the coffee is actually caffeine. Identify absolute and relative information, and comment on their usefulness. (3) Looking at online proces at Loblaws (https://bit.ly/313FsKZ) we find: Presi- dent’s Choice Rosemary (40 g) for $ 2.99 Brussel Sprouts $ 0.44 / 50 g Section 3 Numbers: Quantitative in Quantitative Reasoning 61

King Oyster Mushrooms $ 0.72/25 g Radishes (454 g) $ 1.99 Minced Garlic Jar (128 g) $ 2.99 Which of these items is the cheapest, and which is the most expensive?

Assume that a quantity changes from A to B. How can we describe by how much it changed? We can calculate the absolute change B−A (i.e., new value minus old value), or relate the absolute change to the starting value A to compute the relative change B − A A Consider the following situation: on island X there are 40 monkeys, and on island Y there are 400 monkeys. A year later, the counts were 50 monkeys on island X and 410 monkeys on island Y. Note that the absolute changes on both islands are the same: on island X: 50 − 40 = 10, and on island Y : 410 − 400 = 10. However, we feel that these numbers do not tell the whole story, so we put them into context: the relative change on island X is 50 − 40 10 = =0.25 = 25% 40 40 and on island Y it is 410 − 400 10 = =0.025 = 2.5% 400 400 Thus, island X experienced a much dramatic change in its population of monkeys. (In this case, this is obvious if we just look at the numbers – but things can be more complicated.) Notes: Information given as percent (percentage) is relative information. When- ever we see precent, our ﬁrst question must be–percent of what? In the above case, it is 25 % of the initial population (40) of monkeys on island X and 2.5 % of the initial population (400) of monkeys on island Y . A fraction (representing division) can be viewed as percent, and is thus relative as well: A If = p, then A is p percent of B B For instance, from 1/4=0.25 we conclude that 1 is 25 percent of 4. Or, we can interpret 40/400 = 0.1 by saying that the initial population of monkeys on island X was equal to 10 percent of the initial population of monkeys on island Y.

Sensitivity

We have seen that rounding numbers off does not change the value of a number by much. But what happens if we round off numbres which enter calculations? We will answer a slightly different – but related – question: can a small change in a number produce a large change in the value of a number? We use examples to investigate what happens when we change one number involved in a calculation by 5 percent. Addition: adding 458 and 5100 we obtain 458 + 5100 = 5558. If we increase 5100 by 5%, we obtain 5100 ∗ 1.05 = 5355 and this time, the sum is 458 + 5355 = 5813. The two sums are of the same order of magnitude; from 5813/5558 = 1.0458798 we conclude that the sum increased about 4.59%. 62 NUMERACY

Multiplication: pick any two numbers x and y, and multiply them to get xy. When we increase x by 5%, we get 1.05x, and this time, the product is 1.05xy. Compare: 1.05xy/xy =1.05, i.e., the product increased by 5% as well. Division: When we increase the numerator in x/y by 5%, we get 1.05x/y. Dividing the two numbers, we get 1.05x/y 1.05x y = · =1.05 x/y y x Thus, the quotient mimics the change in the numerator. When we increase the denominator in x/y by 5%, we get x/(1.05y). Dividing the two numbers, we get x/(1.05y) x y 1 = · = =0.95239 x/y 1.05y x 1.05 Thus, the value of the fraction decreases by about (1 − 0.95239 = 0.04761) 4.76%. Powers: consider 108. If we increase the base by 5%, we obtain 10.58. From 10.58 =1.47745 108 wse conclude that the increase is signiﬁcant - about 47.75%. If we increase the exponent by 5%, we obtain 108.4. From 108.4 =1.47745 108 we conclude that the increase is very large - about 151.19%.

To summarize: a 5% change in a number produces teh same, or about teh same change in the result of addition, multiplication and division. However, when the powers are concerned, the changes are much larger.

Visualizing Numbers

We can often visualize, and develop good intuition, about numbers which are on the ‘human scale’ (what exactly that means is not clear, and is not possible to specify). When we see a group of people we can immediately tell whether there are 30 or 100 people in it. Driving a car, we can feel the diﬀerence between moving at 20 km/h or at 70 km/h. We can tell when we carry 2 kg of bananas, compared to 10 kg of bananas. But what about small and large numbers? Let’s look at a few examples.

Example 3.23 Visualizing Large and Small Numbers. (1) On 2 July 2016, The Guardian (https://bit.ly/2hGTueV) published the article We dont know if your babys a boy or a girl: growing up intersex about a child named Jack who was diagnosed with mixed gonadal dysgenesis. As a consequence, Jack’s sex couldnt be determined at birth, and doctors needed time to assign it. We read: “Jacks specific diagnosis is rare, but being born with a blend of female and male characteristics is surprisingly common: worldwide, up to 1.7% of people have intersex traits, roughly the same proportion of the population who have red hair, according to the Office of the United Nations High Commissioner for Human Rights. ” (2) In the piece The North and the great Canadian lie published in Macleans (11 Sept 2016, at https://bit.ly/2OpKYWH) the author, attempting to describe the size of the Canadian population living in the North, writes: “How many Canadians actually live up north? Approximately 118,000. Thats one-third of one per cent of the national population. To put it another way, about as many Canadians live in Australia as live in Nunavut. If the entire population of the Northwest Territories Section 3 Numbers: Quantitative in Quantitative Reasoning 63 decided to attend an Edmonton Eskimos game, Commonwealth Stadium would still have 10,000 empty seats.” (3) This clock, taken from Wikipedia (https://bit.ly/1VkJrbO) shows geological development of our Earth classified by eons and eras, from its creation (about 4.6 billion years ago) until today. In the picture, Ga (giga) = billion yrs ago and Ma (mega) = million yrs ago.

(4) On 12 Dec 2016, CBC News reported on a large amount of raw sewage that flows into Canadian rivers, lakes and oceans. The article, titled Billions of litres of raw sewage, untreated waste water pouring into Canadian waterways (https://bit.ly/2YsjEXN) states that “[...] the amount of untreated waste water, which includes raw sewage and rain and snow runoff, that flowed into Canadian rivers and oceans last year would fill 82,255 Olympic-size swimming pools an increase of 1.9 per cent over 2014.” (5) The chance of dying in a plane crash has ben estimated to about 1 in 7 million, i.e., 0.00000014 (see [1], page 117). This means that if a person picks one (random) flight every day it would be 19,000 years before they could expect to die in a plane crash. (6) In Humans Have Bogged Down the Earth with 30 Trillion Metric Tons of Stuff, Study Finds (Smithsonian.com, 9 December 2016, https://bit.ly/2ZgULj5)weread “Humans have produced a lot of stuff since the mid-20th century. From America’s interstate highway system to worldwide suburbanization to our mountains of trash and debris, we have made a physical mark on the Earth that is sure to last for eons. Now a new study seeks to sum up the global totality of this prodigious human output, from skyscrapers to computers to used tissues. That number, the 64 NUMERACY

researchers estimate, is around 30 trillion metric tons, or 5 million times the mass of the Great Pyramid of Giza. And you thought you owned a lot of crap.”

Examples 3.24 Scaling Scaling is a powerful way of visualizing data. Here we explore several examples. (1) You are making a scale model of our Solar System and placed the Sun into Hamilton Hall and Earth into MUSC (assume that the distance between Hamilton Hall and MUSC is 150 metres). How far from Hamilton Hall would you have to place our (newly declared) dwarf planet Pluto?

????

(2) In [2] we find estimates on the number of species (described and undescribed) on Earth, by group: chordates (80,500) plants (390,800) fungi (1,500,000) inverte- brates (6,755,830). In a 50-minute lecture, you need to talk about the four groups in such a way that the time that you lecture about a group is proportional to the size of that group. For each group, compute how long (in minutes and seconds) you will be lecturing about it. (3) There are 640 students registered in this course. Assuming that the students in the course have been selected to reflect the demographics of the entire human Section 3 Numbers: Quantitative in Quantitative Reasoning 65 population, how many Canadians would be in this class? How many Indians (citizens of India)? (4) You decide to paint the floor in your room (of area 15 m2) in a combination of blue and brown, where blue represents oceans and brown represents land on Earth. If you were to paint it to reflect the actual ratio of oceans to land what area of your floor would be painted brown?

Notes and Comments: (1) People usually have more problems with small numbers than with large numbers. One reason lies in the way we write numbers: for large numbers, we use comma to separate digits into groups of three, but there is no such separation for small numbers. Consequently, we can easily see read 15, 000, 000 as 15 million, but need to think when reading 0.000015 (so we count zeroes in 0.000001 to see that it is one millionth, and so 0.000015 is 15 millionths). Another difficulty stems from the fact that small numbers suggest division (as in 0.000015 = 15/1, 000, 000) and, of all algebraic operations, division is the least intuitive. (2) Small quantities, by repetition/multiplication, can generate large quantities, and thus have significant effects: think of the pollution coming from one car, and then multiply by one billion to estimate the pollution coming from an estimated 1 billion cars on our planet. One car does not cause a traffic jam, but 10 thousand cars do. One student refusing to use a plastic fork at MUSC for one lunch they bought does virtually nothing to reduce plastic waste. However, all Mac students routinely refusing to use plastic cutlery (and, for instance, bringing their own cutlery) would make a difference. (3) Recall that the multiplication principle can produce huge numbers by combining quantities which are fairly small. Two more examples: there are 2,235,197,406,895,366,368,301,560,000 = 2.235· 1027 ways to divide a strandard deck of 52 cards among four players. there are 120 permutations (orderings) of five objects (if we label them by A, B, C, D, and E, then ABCDE is a permutation, and so are ACBDE, DECAB, CEABD, and so on). There are over 3.6 million permutations which involve 10 objects, about 2.4 · 1018 permutations of 20 objects, about 3.0 · 1064 permutations of 50 objects, and about 9.3 · 10157 permutations of 100 objects (for comparison, there are about 3.28 · 1080 particles (electrons, quarks, etc.) in the entire Universe). (4) Sometimes, small cannot affect big, but does affect another small quantity: if 160 thousand people emigrate from China, it would not affect Chinese population in any major way. However, the same number of emigants would halve the population of Iceland.

Chapter references [1] Niederman D. and Boyum, D. (2003). What the Numbers Say. New York: Broadway Books. [2] Chapman, A. D. (2009). Numbers of Living Species in Australia and the World (2nd ed.). Canberra: Australian Biological Resources Study. pp. 180. 66 NUMERACY

4 Counting and Number Systems

We are so used to our decimal system that we tend to forget that the way we count and write numbers are conventions, based on historic choices. In this chapter we unpack the decimal system, and then discuss other number systems, some of which are used today in major ways (deep down, computers do not work with the decimal system; computer software, Internet, security, et.c all depend on binary and hexadecimal systems (more about these soon). Why does an hour have 60 minutes, and not 100 minutes? Why is the right angle equal to 90 degrees, and not to some power of 10, such as 10 or 100? Why dozen eggs, and not 10 or 15? The decimal system is a positional number system based on powers of 10. Positional means that the location of a digit within a number determines its value: for instance, the ﬁrst (leftmost) occurrence of 7 in 17074 contributes 7000 to the value of the number, whereas the second 7 contributes 70 to its value. Roman numeral system is non-positional: a digit is worth the same, no matter where it is located within a number. Recall that M represents 1000, D is 500, C is 100, L is 50, X is 10, V is 5 and I is 1. For example, MMCCCXXXVIII represents the number 2000 + 300 + 30 + 5 + 3 = 2338. The digits are written from the largest value to the smallest value. There is one exception to this rule: if a smaller value is to the left of a larger value, then it is subtracted: IX is 10 − 1=9, IV is 5 − 1 = 4 (exception to the rule that the same symbol cannot appear more than three times in a group, 4 is sometimes written as IIII), and CM is 1000 − 100 = 900. Historically, there have been many non-positional systems (for instance used in ancient Greece and Egypt), as well as positional systems (Maya in Central America, India, etc.). Non-positional systems become inconvenient when one tries to do algebra (imagine multiplying CCIX by LXXXV); however, the are still used, mostly as labels:

Decimal Number System

To understand how this system works, we deconstruct a number: 38, 912 = 30, 000 + 8, 000 + 900 + 10 + 2 =3· 10, 000 + 8 · 1, 000 + 9 · 100 + 1 · 10 + 2 · 1 =3· 104 +8· 103 +9· 102 +1· 101 +2· 100 Section 4 Counting and Number Systems 67

Because of the last line, we say that the decimal system is based on the powers of 10; and the numbers multiplying these powers are the digits of the original number. (Keep in mind that 100 =1and101 =10.) One more example: 509 = 500 + 9 = 5 · 100 + 0 · 10 + 9 · 1=5· 102 +0· 101 +9· 100 Note that zero does not have a value, however, it is important that it is there as it serves as a placeholder–without it, we would have a diﬀerent number, 59. Thus, i order to write a number in decimal system, we use ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. All numbers can be written based on these digits, and with the positioning based on the powers of 10. A decimal number can be analyzed in the same way: 7.830056 = 7 + 0.8+0.03 + 0.00005 + 0.000006 =7· 1+8· 0.1+3· 0.01 + 5 · 0.00001 + 6 · 0.000001 =7· 100 +8· 10−1 +3· 10−2 +0· 10−3 +0· 10−4 +5· 10−5 +6· 10−6 To count in the decimal system, we go: 0, 1, 2, 3,...,8, 9; we reached ten, for which we do not have a single digit, but instead we introduce a new position to the left (tens): 10, 11, 12,...,18, 19; we do not have a new symbol for twenty; instead, we increase the tens digit: 20, 21,...,97, 98, 99 (which is the largest two- digit number); to write one hundred, we introduce a new position to the left (hundreds): 100, 101, 102, and so on. The decimal sysytem is also called base 10 system and when we need to distinguish between diﬀerent number bases, we will write (38, 912)10. However, we will stick to aconvention that a number written without indication of its base isabase10number.

Number Systems

What does numer in base 6 look like? (Base 6 counting has been used by Indige- nous peoples in parts of Canada, as well as in Papua New Guinea.) Base ten numbers need 10 digits–base six numbers need only six: 0, 1, 2, 3, 4, and 5. Thus, 3402 is a valid base 6 number, and we will write (3402)6; however, 3476 is not a valid base 6 number (as it contains 7 and 6 which are not base 6 digits). The value (i.e., the decimal equivalent) of a base 6 number is determied analogously to the base 10 , but this time, it’s the powers of 6: 3 2 1 0 (2315)6 =2· 6 +3· 6 +1· 6 +5· 6

=2· 216 + 3 · 36 + 1 · 6+5· 1 = 551 = (551)10

Here is a convenient way to do this: to convert (235041)6 into a decimal number, write the powers of six above its digits, starting from zero, and going from right to left:

Now write the number using the powers of 6: 5 4 3 2 1 0 (235041)6 =2· 6 +3· 6 +5· 6 +0· 6 +4· 6 +1· 6 =2· 7776 + 3 · 1296 + 5 · 216 + 0 · 36 + 4 · 6+1· 1=

=20, 545 = (20, 545)10 How do we count in base 6?

We start as usual: (0)6, (1)6, (2)6, (3)6, (4)6, (5)6; the next number has a decimal value of 6, but in base 6, (6)10 = (10)6. (Note: in the decimal system, when we cannot go any further with single digits, we introduce a new place, i.e., 68 NUMERACY

we go from 9 to 10; in base 6, the last single digit is (5)6 and the ﬁrst next digit is (10)6 –thesamething!)

So, we count: (0)6, (1)6, (2)6, (3)6, (4)6, (5)6, (10)6, (11)6, (12)6, (13)6, (14)6, (15)6; (now think about what happens when we reach 19 in the decimal system) (20)6, (21)6, (22)6, (23)6, (24)6, (25)6; then (30)6, (31)6, and so on, until we reach the largest two-digit number (it’s 99 in the decimal system, so it must be) (55)6. The next number, and the smallest three-digit number on base 6 is (100)6. The decimal value of this number is 2 1 0 (100)6 =1· 6 +04· 6 +0· 6 =36. Continue counting:

(100)6, (101)6, (102)6,..., (105)6, (110)6, (111)6, (112)6,..., (115)6,

(120)6, (121)6, (122)6,..., (155)6,

(200)6, (201)6, (202)6,..., (205)6, (210)6,..., (555)6,

(1000)6, (1001)6, and so on.

How do we do algebra in base 6? We thinkin bas 10, but write in base 6. In this example we add (42351)6 and (3041)6. Write the numbers one below the other

Add the rightmost digits: 1 + 1 = 2, and write 2 as the rightmost digit of the sum:

Next, 5 + 4 = 9, but we need to write 9 is base 6: (9)10 = (13)6. (Recall how we counted in base 6.) So in the second (from right) position we enter 3 and carry over 1:

Now 1 + 3 + 0 = 4 (decimal value) which is also 4 in base 6. So the third digit is 4.

Adding the remaining digits, we obtain the answer:

So, after all, this is not so much diﬀerent from the decimal system. Section 4 Counting and Number Systems 69

Exercises 4.1 Practice in Base 6

(1) Convince yourself that (100)6 = (36)10, (1000)6 = (216)10, and (10000)6 = (1296)10. What is the decimal value of (100 ... 00)6 if n is a positive number? n

(2) Convert each number to ﬁnd its decimal value: (204)6, (55)6, (1320)6, (555)6, and (10101)6. (3) Convince yourself that the following calculation (addition) is correct:

(4) Add the following numbers, and write their sum as a base 6 number:

(204)6 + (1231)6

(3333)6 + (4444)6

(2)6 + (55555)6

(10352)6 + (43205)6 (5) What is the decimal equivalent of the largest 6-digit number in base 6?

Partial answers: (1) Expand 4 4 (10000)6 =1· 6 + (all other terms are zero) = 6 = (1296)10. 4 3 2 1 0 (2) (10101)6 =1· 6 +0· 6 +1· 6 +0· 6 +1· 6 = 1296 + 36 + 1 = (1333)10

(4) (3333)6 + (4444)6 = (12221)6

(2)6 + (55555)6 = (100001)6

(5) The largest 6-digit number in base 6 is (555555)6. If we continue counting, the 6 6 next number is (1000000)6 =(6 )10 (see (1)). Thus, (555555)6 =6 − 1=46, 655 in the decimal system.

Working with any other number system is analogous to what we have just done. Base 20, or vigesimal number system is based on twenty. (Why would anyone pick twenty? One explanation is that it is the number of human ﬁngers and toes.) This system was used in many countries, and by many cultures: Maya and Aztec in Central and South America, Inuit in Canada and Greenland, and in Africa (see Wikipedia, https://bit.ly/2LQdezA, for a complex system of Yoruba numerals). Base 20 names for numbers appear French, Welsh, Danish, Albanian, and a few other languages. If the base 10 system has 10 digits and the base 6 system has 6 digits, then the base 20 system must have 20 digits, representing decimal values 0, 1, 2, 3, ..., 18, and 19. Inuit peoples have used Kaktovik Inupiaq numerals

1152193 4 5 6 7 8 9 10 11 12 13 14 1617 18 0

Maya peoples used the digits below (left), with a stylized shell as a symbol for zero (for details, see Wikipedia under Vigesimal). 70 NUMERACY

On the right is a stela (vertical stone pillar), on display at the Royal Ontario Museum in Toronto, that was used to record a date (and hence the numbers come with pictorial representations of units used in Maya calendars). To work with base 20 numbers, we will use decimal numbers 0, 1, 2, 3, ...,18, 19, and separate them by a semicolon, as in (9; 14; 0; 18)20. Its decimal equivalent is calculated analogously to what we’ve done before–write the powers above the digits, from right to left:

and then keep in mind that the base is 20: 3 2 1 0 (9; 14; 0; 18)20 =9· 20 +14· 20 +0· 20 +18· 20

=9· 8000 + 14 · 400 + 0 · 20 + 18 · 1 = (77618)10

How did Maya people count in their system? Start with single-digit numbers:

(0)20, (1)20, (2)20,..., (18)20, (19)20;

The next number is 20 in decimal notation; in base 20, it is (1; 0)20. We continue

(1; 0)20, (1; 1)20, (1; 2)20,..., (1; 18)20, (1; 19)20,

(2; 0)20, (2; 1)20, (2; 2)20,..., (2; 18)20, (2; 19)20; The last set of two-digit numbers is

(19; 0)20, (19; 1)20, (19; 2)20,..., (19; 18)20, (19; 19)20; To write the next number we need to introduce a new place (value)

(1;0;0)20; 1 2 By the way, (19; 19)20 =19· 20 + 19 = 399, and (1; 0; 0)20 =1· 20 = 400. Makes sense! A new place value always starts with a new power of the base. Thus, 3 4 (1;0;0;0)20 =1· 20 = 8000, and (1; 0; 0; 0; 0)20 =1· 20 = 160000, and so on. To resume counting:

(1;0;0)20; (1; 0; 1)20, (1;0;2)20,..., (1;0;18)20, (1;0;19)20,

(1;2;0)20, (1;2;1)20, (1;2;2)20,..., (1;2;18)20, (1;2;19)20, and so on ... the largest three-digit number is

(19; 19; 19)20 whose decimal value is 7,999. We know that because it is 1 smaller than the next 3 number(1;0;0;0)20 =1· 20 = 8000. Then, it is

(1;0;0;0)20, (1;0;0;1)20, (1;0;0;2)20, and so on.

Here is an example of algebraic manipulation is base 20: using our usual algorithm, we add (17; 11; 0; 18)20 and (19; 9; 9; 15)20. Section 4 Counting and Number Systems 71

Exercises 4.2 Practice in Base 20 (1) What is the decimal value of (1;0; 0; ...0; 0)20 if n is a positive number? n (2) Convert each number from base 20 into a decimal number:

(3; 19)20

(19; 19; 19; 19)20

(14; 0; 0; 2)20

(10; 10; 10)20 (3) Add the following numbers, and write their sum as a base 20 number:

(4;5;6)20 + (13; 12; 11; 10)20

(18; 0; 19)20 + (17; 0; 16)20

(19; 19; 19)20 + (1; 1; 1)20

(14; 15)20 + (18; 19; 9; 0)20

Partial answers: (1) As usual, above the digits of the given number, write the numbers 0, 1, 2, 3, ..., n from right to left. The only non-zero term is 1 · 20n =20n. Thus, for instance, 3 (1;0;0;0)20 =(20 )10 =(8, 000)10 5 (1;0;0;0;0;0)20 =(20 )10 =(3, 200, 000)10. 3 0 (2) (14; 0; 0; 2)20 =14· 20 +2· 20 =14· 8000 + 21 = 112, 002

(3) (18; 0; 19)20 + (17; 0; 16)20 = (1; 15; 1; 15)20

(14; 15)20 + (18; 19; 9; 0)20 = (19; 0; 3; 15)20

Sexagesimal number system is based on the number 60. Its roots can be traced to ancient Sumerians (3rd millennium BCE), so it is a bit over 5,000 years old. Base 60 looks familiar? Ancient Egypteans divided a day into 24 hours. Baby- lonians, who inherited the system from Sumerans, took it from there and divided one hour into 60 minutes, and one minute into 60 seconds. As well, it is believed that Babylonians divided a full circle into 360 degrees, one reason being that 360 is divisible by 2, 3, 4, 5, 6, 8, 9, 10, and by many more numbers (10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 and 360), and thus it was easy to mentally divide a circle into pieces, i.e., mentally calculate an angle. Geographic coordinates (longitude and latitude) are also given in angle degrees; for instance, McMaster University is located at the latitude of 43.2609 degrees N (North of Equator) and the longitude of 79.9192 degrees W (West of Greenwich, which is deﬁned to be zero degrees). In base 60, we need sixty digits, representing decimal numbers from 0 to 59. Babylonians used the following digits; they did not have a sybmol for zero, and just used blank space (source: Wikipedia article Sexagesimal): 72 NUMERACY

Exercises 4.3 Practice in Sexagesimal System

(1) Count forward from (45; 59; 57)60, by listing 5 numbers that come next.

(2) Count backward from (1; ; ; 2)60, by listing 5 numbers that come before it. (Recall that space represents zero.) (3) Convert each number into its decimal equivalent. (Recall that space represents zero.)

(1; )60

(1; ; )60

(10; 30; 50)60

(1;1;1;1)60 (4) What is the decimal equivalent of the largest two-digit base 60 number?

(5) Find (45; 51; 12)60 + (59; 38; 41; 36)60 Partial answers:

(2)(1;;;2)60, (1;;;1)60, (1;;;)60, (59; 59; 59)60, (59; 59; 58)60, (59; 59; 57)60.

(4) The largest two digit number is (59; 59)60, which is one smaller than 2 (1;0;0)60 =(60 )10 = (3600)10.

Thus, (59; 59)60 = (3599)10.

Hexadecimal and Binary Systems

The hexadecimal system is based on the number sixteen. The digits consist of theusualdecimaldigits0,1,2,3,4,5,6,7,8,9,towhichweaddlettersa(=10),b (=11), c (=12), d (=13), e (=14), and f (=15); sometimes, uppercase letters A, B, C, D, E, and F are used instead. Because we have all digits as single symbols, we can write numbers in the hexadecimal system as we write numbers in the decimal system, one digit next to another (with no separation symbols needed). Thus,

(12)16 =1· 16 + 2 = (18)10 2 (209)16 =2· 16 + 9 = (265)10

(5a)16 =5· 16 + 10 = (90)10 2 (100)16 =16 = (256)10 2 (ef7)16 =14· 16 +15· 16 + 7 = (3831)10 Section 4 Counting and Number Systems 73 and so on. Hexadecimal numbers are widely used in anything related to computers and electronic communication. For instance, MAC (media access control) address is a uniquecodeassignedtoadevicewhichisconnected to the Internet. It consists of 6 groups of two-digit hexadecimal numbers:

Internet protocol IPv6 is a communications protocol that locates and identifies a device on the Internet, so that it can route traffic to and from it. In Network System Preferences of a device we can find IPv6 codes in hexadecimal form

To access certain sites (banking, utilities, Amazon, etc.) we need to enter a password. How does it work? When we register and/or enter a new password, it is converted into a very long hexadecimal number (that number is called hash; it could have 64, or 128 digits, or more). That hash is then saved (our actual password is not stored anywhere). The Python code below shows how passwords ‘a’ and ‘A’ are stored in the memory of the page we are trying to access. Whenever we enter a password, its hash is calculated and compared to the stored hexadecimal hash.

Note that the passwords thus created are case-sensitive: ‘a’ and ‘A’ have two completely diﬀerent hashes.

A screen on any electronic device we use (phone, tablet, laptop, computer, etc.) is divided into pixels. To display a page, a device’s software knows where each pixel is located, and what colour to assign to it. There are several conventions which are used to define colour for screens of electronic devices. One of the most commonly used system is the RGB (red, green, blue) system. In this system, a colour is viewed as an ordered triple of numbers (r, g, b) where the numbers (defined to be between 0 and 255) indicate the intensity of red, green (lime), and blue, respectively. The combination of these intensities gives the colour of each pixel. For instance, (255, 0, 0) is maximum intensity red, no green, and no blue, so it represents pure red colour. Often, the triple is written as a string of six characters RRGGBB, where RR, GG and BB are hexadecimal equivalents of intensity. Thus, instead of (255, 0, 0) we use ff0000 (recall that (ff)16 = 255 in the decimal system). Likewise, 00ff00, or (0, 255, 0) represent green (lime) colour, and 0000ff or (0, 0, 255) 74 NUMERACY

is blue. The picture below shows the three basic colours, as well as three colours derived from the basic red, green and blue colours.

Because colours are deﬁed as numbers, we can do various algebraic operations and/or apply algorithms (and this is exactly what is going on in Photoshop, when we modify a photo, for instance by making the sky more blue than it actually was).

The binary system is based on the number two – so it uses only two digits, 0 and 1 (it does not get any simpler than that!). For instance, 5 4 3 2 1 0 (101011)2 =1· 2 +0· 2 +1· 2 +0· 2 +1· 2 +1· 2

=32+8+2+1=(43)10 Let’s count (and keep in mind that every time we encounter a power of two, we introduce a new place value):

(0)2, (1)2,

(10)2 =(2)10, (11)2,

(100)2 =(4)10, (101)2, (110)2, (111)2,

(1000)2 =(8)10, (1001)2, (1010)2, (1011)2, (1100)2, (1101)2, (1110)2, (1111)2,

(10000)2 = (16)10, (10001)2, and so on. Algebraic operations with numbers in the binary system are simple, and there is very little we need to memorize. In particular, here is the entire multiplication table: (0)2 · (0)2 =(0)2, (0)2 · (1)2 =(0)2, (1)2 · (0)2 =(0)2, and (1)2 · (1)2 =(1)2. The binary system is so simple that it is believed that if there is life somewhere else un Universe, very likely they will be able to ﬁgure out the binary system. Section 4 Counting and Number Systems 75

In 1972 and 1973, two spacecrafts (callen Pioneer 10 and 11) were launched into space (the photograph and the drawing above show Pioneer 11; source: NASA Archive The Pioneer Missions https://go.nasa.gov/2mYEzDw). By 1983, Pioneer 10 left our Solar system, and it sent its last signal to Earth in 1995. In January 2019, it was calculated to be about 16 billion kilometres from Earth. Pioneer 11 was lost (i.e., stopped communicating) in 1995 as well. Both spacecrafts carried identical plaques, in the case that they were found by extraterrestrial life. The plaques contained vital information about humans: what we look like (with the outline of the spacecraft in the background, to show how tall we are); where we live (the star-like diagram on the left shows the location of our Solar system with respect to nearby pulsars, and the distances to them); on the top is a representation of the hydrogen atom, and on the bottom is our Solar system.

All numeric information is given in the binary system! A vertical stroke (|)represents 1, and a horizontal stroke (−)represents0.

The distances between planets and the Sun are given with reference to the distance from Sun to Mercury, which is given as |−|−= (1010)2, whose decimal equivalent is 10. The distance from Earth to the Sun is given as || − |− = (11010)2, or 26 (reﬂecting the fact that Earth is 2.6 times farther from the Sun than Mer- cury). Jupyter is roughly 13.4 times farther from the Sun than Mercury, so the 8 2 information is encoded as |−−−−||−= (10000110)2 =2 +2 + 2 = (134)10. Notes: (1) As quantum computing develops, the classical binary (positive charge- negative charge, that we denote by 0 and 1) is going to be replaced by a qubit, the smallest quantum information unit, which stil has two states–thus the binary system will remain highly relevant. (2) In the conference paper on computer architecture A DNA-Based Archival Stor- age System https://bit.ly/2LSJeTW, the authors from University of Washington and Microsoft discuss DNA to store (archive) information. We read that such storage “[...] is an attractive possibility because it is extremely dense, with a raw limit of 1 exabyte/mm3 (109 GB/mm3), and long-lasting, with observed half-life of over 500 years.” What is interesting is that this technology requires that either base 3 or base 4 numbers are used (these options are discussed in the paper.) 76 NUMERACY

(3) For those interested in the history of math: Plimpton 322 is a Babylonian clay tablet, famous for its content: it shows ratios of sides in triangles of various sizes (recall that that’s exactly what trigonometric functions are). The tablet is an evidence that trig functions are at least a thousand years older than what historians believed. (Source: AncientPages.com Mysterious Clay Tablet Reveals Babylonians Used Trigonometry 1,000 Years Before Pythagoras https://bit.ly/2MmbIoy.)

Mayan Calendar

Maya civilisation, developed by the Maya peoples in the regions of Central Amer- ica, was a highly advanced civilisation, which made major advances in architecture, astronomy, mathematics, and art. Maya people inherited, and further developed, a variety of methods of time keeping. They simultaneously used three calendars, depending on the purpose: The Haab counted days and months, and was used for everyday events, and shorter time-scale events. The Tzolkin consisted of 13 months with 20 days each, and was used to determine the time for various ceremonies and religious events. The Long Count calendar was used to keep track of historic and long-term events. The Haab divided a year into 18 months of 20 days each, and then added 5 days to the end, thus making a year 365 days long. As neither Haab nor Tzolkin kept track of the year, how did people ﬁgure out the year? They did it by an ingenious combination of the two calendars. Place each calendar on a wheel, as shown, and align them so that the wheels touch at the day labelled 1 on both wheels.

259 364 260 365 1 1 Tzolkin Haab 2 2 3 3 4 4

This position marks the date Tzolkin 1, Haab 1. The following day both wheels advance by 1, and the date is Tzolkin 2, Haab 2 Then Tzolkin 3, Haab 3 Tzolkin 4, Haab 4 Tzolkin 5, Haab 5 Section 4 Counting and Number Systems 77 and so on, until Tzolkin 259, Haab 259 Tzolkin 260, Haab 260

258 258 259 259 Tzolkin260 260 Haab 1 261 2 262 3 263

The Tzolkin wheel completed one full revolution, so the following day, it is reset; the Haab continues: Tzolkin 1, Haab 261 Tzolkin 2, Haab 262 and so on. Skipping 101 days, Tzolkin 103, Haab 363 Tzolkin 104, Haab 364 Tzolkin 105, Haab 365 Now the Haab wheel resets: Tzolkin 106, Haab 1 Tzolkin 107, Haab 2 Tzolkin 108, Haab 3 and so on. Obviously, the two calendars are out of phase, and that’s exactly what is desired – that way, we keep getting unique combinations of the two numbers. However, it cannot go on forever - when will the two wheels realign again, i.e., reach their initial position Tzolkin 1, Haab 1? We use a small example to ﬁgure it out: on a piece of paper, draw two calendar wheels, one with 4 and the other with 6 days. How many unique combinations will you get? I.e., if you start with 1,1 (as Mayan calendars) after how many days will the wheels get back to 1,1? (Answer: the least common multiple.) The least common multiple of 260 and 365 is 18,980, which is exactly 18.980/365 = 52 Haab years. This is called a calendar round. In other words, there are 18,980 unique combinations to label the days, and the two calendars synch again after 52 (365-day long) years. Thus, two days with identical Tzolkin and Haab numbers must 52 be years apart, or 104 years apart, or some other multiple of 52 years apart. If two people have a birthday on the same day (say, Tzolkin 148, Haab 279), the are either of the same age, or one is 52 (or 104) years older. The end of one calendar round (the 52-year cycle) and the start of another was a huge event, accompanied by ceremonies, including human sacriﬁces.

The Long Count was an astronomical calendar, used to record historic events, and to track long time intervals. The reason for the word ‘count’ is because the calendar actually counted days, starting with the day Maya astronomers believed the Universe (in its present form) was created. Historians determined that, most likely, that date was Monday, 11 August 3114 BCE (according to the Gregorian calendar, adjusted to reﬂect dates before it was introduced in 1582). 78 NUMERACY

The counting system used in the Long Count calendar was a modiﬁed base 20 system, with (of course) a day as a basic unit. The longer units are: 1Uinal=20days 1 Tun = 18 Uinals = 360 days 1 Katun = 20 Tuns = 7,200 days 1 Baktun = 20 Katuns = 144,000 days (about 394 (365-day long) years) 13 baktuns (1,872,000 days = 5,125.36 years) form a Great Cycle. Mayan people believed that at the end of each great cycle the existing Universe is destroyed, and a new one begins its life. So how did the count work? A number was assigned to each of the ﬁve units:

The day when the existing Universe was created, and the start of the ﬁrst baktun, was labelled as

and the following day was recorded as

It is common to write these as numbers separated by dots. Thus, the ‘Creation Date’ is the sequence of zeroes: 0.0.0.0.0 (also labelled as 13.0.0.0.0) From that day, then the long count advances: 0.0.0.0.1 0.0.0.0.2 0.0.0.0.3 ... skipping 16 days ... 0.0.0.0.19 0.0.0.1.0 (looks familiar? Addition in base 20!) 0.0.0.1.1 0.0.0.1.2 ... skipping many days ... 0.0.0.17.19 (day 360) 0.0.1.0.0 (keep in mind that the second place is like base 18, not base 20) 0.0.1.0.1 ... skipping many, many days ... 0.0.19.17.19 (day 7,200) 0.1.0.0.0 0.1.0.0.1 and so on, until: Section 4 Counting and Number Systems 79

0.19.19.17.19 (last day of the ﬁrst baktun; day 144,000) 1.0.0.0.0 (ﬁrst day of the second baktun) and it keeps going. This calendar does not care about the length of a year, since it counts days (so there is no need to make adjustments for leap years). Smithsonian National Museum of the American Indian, on its page Living Maya Time (https://s.si.edu/2n06D4N) keeps track of the long count calendar. Here is its report for 3 January 2020, showing Maya numerals as they would have been (associated with various gods)

Based on it, we compute that 3 January 2020 is the day number 13 · 144, 000 + 0 · 7, 200 + 7 · 360 + 2 · 20 + 9 · 1=1, 874, 569

Why was the word supposed to come to its end on 20 December 2012?

According to the Long Count calendar, 20 December 2012 was the last day of the 13th baktun (note the the ﬁrst baktun is labelled by 0; the second bactun is then labelled by 1, and the thirteenth battun labelled by 12). 80 NUMERACY

In other words, that was the last day of the Great Cycle! According to Maya peoples’ beliefs, the Universe as we know it should have ended. (By the way, Great cycle resets itself once every 5,125.36 years, and we witnessed it, and survived!) The Long Count recorded 20 December 2012 as

The following day was the beginning of a new Great Cycle and a new baktun, 21 December 2012

which is also recorded as 0.0.0.0.0.

Notes: (1) Variations of the calendars we mentioned are still used in Central America, for instance in parts of Guatemala. (2) If you’re interested: the baktun we are in at the moment will end on Sun- day, 25 March 2407; its long count will be 0.19.17.19.19 (sometimes written as 13.19.19.17.19), and the long count for Monday, 26 March 2407 will be 1.0.0.0.0.0 (or 14.0.0.0.0). (3) Why did Maya astronomers deﬁne a month to be 20 days long? Historians are not sure, one possible explanation is that they took 260 (=13 · 20) as it is the interval (actually approximate interval) between conception and birth of a baby. Plus, Mayans had a thing for number 13. Section 5 Proportional Relationships 81

5 Proportional Relationships

We have already seen some ways of talking about quantities: stating a number to describe it (absolute), relating that number to something else (relative; as in per minute, per 100,000 people); using orders of magnitude; comparing to human size, using our intuition, or visualizing by comparing to something we know, or have experience with. To describe how quantities change, we often look for patterns. Understanding patterns and what they say about the change is an essential part not just of mathematics, but of the way we think mathematically about almost anything that surrounds us. How do we know that, unless we all decide to make a change and actually do something, the global temperature will rise by about 1.5 degrees Celsius? First, scientists defined what the term global temperature means, and then started measuring and recording its values (easier said than done). By looking at these (historic) values, scientists have been trying to identify a pattern, and then apply that pattern to look into future–predicting what would happen 10, 20, or 50 years from now, and what the consequences would be. In its report Global Warming of 1.5 degrees C the IPCC (= Intergovernmental Panel on Climate Change, a United Nations body which studies climate change) states (http://www.ipcc.ch/report/sr15/) “Why is it necessary and even vital to maintain the global temperature increase below 1.5C versus higher levels? Adap- tation will be less difficult. Our world will suffer less negative impacts on intensity and frequency of extreme events, on resources, ecosystems, biodiversity, food security, cities, tourism, and carbon removal.”

Pattern: Proportional Quantities

We say that two quantities A and B are proportional if the ratio of their values A/B (or B/A, does not matter) remains ﬁxed as the quantities change. For instance, consider the measurements of A and B taken once a second for 6 seconds:

Initially, when the time is 1, the ratio is A/B =9/6=1.5. Note that the same ratio is kept throughout the changes in A and B: when the time is 3, A/B =3/2=1.5, when it is 6, A/B =12/8=1.5 (check the remaining values). Thus, we conclude that A and B are proportional. However, the quantities A and C are not proportional: initially, when the time is 1, A/C =10/2=5, whereas when the time is 4, A/C =5/6 ≈ 0.7833 and when thetimeis5,A/C =8/10 = 0.8.

Case Studies 5.1 Proportional Relationships (1) The excerpt below discusses two major electoral systems: Proportional representation is an electoral system where the proportions of votes reﬂect in the 82 NUMERACY

composition of the elected body (such as a parliament). Under the first-past-the- post system each voter indicates the candidate of their choice, and the candidate with the most votes wins. That candidate (their party) then wins most seats in elected body (thus forming, for instance, a majority government). (Read more about these – as they are important for Canada – in Wikipedia articles ‘Propor- tional representation’ and ‘First-past-the-post voting’). In Why Trudeau’s broken electoral reform promise could rebound on him (CBC News, 26 July 2019 https://bit.ly/2YqqDW3) we read: “In June 2015, Trudeau vowed that the federal election of that year would be the last conducted under the first-past-the-post system. In February 2017, as prime minister, he decided to walk away from that commitment. Whatever the merits of that decision (Trudeau had misgivings about the ramifications of moving toward proportional representation and feared that a national referendum would be divisive), electoral reform is easily classified as a ‘broken’ promise.” Related to a possible change in electoral reform, on 20 Decemcer 2018, Global New reported that British Columbians reject proportional representation, vote to stay with first-past-the-post (https://bit.ly/2KlwFNG). (2) In the report Chronic Anemia May Affect White Matter Volume in the Brain and Cognitive Performance published on 1 August 2019, Hematology Ad- visor (https://bit.ly/2Kf9Psl) reports that “According to results published in the American Journal of Hematology, lower hemoglobin levels appear to be associated with reduced white matter volume throughout the brain, regardless of patients sickle cell disease (SCD) status. Thus, the severity of anemia, not disease state, predicts white matter volume.” It then continues, by qualifying the relationship as proportional: “ Patients with chronic anemia demonstrated a decrease in brain white matter volume proportional to anemia severity.” (3) In some situations (could be quite often) the word proportional is used in a more loose sense, where it jus means related. For instance, in the article 12 Things Homebuyers Should Look For In A Real Estate Agent published in Forbes, at https://bit.ly/2ZvAsOV the author writes “I have found a proportional relationship between the amount of time it takes an agent to get back to you and their level of professionalism. If an agent doesn’t miss emails or calls, this is an indication that they are on the ball and detail-oriented. Conversely, agents that are inattentive tend to be equally sloppy in details of the transaction.”

To emphasize: in a proportional relationship, one variable mimics the (multiplicative) changes in the other variable. (Another way to say this is to state that boith variables react the same under scaling.) Assume that A and B are proportional. This means that if A doubles, so does B;ifA decreases by 75%, then B decreases by 75% as well. If B quadruples, then A quadruples. In a more technical language, A and B are called proportional if they are multiples of each other, i.e., there is a number m (which is not zero), such that A = mB Of course, then A/B = m, which shows that the ratio of A to B remains constant, and equal to m. What does it look like visually? Consider the table below which establishes a proportional relationship between quantities A and B (recall that A/B =1.5) and graph the points in a coordinate system: Section 5 Proportional Relationships 83

(8,12)

(6,9)

(3,4.5) (4,6) (2.5,3.75) (2,3)

Note that B is on the horizontal axis, so the ﬁrst coordinate of each point is the value of B. So, all points lie on the same line, and the slope of that line is 1.5, which is equal to the constant of proportionality m.

Example 5.2 Proportional Reasoning: Distance and Speed It is good to remember that 1 mile is approximately 1.6 kilometres (as a matter of fact, 1 mile = 1.60934 kilometres, but for all practical purposes we can say that 1 mile = 1.6 km). Then 2 miles is 2 · 1.6=3.2 kilometres, 10 miles is 10 · 1.6=16 kilometres, and so on. Thus, the relationship is proportional, and we can write (use = but keep in mind that it’s an approximation) x miles = 1.6x kilometres Thus, 390 miles (roughly the distance from Buﬀalo to New York City) is approximately 390 · 1.6 = 624 km. Multiplying 1 mile = 1.6 km by 60, we obtain that 60 miles = 1.6 · 60 = 96 ≈ 100 kilometres. Dividing both sides by hours (thus getting the speed) we obtain another useful thing to remember: 100 km/h is roughly 60 mph This relationship is also proportional, so we can halve it to obtain 50 km/h is roughly 30 mph Computing 80% of it, we obtain 80 km/h is roughly 48 mph ≈ 50 mph Multiplying by 1.2 gives 120 km/h is roughly 60 · 1.2=72mph≈ 75 mph and so on. Using the conversion rate we just discussed, you can check that all entries in this chart from tripsavvy (https://bit.ly/31fJiAD) are correct, or approximately correct: 84 NUMERACY

Example 5.3 Proportional Reasoning: Scaling Scientists have calculated that Earth is about 4,540 million (i.e., about 4.5 billion) years old, with a margin of error of about 50 million years (thus, the range for Earth’s age can be written as 4, 540 ± 50 million years). Dinosaurs lived through the Mesozoic Era, which started about 245 million years ago, and whose end is marked with their extinction about 65 million years ago. It is hard to determine when humans emerged, because it is hard to deﬁne what ‘human’ actually means. For the sake of this example, we deﬁne it as the emergence of homo erectus about 1.8 million years ago. 4.5 billion, 65 million, 1.8 million – how do we make sense, how do we visualize these numbers? One way is to scale them to something that we are all familiar with–for instance, the length of a day, 24 hours. So we take 24 hours to represent 4,540 million years. Dinosaurs went extinct 65 million years ago, whose ratio with regards to the age of the Earth is 65/4, 540 = 0.014317 Now apply this same ratio to 24 hours, to get 24 · 0.014317 = 0.34361 hours. Thus, dinosaurs went extinct less than half an hour before the clock turned to 24. More precisely: 0.34361·60 = 20.6166 minutes, which is 20 minutes and and 0.6166 · 60 = 36.996 ≈ 37 seconds. We conclude that the dinosaurs went extinct at approximately 23:39:23. Repeat the calculation for the time of 1.8 million years: 1.8/4, 540 = 0.000396 Apply this ratio to 24 hours, 24 · 0.00033 = 0.009504 hours. This is 0.009504 · 60 = 0.57024 minutes, which is 0 minutes and and 0.57024 · 60 = 34.214 ≈ 34 seconds. We conclude that homo erectus appeared on Earth at approximately 23:59:26. Now we can easily see that, compared to Earth’s geological age, even the extinction of dinosaurs is a recent event.

24:00:00 (today)

23:39:23 00:00:00 23:59:26

To practice this thinking about proportionality, a couple of examples (we will do many more in the following sections). (1) A 30 kg bag of cement costs $ 7.80. How much does 5 kg of cement cost? Answer: 5 kg is one-sixth of 30 kg, so the price must be one-sxith of $ 7.80, which is $ 7.80/6 = $ 1.30. How much does 100 kg of cement cost? Apply the ratio 100/30 of weights to the prices (as there quantities are proportional, the same 100 · ratio applies), to get 30 7.80 = $ 26. How much cement can we buy for $ 3.90? Answer: Since $ 3.90 is one half of $ 7.80, we can buy (the same ratio, i.e.,) one half of 30 kg, which is 15 kg. Section 5 Proportional Relationships 85

(2) You are driving from Hamilton to Montreal (driving distance of about 610 km) and just passed Oshawa, which is about 128 km from Hamilton. If you were to represent the distance from Hamilton to Toronto with a 10 cm piece of string, wherewouldyouhavetoplaceOshawa? Answer: At the moment when we are in Oshawa, we covered 128/610 of the 128 · distance to Montreal. Applying the same ratio to the string, we get 610 10 = 2.098. Thus, we would have to place Oshawa just a bit over 2 cm (2.089 cm) from the end of the string which represents Hamilton.

Next, we study several important proportionality relations, and then we investigate other common relationships. Just to contrast with proportional things we have studied–what is not proportional? The area of a square with side 3 is 9. The area of the square with side 6 (double the side) is 36, which is not the double of 9. Thus, the area of a square and its side length are not in a proportional relationship. Human bones do not grow in a proportional way, in the sense that the length of a bone and its diameter (thickness) are not proportional. Studying bones of various animals, Galileo (16/17th c. CE) realized that that was the case. His illustration shows that a bone three times longer is not three times thicker, but more than that. (Note: he exaggerated the thickness - the longer bone should be about 5.2 times thicker than the shorter one).

Currency Exchange

When we asked Google (on 4 August 2019 around 1pm) what the exchange rate between Canadian Dollar and Euro is, we saw this:

Thus 1 Euro equals 1.47 Canadian dollars, which we think of as A Euro = 1.47 · A Canadian dollars 86 NUMERACY

Thus, 100 Euro is worth 1.47 · 100 = 147 Canadian dollars, and 5 Euro is worth 1.47 · 5=7.35 Canadian dollars. So, this currency conversion is a proportional relationship, where the constant of proportionality is the exchange rate of 1.47. Note that the diagram on the right shows that the exchange rate is not constant, but changes over time. Dividing both sides of A Euro = 1.47 · A Canadian dollars by 1.47, we obtain the reverse exchange rate: 1 · A Euro = A Canadian dollars 1.47 i.e., A Canadian dollars = 0.68 · A Euro

So, if we wish to buy 100 Euro, we need to pay $ 147 (Canadian), and to buy 100 $ (Canadian), we need 68 Euro. Mathematically this is correct, but in real life banks and other ﬁnancial institutions involved in currency exchange need to make money. Thus, when currency exchange is involved, there are two rates, called buy rate and sell rate. The buy rate refers to the rate at which a bank (ﬁnancial institution) buys currency from customers, and the sell rate is the rate at which they sell it to their customers. Here is a photo of the buy and sell rates from the currency trader ice at Pearson airport in Toronto (taken in 2018):

Note that buy rates are smaller than sell rates. If we wish to buy 100 Euro, we look at the sell rate of 1.805410, and compute that we will need 1.805410 · 100 = 180.54 $ (Canadian). But if we have 100 Euro andwishtosellit,ice will give us 1.316786 · 100 = 131.68 $ (Canadian). The difference is the profit that ice makes. Sucks! Of course, the smaller the difference between buy and sell rates, the less money currency exchange people/institutions make. When travelling, keep in mind that different entities (banks, financial institutions, money exchange kiosks) offer different exchange rates, and that the rates offered at certain places (airports, train stations) are often lot more extreme than the ones a bank in the centre of a city will offer. So don’t change your money at airports (unless you really have to); as well, avoid money exchange kiosks, as sometimes they offer exchange fees on top of unfavourable buy and sell rates (some of these places are really shady and do not fully discolse their rates untl it’s too late). Section 5 Proportional Relationships 87

Sometimes the currency exchange places will post the rates, without identifying buy and sell rates. How do you know which is which? Just ﬁgure out which works against you. Assume that, in Vancouver, you see two rates for Australian Dollar: 0.8547 and 0.9216. Assume that 0.8547 is the sell rate (the rate at which the currency exchange kiosk sells to us): we need $ 85.47 (Canadian) to buy $ 100 (Australian). Now we sell Australian dollars back: applying the buy rate (i.e., the rate at which the kiosk buys currency from us) of 0.9216 we get $ 92.16 (Canadian). So, in this processwemade92.16 − 85.47 = $ 6.69 (Canadian). And then, if we repeat this hundred times, we would make over 669 dollars! Of course this cannot happen – it’s the other way around: 0.8547 is the buy rate and 0.9216 is the sell rate. By the way, these days there is an eﬃcient way to avoid changing actual (paper) money – use credit or cash cards (bit more about it soon), as they are accepted in almost every country in the world. However, there might be places which are still cash only (on the other extreme, there are places which no longer accept cash).

One more thing to keep in mind when abroad: it is becoming more and more common that shops (especially duty free shops at airports), restaurants, ATMs, varuous businesses oﬀer conversions, so that you pay in your home currency, instead of the local currency. Here is a screen from an ATM at Heathrow airport:

(Source: The Conversation article Save money when traveling abroad by thinking like an economist at https://bit.ly/2YKXfJj). If you plan to travel, read the above article! It stronlgy discourages you from paying in your home currency (Continue with conversion option), as the exchange rates are unfavourable, and there are extra charges involved (which again, might or might not be clearly spelled out). In any case, when abroad, always select to pay in local currency. In the above article from The Conversation, we read: “Increasingly, however, retailers, restaurants and ATMs are offering travelers the option to pay or with- draw money in terms immediately converted into their home currency. Companies offering the service call it ‘dynamic currency conversion.’ For example, an Ameri- can tourist visiting Paris is able to use her credit card to pay for a fancy meal at a French bistro in U.S. dollars, instead of euros. This may seem innocuous – or even convenient – but agreeing to use your home currency in a foreign land can significantly inflate the cost of every purchase.” The article goes on explaining what the extra charges are, concluding that these can add a significant amount to our bill. We read: “[...] Even some bankers warn against consumers doing this [i.e., paying in one’s home currency when abroad] because the exchange rate used is much worse than the one your bank would offer.” 88 NUMERACY

The article Holiday money: how to find the best cards and currency rates published in the Guardian (23 June 2018, https://bit.ly/2lCNAzt) offers identical advice, and provides a “[...] guide on avoiding bank charges and making your cash go further this summer.” Read these articles, and consult an online currency converter to get a good feel about the currency of the country you are travelling to. Think about this before you go, as quite often, when things actually happen (say, rushing through duty free because you have to catch your flight, or standing in front of an ATM with 10 people in line behind you) you will not have time to reflect and make a good decision. Keep in mind that exchange rates depend on many factors, including the time of day (the rates are never stable), who is involved (bank, financial institution), what we are paying with (cash, travellers cheques), and the type of a card we use (credit, debit).

Conversion Between Units

To convert metres to inches, we multiply by 39.3701 (or, very often, by 39.37): so three metres are equivalent to 39.37·3 = 118.11 inches, half a metre is 39.37·0.5= 19.685 inches, and so on. A general conversion between units has (by now, well known) form of a proportional relationship: unit = conversion factor · another unit One kilogram is equivalent to 2.20462 pounds (which is often rounded oﬀ to 2.2 pounds). In general, A kg = 2.2A lb As any proportional relationship, we can visualize it as a line (through the origin, since 0 kg = 0 lb). The slope of the line is the conversion factor 2.2.

(10.7,23.54)

(6.4,14.08)

(3,6.6) (1,2.2)

To compute the reverse conversion, we divide by 2.20462 : 1 A lb = A kg = 0.45359A kg 2.20462 In other words, to ﬁgure out the mass of something in ponuds, we multiply its mass in kilograms by 0.45359.

Exercises 5.4 Conversion Between Units (1) If x metres is converted to y kilometres, which number is larger, x or y? (2) If x mph (miles per hour) is converted to y km/h, which number is larger, x or y? (3) If x km/h is converted to y km/s (kilometres per second), which number is larger, x or y? (4) How many litres are there in a cubic metre? Section 5 Proportional Relationships 89

(5) Human hair grows at about 1.25 cm per month. Convert this speed into metres per second. Partial answers: (1) Say we walk for 400 metres - we still have not walked for · 1 1 kilometre (to be precise, 400 metres is 400 1000 =0.4 kilometres). Thus, x (the number of metres we walked) is larger than y (the number of kilometres we walked). (3) Driving at 120 km/h, we cover the distance 120 kilometres in one hour. Deﬁ- nitely we will cover a much smaller distance in one second, so x is smaller than y. If we’re not convinced (keep in mind that 1 hour = 3600 seconds): km 1 km 1 km 120 = 120 · = h 3600 s 30 s (so x = 120 and y =1/30). (4) A good visual to remember: one litre of water ﬁlls a 10 cm × 10 cm × 10 cm cubic box. In other words, the volume of 1 litre is equivalent to the volume of (10 cm)3 = 1000 cm3. Now multiply the relation 1 m = 100 cm by itself three times (i.e., cube it): 1m3 = 1000000 cm3 and thus 1m3 = 1000 · 1000 cm3 = 1000 litres.

(5) Assume that a month has 30 days. Then cm 0.01 m 1.25 =1.25 month 30 · 24 · 60 · 60 s Here we used the fact that 1 metre has 100 centimetres, and thus 1 centimetre = 1/100 = 0.01 metres. In the denominator, we converted days into seconds. Using a calculator, we ﬁnd that the number on the right is 4.8225·10−9 metres per second. Although this number is not of much help, we learned something: one way to make a small (or a large) number more manageable size is to change the units involved. 4.8225 · 10−9 metres per second is not what many of us can comprehend, however 1.25 cm per month we can visualize and work with.

Notes: (1) There is a number of online units converters that we can use. When we do, we realize how many different units are there, and it’s all quite chaotic. That is why countries around the world have been adopting the International System of Units (known as the SI system), which is based on metre (length), kilogram (mass), second (time), kelvin (temperature), ampere (electric current), mole (amount of substance) and candela (luminous intensity). However, for many practical purposes we use other units (imagine a weather forecast for an August day given as 303.15 degrees Kelvin; so we stick to degrees Celsius or Fahrenheit). A pint is lot more common in some contexts than litres or millilitres (Challenge: try to order 568 millilitres of beer. By the way, 568 ml is called a British pint. American pint is smaller, and contains 473 ml). (2) Why is it important to have a unique, clearly defined system of units? Buying one pound of bananas instead of one kilogram is usually not a big deal. How- ever, sometimes the errors caused by messing up units could be (and are) deadly. Wikipedia article Korean Air Cargo Flight 6316 (https://bit.ly/2GC6yRu) recalls the 1999 cargo plane crash which killed three people on board. We read: “[...] investigation carried out by CAAC (= Civil Aviation Administration of China) showed that the first officer had confused 1,500 metres, the required altitude, with 1,500 feet, causing the pilot to make the wrong decision to descend.” On 23 July 1983, Air Canada flight from Montreal to Edmonton run out of fuel mid-air. The plane involved was the first Air Canada’s plane that used 90 NUMERACY

metric (SI) measurements; see Gimli Glider https://bit.ly/1T25uTu). We read: “The Aviation Safety Board of Canada (predecessor of the modern Transportation Safety Board of Canada) reported that Air Canada management was responsible for ‘corporate and equipment deficiencies.’ Their report praised the flight and cabin crews for their ‘professionalism and skill.’ It noted that Air Canada ‘neglected to assign clearly and specifically the responsibility for calculating the fuel load in an abnormal situation.’ [...]” (3) Not all conversions are given by a proportional relationship. We will see soon that conversion between degrees Celsius and degrres Fahrenheit involves a non- proportional relationship. (4) Numbers that come with units behave in ‘weird’ ways: for instance, mathematically correct equation 1 = 1 might not be correct when units are involved: for instance, 1 metre is not equal to 1 centimetre. As well, mathematicall incorrect statement 1 = 2.54 can be made correct if we add units: 1 inch = 2.54 cm. (5) In Canadian grocery stores, prices for both pounds and kilograms are displayed, but in many cases the prices per pound are in larger print. Below are screenshots from a Sobeys flyer:

One reason for this, as researchers tell us, is psychological: shoppers are more sensitive to prices than to quantities. So, as long as the price is lower (even if the quantity is smaller), they are more likley to buy. Read more about this in the BBC News article The food you buy really is shrinking https://bbc.in/2OBQzJJ.

Percent

Recall that to convert a number into percent we multiply by 100 (move the decimal point two places to the right), and to convert percent into number we divide by 100 (move the decimal point two places to the left). Thus, 0.45 = 45 % 12.4% =0.124 1.28 = 128 % 1% =0.01 0.07 = 7 % 0.8% =0.008 0.0004 = 0.04 % 231 % = 2.31

If A is a quantity, then a percent of A is calculated by multiplication (hence it’s a proportional relationship): 16 % of A is 0.16A, 1.6% ofA is0.016A, and 160 % of A is 1.6A. Recall that a proportional relationship means that a multiplicative input generates identical response (output). Computing a percent is proportional in two Section 5 Proportional Relationships 91

diﬀerent ways: with respect to the percent, and with respect to the quantity involved. Assume that 15 % of some quantity A is equal to 80. Then (triple the relationship) 45 % of A is equal to 240 (triple the value), (halve the relationship) 7.5% of A is equal to 40 (halve the value), (take one tenth of the relationship) 1.5% of A is equal to 8 (one tenth the value), and so on. Again, assume that 15 % of some quantity A is equal to 80. Then (triple the relationship) 15 % of 3A is equal to 240 (triple the value), (halve the relationship) 15 % of A/2 is equal to 40 (halve the value), (take one tenth of the relationship) 15 % of A/10 is equal to 8 (one tenth the value), and so on.

The most important thing about percent is that it is relative – so whenever we work with information presented as percent, we must ask percent of what? Although this might seem obvious, there are many cases when we forget about it, or various sources (purposely or not) omit the reference information. For instance, information contained in the phrase “The unemployment rate in Manitoba rate fell by 0.2 %” is useless, until we make clear what the reference is, such as “compared to the last year,” or “when compared to the first quarter of this year.” Does 10 % + 10 % equal to 20 %? It depends! The first thing we need to be clear about is the meaning – 10 % of what? 20 % of what? If the above phrase means 10 % of A + 10 % of the same quantity A then this sum is 0.1A +0.1A =0.2A, which is 20 % of A. Otherwise, it could be many different things. Is 10 % larger than 4 %? Again it depends!

If a quantity A grows by 34 %, then its value is A +0.34A =1.34A. We can think of this using the so-called 1plusrule:1.34 = 1 + 0.34, where 1 represents the original quantity, and 0.34 is the 34 % increase. Thus, 1.5A can be interpreted as (1.5=1+0.5) a 50 % increase in the value of A, and a 3 % percent increase in A can be written as 1.03A. Likewise, if a quantity loses 18 % of its value, then its new value is A−0.18A = 0.82A, i.e., 82 % of its original value. We can think of this using the so-called 1 minus rule:0.82 = 1 − 0.18, where 1 represents the original quantity, and 0.18 is the 18 % loss. Thus, 0.59A can be interpreted as (1 − 0.41 = 0.59) a 41 % decline of the value of A. If A loses 78 % of its value, what is left is 0.22A. The 1plusruleand the 1minusrulecan help us ﬁgure our more complex situations. Look at the following example.

Example 5.5 Using 1 Plus and 1 Minus Rules Assume that A represents the price of something. (1) If A is discounted by 7 %, and then the new price is further discounted by 12 %, then the final price of A is 0.88(0.93A)=0.8184A, i.e., the total discount is (1 − 0.8184 = 0.1816), i.e., 18.16 %. (2) If A is discounted by 7 %, and then the new price is increased by 12 %, then the final price of A is 1.12(0.93A)=1.0416A, i.e., the final price is 4.16 % higher than the original price. (3) Reverse the situation in (2): if A is first increased by 12 %, and then the new price is discounted by 7 %, then the final price of A is 0.93(1.12A)=1.0416A, i.e., the final price is 4.16 % higher than the original price – the same as in (2). (4) If A is increased by 12 % and the new price in increased again by 12 %, is the total increase going to be 24 %? 92 NUMERACY

No, it will be more than 24 %. The second increase is calculated not based on A, but based on the increased value of A:1.12(1.12A)=1.2544A, which is a 25.44 % increase. (5) If A is increased by 12 % and the new price in discounted by 12 %, is the final price going to be A? No, it will be less than A, because the 12 % discount is calculated not based on A, but based on the larger price (A increased by 12 %). To check, we compute: 0.88(1.12A)=0.9856, i.e., the final price is not A, but 98.56 % of it (equivalently, the final price is A discounted by 1.44 %).

Case Study 5.6 Caesarean Births are ‘Affecting Human Evolution’ In the report Caesarean births ‘affecting human evolution’ (BBC News, 7 Decem- ber 2016, https://bbc.in/2g4kpQv) we read: “The researchers estimated that the global rate of cases where the baby could not fit through the maternal birth canal was 3 %, or 30 in 1,000 births. Over the past 50 or 60 years, this rate has increased to about 3.3 − 3.6 %, so up to 36 in 1,000 births. That is about a 10-20 % increase of the original rate, due to the evolutionary effect.” Let us check the math: is it really 10-20 % increase of the original rate, as claimed? First, we look at the lower end of the range: 3.3 % is 33 per 1,000. Thus, the increase is from 30 to 33. The absolute change is 3 births, and the relative change is 3/1, 000 = 0.003, or 0.3 %. However, if the relative change is computed based on the initial count of 30, then it is 3/30 = 0.1, or 10 %. Similarly, the upper end of the range is 3.6 %, or 36 per 1,000. The absolute change is 6 births, and the relative change could be 6/1, 000 = 0.006, or 0.6%,if compared to the 1,000 births, or 6/30 = 0.2, or 20 %, if compared to the initial rate of 30 (in 1,000 births). Thus, mathematically both ranges 0.3-0.6 % and 10-20 % are correct. Very likely, the author picked the 20 % angle to make a stringer point.

Case Study 5.7 (Mis)interpreting Risk: ‘Pill Scare’ This case study refers to the risk of developing blood clots (venous thromboembolism) when taking third- and fourth- generation contraception pills (called CHC = combined hormonal contraception), which have been introduced to the general public in 1990s and 2000s, respectively, and which are in widespread use. On its web page, Hormonal Birth Control and Blood Clot Risk, U.S. organization National Women’s Health Network (https://bit.ly/2yDvAeL) states that “[...] CHC methods are birth control methods containing the hormones estrogen and progestin. Tens of millions of people safely use CHCs – including birth control pills, patches, and vaginal rings – to help space births and prevent unintended pregnancy.” Later, on the same page, we ﬁnd risk statistics: “Blood clots are generally rare but sometimes occur in otherwise healthy people, even those not taking CHCs. Between 1 and 5 of every 10,000 women (who are not pregnant and not using CHCs) will experience a blood clot in any given year. This number increases slightly if the person uses CHCs. Between 3 and 9 of every 10,000 CHC users will experience a blood clot in any given year.” Note that the article states that the risk increases slightly for a person who uses CHCs. Let us analyze this change from 1-5 per 10,000 (no CHC, no pregnancy) to 3-9 per 10,000 (CHC) in a diﬀerent way. Section 5 Proportional Relationships 93

The average of the interval 1-5 is (1+5)/2=3, and the average of the interval 3-9 is (3 + 9)/2=6. Thus, we can say that the average risk doubles. Alternatively, the change from 1-5 per 10,000 to 3-9 per 10,000 could be interpreted as double the rate, as the doubles of numbers from 1 to 5 are 2 to 10, which is close to the 3-9 range. Unfortunately, the news that the risk of developing blood clots doubles for CHC users found its way into media on both sides of the Atlantic. For instance, on 14 November 2012, ScienceNordic (an Independednt news organization which focuses on writing about research in Scandinavian countries) published the ﬂoow- ing article:

(Source: https://bit.ly/31b8tV8.) ScienceNordic states that “Now a Danish literature review conﬁrms what has been suspected for years: that the so-called fourth-generation OCP (= oral contraceptive pill) doubles the risk of blood clots compared to second-generation OCPs.” Later in the same article, we ﬁnd the following statistics:

The narrative states that there is no cause for panic. However, in the last line, it says that the risk (fourth generation OCPs) increases six-fold, as the authors compared it not to the women who take OCPs but to whose women who do not take contraception. In the piece Some contraceptive pills said to have DOUBLE the risk of blood clots published in Cosmopolitan (27 May 2015, https://bit.ly/31jj7t2)weread:“A new study has revealed that some third-generation combined contraceptive pills are twice as likely to cause blood clots in the arm or leg than older brands – but it’s not quite time to panic yet.” The messages “no cause for panic” and “it’s not quite time to panic yet” did not catch people’s attention nearly as much as the claims about doubling the risk or six-fold increases in the risk.

The key observation is that (even when the risk is doubled, or increased six- fold) the numbers are still very small, i.e., the actual risk of developing blood clots when taking new generations oral contraceptives is still very small. However, it was too late. Many women decided to stop taking the pill, and some switched to other methods of contraception. The consequences were serious. 94 NUMERACY

In the report Oral Contraceptives and the Risk of Venous Thromboembolism: An Update published in 2010 by The Council of the Society of Obstetricians and Gynaecologists of Canada (https://bit.ly/2yFTWV2)weread: “Recent contradictory evidence and the ensuing media coverage of the venous thromboembolism (= blood clots) risk attributed to [...] certain newer oral contraceptive products have led to fear and confusion about the safety of oral contraceptives. ‘Pill scares’ of this nature have occurred in the past, with panic stopping of the pill, increased rates of unplanned pregnancy, and no subsequent decrease in venous thromboembolism rates.” The abstract of the paper Pill scares, an avoidable side effect published in The European Journal of General Practice (https://bit.ly/33hiaTY)echoesthe Canadian report: “Whilst their [oral contraceptives] benefits have been obvious to women, the majority of medical publications have focused on the risks that may be associated with their use. ‘Pill scares’ are becoming a part of contraceptive practice – they worry women and their families, cause considerable morbidity (through termination of unwanted pregnancies) and complicate the work of doctors and other health professionals.” Epilogue: the 1995 ‘pill scare’ in England and Wales resulted in additional (i.e., over the yearly average) 26,000 unplanned pregnancies, with a bit over one half (13,600) resulting in abortions (see the diagram on the left; source: [1], page 54). Subsequent ‘pill scares’ have had similar consequences elsewhere. The diagram on the right shows the large upward trends in both the number of abortions and the abortion rates due to the two pill scares in France in 2000 and 2012 (source: Institut National d’Etudes´ Démographiques (INED), https://bit.ly/2GMP8Si).

Notes: (1) The two case studies discussed here are really important: the first one (Caesarean Births) shows how the same information can be presented in different ways (this is called frame or framing, and we will talk more about it), and, consequently, can generate different reactions from people. The second case study (‘Pill Scare’) is based on a conflict between two frames: small increase in risk, vs. doubling the risk. It shows how a wrong choice of framing the given statistics can lead to very serious negative consequences. Here is a classical example which underlines the importance of framing. A physician is describing the risk of death from a surgery to their patient, who is considering undergoing the surgery. The physician can say “You have 10 percent Section 5 Proportional Relationships 95 chance of dying from the surgery” (this is called a negative frame)or“Youhave 90 percent chance of surviving the surgery” (positive frame). Research shows that a patient who is presented with a positive frame is more likely to give consent to undergo the surgery. (2) To repeat the message of the ‘Pill Scare’ story: a small number, doubled, is still a small number. But again, it’s psychology – we tend to exaggerate small chance, rarely occuring events. For instance, a story about people dying from smoking is not a front page news, unlike a report on a plane crash (but it should be: in terms of morbidity, smoking deaths = three standard size passenger plane crashes daily).

Chapter rerefences [1] Gigerenzer, G., Gaissmaier, W., Kurz-Milcke, E., Schwartz, L. M., & Woloshin, S. (2007). Helping Doctors and Patients Make Sense of Health Statistics. Psycho- logical Science in the Public Interest, 8(2), 5396. https://doi.org/10.1111/j.1539- 6053.2008.00033.x 96 NUMERACY

6 Linear and Non-linear Relationships

When the UN Arctic Chief Jan Dusik said “Climate change isn’t linear – it’s accelerating” (source: https://bit.ly/2FKIXQe) what did he mean? First – what is linear, what causes linear behaviour? Consider an example: the population of Ontario was 14.32 million in 2018, and the current rate of increase is 0.258 million (258,000) people per year. To make a prediction about the future population count, we assume that the rate of increase will remain unchanged. Thus: 2018: 14.32 million 2019: increase of 0.258 million; the population is 14.32 + 0.258 = 14.578 million 2020: another increase of 0.258 million; the population will be 14.32 + 2 · 0.258 = 14.836 million 2021: another increase of 0.258 million (the third since 2018); the population will be 14.32 + 3 · 0.258 = 15.094 million 2022: further increase of 0.258 million; the population will be 14.32 + 4 · 0.258 = 15.352 million 2023: the population will be 14.32 + 5 · 0.258 = 15.610 million As every year the same number of people is added, we see a pattern emerging: Population in year x = population in 2018 + (number of years since 2018) · (yearly increase) =14.32 + (number of years since 2018) · 0.258 Using P to denote the population of Ontario and t the number of years since 2018, P =14.32 + t · 0.258 The rate of increase of 0.258 million people per year is called a marginal change. In general, a marginal change is the change in the value of a quantity from one moment to the next, i.e., over a small interval (in mathematics, we insist on really very small intervals, called inﬁnitesimally small). A quantity is called linear,orissaidtochange linearly if all marginal changes, measured over intervals of identical length, are equal. The graph of a quantity which changes linearly is a line. In the picture below, the population is shown in a blue line. All intervals are of length one (so ‘from one moment to the next’ in this case is one year), and all corresponding changes in the population (red vertical segments) are equal.

population (million) 0.258 = marginal change 14.32

04152 3 time (year) 201820192020 2021 2022 2023 Section 6 Linear and Non-linear Relationships 97

A linear quantity is represented using a linear function,asinP =14.32+0.258t. Recall that a linear function is of the form y = b + mx, where b represents the vertical intersect (the value of the quantity y corresponding to the value 0 of x), and m is the marginal change, also called the slope. The picture below (left) shows a quantity which increases over time, as its marginal changes (slope) are positive. On the right, the marginal changes (slope) are negative, and the quantity loses its value.

Thus, a quantity that changes by accumulation, where, at identical intervals, a fixed amount (marginal change) is added, is a linearly increasing quantity. If a fixed amount is removed at identical intervals, the quantity is decreasing linearly. If a quantity experiences non-equal marginal changes, then it is called non- linear. For instance, the graph below shows a non-linear quantity: its marginal changes are of different sizes; as well, some are positive (see the change from 1 to 2 or from 10 to 11), and some are negative (see the change from 4 to 5 or from 6 to 7).

011152123 4 6 7 8 9 10

Back to the start: now we know that ‘Climate change isn’t linear – it’s accelerating’ means that the way climate changes cannot be represented by a line. The part ‘it’s accelerating’ suggests that the marginal changes are growing, as the quantity in the above graph between 9 and 12.

Example 6.1 Measuring Temperature To most commonly used units for measuring temperature are deﬁned based on the Celsius and the Fahrenheit scales. The Celsius scale divides the temperature between the freezing point of water (labelled 0◦C) and the boiling point of water (labelled 100◦C) into 100 equal parts. The Fahrenheit scale divides the temperature between the freezing point of water (labelled 32◦F) and the boiling point of water (labelled 212◦F) into 180 equal parts. By deﬁnition, both scales are linear. Note that the degrees Fahrenheit change ‘more quickly’ than the degrees Cel- sius, as it takes 180 of them to reach the boiling point of water from its freezing point, whereas it takes only 100 degrees for the Celsius to do the same. 98 NUMERACY

Based on the deﬁnition, conversion formulas have been calculated. In particular, 9 F = C +32 5 converts degrees Celsius into degrees Fahrenheit. Note that the slope is 9/5=1.8, meaning that for each one degree change in the temperature measured in Celsius, the temperature measured in Fahrenheit increases by 1.8 degrees (this quantiﬁes the ‘more quick’ change mentioned above). Thus a change of 5 degrees Celsius corresponds to a change of 5 · 1.8 = 9 degrees Fahrenheit, a change of 10 degrees Celsius corresponds to a change of 10 · 1.8 = 18 degrees Fahrenheit, and so on. ◦ 9 ◦ ◦ For instance, 12 Cisequivalentto 5 (12)+32 = 53.6 F, and 25 Cisequivalent 9 ◦ to 5 (25) + 32 = 77 F. degrees F 212

32 −40 0 100 −40 degrees C

Thus, we can write 25◦C=77◦F. Subtract (keep in mind the above comment about the slope) 5◦C=9◦F from it, to get 20◦C=68◦F, subtract again, to get 15◦C=59◦F. Add 10◦C=10◦Fto25◦C=77◦Ftoobtain35◦C=95◦F, and so on. The reverse conversion is given by 5 C = (F − 32) 9 ◦ 5 − ◦ − ◦ Thus, 104 Fisequivalentto 9 (104 32) = 40 Cand 13 Fisequivalentto 5 − − − ◦ 9 ( 13 32) = 25 C − ◦ 5 − − − ◦ Note that 40 Fisequivalentto 9 ( 40 32) = 40 C, so the two scales meet there: −40◦C=−40◦F.

Linear vs Proportional

Recall that two quantities are proportional if they are multiples of each other. Naming the quantities x and y, we say that they are proportional if there is a non-zero number m such that y = mx. Note that this is the line y = mx + b, but without b (i.e., b =0).Thusa proportional relationship is linear, and is represented by a line going thorugh the origin (because b =0). A linear relationship, however, is not proportional (except when b =0).Con- sider the temperature measured in degrees Celsius and degrees Fahrenheit in the table on the left. The scaling (a tell-tale sign of proportionality) does not work: for instance, doubling the temperature in degrees Celsius does not double degrees Fahrenheit: 10◦C=50◦F, but 20◦C is not equal to 100◦F. As well, look at the graph above - it does not go through the origin. Section 6 Linear and Non-linear Relationships 99

To summarize: proportional is linear, but linear is not proportional (unless b =0 in Ey = mx + b).

Linear Regression

It is easy to ﬁgure things out when we are told what to expect, such as that a relationship is linear (as in the conversion between Celsius and Fahrenheit). In general, we do not have such information – but still, somehow, need to describe the relationship between the variables from the data which is available to us. In this course, we focus on phenomena (data) which consist of one independent variable and one dependent variable. Depending on the context, the two variables are also referred to as input (variable) and output (variable), or cause (variable) and eﬀect (variable),orexplanatory variable and response variable, respectively. The data (also called data points) we have to work with could look like this: output (response variable)

input (explanatory variable)

It is (nearly) impossible to figure out a formula for a curve that would go through all these points (or even if we could do it, very likely such a formula would not be helpful). Why do we need to find a pattern, or, as is often called, model the relationship between the variables? The known data points determine the range of the independent variable (explanatory variable, input). We might need to know more about the data within the range (this is called interpolation), or about the data outside the range (called extrapolation. Assume that we computed a model for the population of Canada based on the census data for 1996, 2001, 2006, 2011 and 2016 (thus, the range of data is 1996-2016). Figuring out an estimate for the Canadian population in 2008 is a matter of interpolation, whereas predicting the population in 2025, or figuring out what it was in 1972 is a matter of extrapolation.

interpolation response

extrapolation extrapolation

range

explanatory

One of the simplest models are curves – and this is what we are going to do. In other words, we will try to ‘explain’ a relationship between the variables hidden in the given data by using mathematically well understood curves (such as lines or exponential graphs). 100 NUMERACY

To ‘explain’ means to approximate (in the sense that we will explain soon) the given data with a curve. If that curve is a line, the modelling process is called linear regression. In the case of one explanatory variable, it is also referred to as simple linear regression. For instance, the line that approximates the data given at the start of this section could look like this: output (response variable)

input (explanatory variable)

Here is how we will do linear regression: using software, we will model the data by a line (i.e., we will identify m and b in y = mx + b) and then (again, using software) ﬁgure out a number that will tell us how well the relationship given by the data can be explained by a linear function.

Example 6.2 Linear Regression Problem, Part 1 By the end of this section, we will ﬁgure out the question related to this data set:

Math 1LS3 course instructors are worried about the success of students in their course. Hence they are asking the following question: when we are half-way into the course i.e., looking at the grades on the ﬁrst two term tests, how well can we predict students’ ﬁnal grades? Section 6 Linear and Non-linear Relationships 101

The horizontal axis (explanatory variable) represents grades on the first two tests, and the vertical axis (response variable) are the final course grades, given as percent. For instance, students who had between 20 and 40 percent on tests 1 and 2 combined either failed the course, or received a grade below 60 % (except for one student, whose final grade was a bit over 70 %). We see that there is some kind of a trend - the better the grades on the first two tests, the better the final course grade. But how confident can we in claiming that, can we quantify our confidence?

We start small, with a data set consisting of two points: (0, 32) and (100, 212) (looks familiar?). We assume (have reasons to believe) that this data comes from a linear relationship, so we are looking for a linear model. We use the online linear regression calculator https://bit.ly/31qZf7s for all regression-related calculations. Of course, in this case we can just compute the equation of the line which goes through the two points – but we are learning how to use the calculator. (An alternative to using one of many online regression calculators is Excel.) Note that the calculator writes the line in the form y = A + Bx (and not the one we are used to: y = mx + b). Enter the data

and after you press Execute you will obtain the following output

From the table on the left we read the values of A and B, to identify the line y = A + Bx =32+1.8x. Note that this is the Celsius to Fahrenheit conversion formula! Compare the graph on the right with the one we drew earlier. (Soon we will explain what the correlation coeﬃcient is; we will not use the information about the means of the data points). In our next example, we consider a data set which consists of six data points: (2, 5), (3, 4), (1, 2.5), (6, 6), (5, 4.5), and (1, 5). First we plot them in a coordinate system: 102 NUMERACY

6 5 4 3 = response variable y 2 1

0 1 263 4 5 x = explanatory variable

Clearly, there is no line that goes through all six points, so instead we will try to identify the line that ‘best ﬁts’ the data points. The ‘best ﬁt’ will be found by considering all lines, and picking the one for which the sum of the distances (‘errors’) from the points to the line are the smallest (i.e., the sum of the lengths of the blue segments in the picture below).

6 5 4 3 = response variable y 2 1

0 1 263 4 5 x = explanatory variable

In other words, we assume that the data points are random deviations from a linear relationship between the explanatory and response variables. The line that we pick is the one (among all lines) for which the sum of these deviations is the smallest. That line is called a line of regression or a trendline.Inourcase, the line of regression is

6 5 4 3 = response variable y 2 1

0 1 263 4 5 x = explanatory variable

How did we ﬁnd the line of regression? We entered the six data points into the online regression calculator Section 6 Linear and Non-linear Relationships 103

and obtained the following output:

From it, we read the equation of the line of regression (with the coefficients rounded off to two decimal places): y = A + Bx =3.54 + 0.32x. Now we can interpolate: when x =4, a likely value for y is y =3.54 + 0.32 · 4=4.82 as well as extrapolate: when x =8, a likely value for y is y =3.54 + 0.32 · 8=6.10. How well a line of regression describes the data that generated it, or equivalently, how well a linear model explains the relationship between the variables is given by the correlation coefficient, which is usually denoted by r. In other words, the correlation coefficient measures the strength of a linear relationship between the variables. In the above case, r =0.56 (if you are interested: scroll the results page in the online calculator to find the formulas used to calculate the line of regression and the correlation coefficient r). What is important for us to know is that r and the slope of the line of regression (B) are of the same sign (i.e., if one is positive then the other is positive; if one is negative, the other is negative; and if one is zero the other is zero as well). The range of the values of the correlation coefficient is between −1and1, where 1 and −1 indicate strongest possible agreement with the linear model (go back to the example of two points – in that case we found the line that goes straight trough them, so the errors (deviations) are zero). To judge the strength of a relationship, we use the following convention; there is no general agreement on this, and what we state is only one possible interpretation. No need to memorize all this, just memorize the first and the last lines (i.e., the cases of strong correlation). ∗ When r>0.7, the correlation is called strong positive, and the line of regression is deemed to well represent the relationship between the variables. The line of regression has a positive slope and describes a positive correlation. ♦ If 0.4

Example 6.3 Illicit Drug Overdose Deaths in B.C. In the report A year of overdoses: 7 charts that show the scope of B.C.’s drug crisis (CBC News, 26 December 2016, https://bit.ly/2M43PER)wereadthat “despite declaring a public health emergency in the spring, despite more and more 104 NUMERACY

funding announcements and despite naloxone becoming more and more available, the number of illicit drug overdoses in B.C. grew to record heights this year.” Let us look at the numbers – on the same we page, we ﬁnd a diagram, part of which we reproduce here:

Assume that it is 2014, as part of identifying strategies to improve the situation, we need to estimate (predict) the number of deaths in 2016. (In other words, we are asked to extrapolate from the given data). To simplify, instead of taking all data points, we take the data from 2008 to 2014 (table on the left). As the data points seem to be close to a line (look at the graph above), we decide to model the relationship between the time and the number of deaths using a linear model; thus, we will do linear regression. Before we enter the data into our online calculator, we make one simpliﬁcation (not necessary, but useful): to make the numbers smaller, we re-label the years, so that the earliest data point (2008) is assigned the label 0. Then 2009 is 1, 2010 is 2, and so on, 2014 is 6; we are interested what happens in the year 8, which is a label for 2016. The input, as well as the results of calculatios are shown below.

Look at the input data to see how it was entered. The correlation coeﬃcient r =0.96 is very close to 1, so the two variables, the time (years) and the number of deaths, are strongly positively correlated. Positive correlation means that as one variable increases (i.e., the time increases from 2008 on), the other variable increases as well. Strong means that their relationship is very close to a linear – look at the graph (above right) to see that indeed the data points lie close to the line. The line of regression is given by y = A + Bx = 171.89 + 30.89x, which we interpret as number of deaths = 171.89 + 30.89 · year where (recall out labels) year = 0 represents 2008, and so on. Thus, the prediction for the number of deaths in 2016 is number of deaths in 2016 = 171.89 + 30.89 · 8 = 419.01. Section 6 Linear and Non-linear Relationships 105

We have data for 2016, so we can compare:

In 2016, there were over 900 deaths (914); so our model, predicting 419 deaths, severely underestimated the actual ﬁgures. What does this mean? Our extrapolation was based on the assumption that the phenomenon we studied would continue its (linear) trend. However, this did not happen, as the number of deaths increased in 2015, and then shot upward in 2016. In other words, this means that a linear model was not adequate as a predictor of what would happen in 2016. Of course, it’s easy to criticize once we know what happened. In reality, in 2014, people would try (and did) all kinds of models (we will explore exponential regression), and then look critically at all diﬀerent answers to try to make a good prediction. Recall that we picked only some of the available data. Of course, to have a better chance of obtaining a more suitable model, we should use as much data as available (however in this case, that would not have helped, as nothing in historic data suggests a large jump that occurred between 2015 and 2016).

Example 6.4 Linear Regression Problem, Part 2 Now that we know about linear regression, let’s quantify the relationship between the grades on the first two tests (explanatory variable) and the final course grades (response variable). Importing the Excel file into a regression calculator (or doing it in Excel), we find the line of regression y = A + Bx =18.95 + 0.7565x i.e., final course grade = 18.95 + 0.7565 · grade on the first two tests The correlation coefficient r =0.89588 tells us that the two variables are strongly positively correlated (thus, the better the first two tests, the better the final grade, and vise versa: the lower the first two tests, the lower the final grade). This, of course, is a general trend, and does not apply to every student. The above formula tells us that a student who has 70 % on the first two tests can expect that their final course grade = 18.95 + 0.7565 · 70 = 71.9 Someone who has 45 % on the first two tests can expect that their final course grade = 18.95 + 0.7565 · 45 = 52.99 106 NUMERACY

Notes: (1) Regression is one possible (but not necessarily reliable) way of making sense of a given data set – it gives a model based on a function that we choose (in our case, it was a linear function; later, we will use exponential regression). Keep in mind that regression is based on approximations and can in no way generate true values of the variables. Another way of modelling a data set is to use curve ﬁtting, where we compute a formula for a function whose graph contains all (or most) data points. Although it is of some use, curve ﬁtting cannot explain underlying mechanisms, so is not reliable when it comes to extrapolation. (2) Linear regression is one of the most commonly used tools (we will use it again when we discuss climate change), but its limitations are often not recognized, or worse yet, known but ignored. In particular, blindly applying extrapolation (say, based on a line of regression) could lead to wrong conclusions. To avoid this, we need to know something about the situation we are modelling. Here is an illustration. Consider the following data set which shows how some variable called Q changed over time, from 1965 to 2015. The second column, is the usual re-labeling of time (time = 0 is assigned to the earliest measurement), so that we work with smaller numbers.

Using our linear regression calculator https://bit.ly/31qZf7s, we obtained the line of regression y = A + Bx =38.624 − 0.248x, i.e., we modelled the quantity Q by the trendline Q =38.624 − 0.248 · (time since 1965)

The correlation coeﬃcient r = −0.992 signals a strong negative correlation. By extrapolating, we obtain predictions for future values of Q: in 2025, it will be Q =38.624 − 0.248 · 60 = 23.744, in 2055, it will be Q =38.624 − 0.248 · 90 = 16.301, in 2085, it will be Q =38.624 − 0.248 · 120 = 8.864, in 2115, it will be Q =38.624 − 0.248 · 150 = 1.424, and in 2145, it will be Q =38.624 − 0.248 · 180 = −6.016. Now we reveal that Q is the percent (ratio) of children years 14 or younger as part of the total world population. The data has been taken from the World Bank report Population ages 0-14 (% of total population) https://bit.ly/2scII7k. The extrapolated predictions do not make sense! Actually they might, for some time, but it is hard to believe that in 2115, less than 1.5 percent of the entire world population will be children age 14 or younger. Of course, the prediction for 2145 makes no sense. The problem here is that linear regression does not care what the data points actually are. These points could represent the ratio of children in the world population, or the average number of times dogs in Westdale bark in a single night, or the maximum grades on an Oxford University math test – the line of regression will be the same. As well, a correlation can be strong (as is the case here), however, extrapolation still turns out to be useless, inappropriate or unrealistic. Section 6 Linear and Non-linear Relationships 107

(3) Here is another example that illustrates the point we made in (2). Looking at the data points on the left, we can visualize the line of regression (centre graph), and extrapolate that whatever quantity is modelled, it will keep growing.

Now assume that the data on the left represents the altitude of a plane as it took oﬀ from an airport. Of course, the altitude cannot be increasing forever – at one point it will stabilize (cruising altitude), and then start decreasing as the plane will be landing. (4) In the paper Chocolate Consumption, Cognitive Function, and Nobel Laureates published in New England Journal of Medicine (see [1] https://bit.ly/2ZJ9ymx, available from within McMaster) we read a report based on linking data about the consumption of chocolate in certain countries and the Nobel prize winners from the same countries (the data is real, this is not a hoax). We read: “There was a close, signiﬁcant linear correlation (r =0.791) between chocolate consumption per capita and the number of Nobel laureates per 10 million persons in a total of 23 countries.”

The article continues: “When recalculated with the exclusion of Sweden, the correlation coeﬃcient increased to 0.862. Switzerland was the top performer in terms of both the number of Nobel laureates and chocolate consumption. The slope of the regression line allows us to estimate that it would take about 0.4 kg of chocolate per capita per year to increase the number of Nobel laureates in a given country by 1. For the United States, that would amount to 125 million kg per year.” Makes sense? 108 NUMERACY

First of all, just because two quantities are correlated does not mean that one causes the other. So people in the U.S. might eat all the chocolate in the world without seeing any increases in the Nobel laureates from their country. As well, a correlation could be due to a coincidence, and so there might be no relation between the variables at all. (5) Keep in mind that uncorrelated does not mean that the two variables are not related – it means that they are not related in the particular way (such as being in a linear relationship) that we tried to detect.

Exercise 6.5 Judging Correlations In each case, state whether the correlation is strong or weak and positive or negative, or alternatively, there is no correlation.

Partial answers: top left: strong positive correlation (the slope of the regression line is small but that’s not relevant for the strength of the correlation). Top right: no correlation (horizontal line represents a constant, i.e., a variable whose values do not change; as such it cannot represent the given data, where the variables involved do change). Bottom centre: weak positive correlation. Bottom right: weak negative correlation.

Example 6.6 Recognizing an (Approximately) Linear Quantity This left column in the table shows the actual values of a quantity, and in the middle column these values have been rounded oﬀ.

Judging by the actual values, the quantity is linear – all marginal changes (not shown in the table) are equal to 1.1. But when we look at the rounded off values, Section 6 Linear and Non-linear Relationships 109 and compute marginal changes (rightmost column), they are not all equal. Note that this happened because the numbers were rounded off. In reality, numeric data is often rounded off, and it is hard (or impossible) to know the true values. So we work with approximations, and in that sense we can say, based on looking at the centre and left columns in the table, that this quantity is approximately (almost, nearly) linear.

Quadratic and Cubic Relationships

Consider the infographics which accompanies the page Stopping distances: speed and braking (at https://bit.ly/2IZi9tN, produced by Queensland Government in Australia). The diagram shows how the reaction distance and the braking distance on dry and wet roads depend on the speed of a car. Reaction distance is the distance the car travels from the moment the driver realizes that they must stop the car until they hit the brakes. Braking distance is the distance the car travels from the moment the brakes are applied until it comes to a full stop.

Look at the reaction distance ﬁrst – is there a pattern?

It is approximately linear, as in Example 6.6. As a matter of fact, the relationship between the speed of a car and the driver’s reaction distance is linear, so 110 NUMERACY

this discrepancy is due to approximations coming from rounding oﬀ. The above page states that “In an emergency, the average driver takes about 1.5 seconds to react.” Thus, the reaction distance is (time times speed) = 1.5· speed, which is a proportional relationship, and thus linear! (Note: to check the values in the table we would have to convert units, as we cannot multiply seconds and km/h.) Now look at the braking distances on dry road, and compute the marginal changes in this case:

This time, the marginal changes are not equal (nor approximately equal), so the way the braking distance depends on the speed of the car is not linear. But we do see a pattern in the marginal changes – it’s approximately linear! There is a name for such quantities. A quantity changes quadratically if its marginal changes change linearly. (Keep in mind that for the marginal changes to be meaningful – so that we can compare them – the measurements for the variable have to be taken at equally spaced time intervals.) Here is the model of a quadratic relationship – squaring a quantity; in math terms, we write y = x2 or A = B2, whatever the names for the variables are.

Check that the calculations for the marginal changes in the table are correct. Thus, a quick way to check if a quantity is changing quadratically is to calculate the second marginal changes and show that they are all equal (or approximately equal).

Exercise 6.7 Recognizing a Quantity Changing Quadratically (1) Calculate the marginal changes and the second marginal changes for the quantity given below, to convince yourself that it is a quadratically decreasing quantity.

(Note that the times when measurements have been taken are given – to make Section 6 Linear and Non-linear Relationships 111

explicit the fact that they are equally spaced.) (2) Analyze the breaking distance on a wet road to show that it is approximately quadratic with respect to the speed of a car.

Note: the stopping distances infographics states that “Stopping distances increase exponentially the faster you go.” As we just saw, this is not the case – the increase is quadratic! There are many different patterns that can be used to model how a quantity changes, and some increase slower (or faster) than others. In the above sentence, the word ‘exponential’ has likely been used in to suggest a ‘rapid increase’, rather than to describe a specific, precisely defined pattern (exponential).

Example 6.8 Quantities Which Change Quadratically (1) The area of a square of side x is x2. For instance, the area of a square of side 3 is 9. If we double the side, the area is 62 =36, which is a 4-fold increase. If we triple the side, the area is 92 =81, which is a 9-fold increase. If we take one-third of the side, the area is 12 =1, which is one-ninth of the original area (note that (1/3)2 =1/9). Thus, the area increases (or decreases) as the square of the scaling that has been applied. In general, if we scale the dimensions of a two-dimensional object by the same constant, its area changes by the square of the scale. Take a 2 by 5 metres rectangle (thus, its area is 10 metres squared). If we scale its width and height by a factor of 3, the dimensions are 6 by 15, and the area 6 · 15 = 90 is a scale squared (i.e., 32 = 9-fold) increase of the original area. (2) In the case of no air resistance, the altitude of an object acting on by gravity only changes quadratically (could be an object falling, or thrown upward from, say, Earth’s surface). In reality, due to air resistance, a free falling object reaches its terminal velocity after some time, i.e., its velocity cannot increase beyond a certain bound (unlike what is suggested by the quadratic growth pattern). (3) A quantity which changes quadratically can be represented as a parabola, which is sometimes referred to as a U shaped curve. The picture below shows a standard parabola where the output values (i.e., the values on the y-axis) are the squares of the input values (i.e., the values on the x-axis).

(4) Psychologists have determined that life satisfaction, or equivalently, perception of well-being viewed as depending on age assumes a roughly quadratic shape: 112 NUMERACY

(Source: World Economic Forum report At what age does happiness peak? of 12 November 2015, at https://bit.ly/2TR5miR Try to explain the left and the right ends of the curve, as well as ﬁnd reasons for its lowest points. Here is a graph (see [2]) showing the WB (= well being) ladder, i.e., well-being scale with similar features. Think how you could deﬁne ‘mid-life crisis’ based on the two diagrams.

(5) Another way to characterize a parabolic shape is based on the properties of reﬂection by a mirror shaped as a parabola: rays (such as Sun rays or light rays) travelling parallel to each other and hitting a parabolic mirror reﬂect through a single point, called the focus of a parabola. (Figure source: OpenStax CNX Project: University Physics Volume 3, section 2.2 Spherical Mirrors, at https://bit.ly/30daq37.) Solar furnaces, automobile headlights, satellite dishes, etc. operate based on this principle. Section 6 Linear and Non-linear Relationships 113

(6) The article The U-shaped Life Cycle of Happiness (The Conglomerate, 20 Au- gust 2018, https://bit.ly/2Z0hoaq) discusses the inverted U curve approximating the probability of depression based on the age of the UK labour force.

(7) In Money can buy happiness, but only to a point CNBC News of 14 December 2015 (https://cnb.cx/2GMhrxg) shows the following diagram:

Look at the red curve in the diagram. What does an increasing piece of the curve represent? A decreasing piece? A level piece (i.e, the one that appears to be horizontal)? Interpret the piece of the curve that corresponds to the income of $ 80 and higher. The meaning of “95 % conﬁdence interval” is the following: the red curve was obtained by surveying a certain number of families and analyzing their replies. If the identical survey is repeated (with a diﬀerent choice of families) and the corresponding curves drawn, it is expected that in 95 % of the cases these curves would fall within the shaded range. I.e., if we repeat the survey 20 times and draw 114 NUMERACY

curves, we expect 19 of them to be within the range (since 19 out of 20 is 95%). That’s all we know – we cannot tell which of the 20 curves is not in the range, or if there is such a curve). (8) Imagine that you kick a ball upward with exactly the same force on four different celestial bodies, and plot its height (in metres) against the time (in seconds) the ball takes to fall back to the surface from where it was kicked. The plot is a reversed parabola, showing that the ball would reach different heights before starting to fall back. From physics we know that the bigger the mass, the stronger the gravity, and thus the stronger the force that prevents the ball from moving higher. The four diagrams represent the ball’s movements on Jupiter, Earth, Mars and Earth’s Moon. Note how big the differences are: the same force that you applied to kick the ball 10 metres high on Jupiter would send the ball a bit higher than 50 metres on Mars and about 120 metres on Earth’s Moon.

(9) Consider a relationship between the revenue a government receives from taxes and the tax rate. (In this example we consider a simple case where everyone pays the same percent of their income for taxes.) What curve could possibly represent such a relationship? Of course, collecting zero percent taxes yields no revenue. As the tax rate increases, so does the revenue (below, left). revenue revenue

02010tax rate (percent) 010 20 100 tax rate (percent)

But this trend cannot continue forever. Consider the other extreme: when the tax rate is 100%, we pay all of our income for taxes, and make nothing. In such a scenario, very likely no one would work (at least not for a job where they have to pay taxes), and the government revenue would be zero. Thus, there must be a point where increasing taxes starts to diminish the revenue, and so the relationship could look like the one above, right. Section 6 Linear and Non-linear Relationships 115

(10) The strength of a bone or a muscle is proportional to their cross-sectional area, and not to their length or mass. Weight-lifters, bodybuilders and Arnold Schwarzenegger know that to increase their strength, they have to work out to make their muscles wider, and not longer or just arbitrarily larger.

(Photograph source: CNN.com article Before Arnold Schwarzenegger was the ’Ter- minator’ at https://cnn.it/31ZttPb)

In the table below we recorded several values of quantities which are linear, quadratic, and – a new one – cubic. To obtain a value in the last column, look at the corresponding value in the ﬁst column and multiply it by itself three times (for instance, 6 · 6 · 6 = 216).

All three quantities increase, and as we can see, at diﬀerent rates. A cubic quantity changes much more quickly than others: for instance, when a linear quantity reaches 6, the quadratic is at 36, and the cubic is 6 times larger, at 216. Let’s look a bit closer at a cubic quantity by computing its marginal changes:

We might not recognize the pattern in marginal changes, but we certainly recognize the linear patern of second marginal changes. Recall that linear marginal changes indicate a quadratic quantity! Thus, a quantity is cubic if its marginal changes change quadratically. Quadratic and cubic are examples of power relationships, which we formally write as y = xn: n = 1 (linear), n = 2 (quadratic), n = 3 (cubic), n =4(fourth power), and so on. We use diﬀerent powers as one possible model for (i.e., to distinguish between) quantities which change at diﬀerent rates. 116 NUMERACY

Example 6.9 Cubic Relationships (1) The volume of a cube of side x is x3. For instance, the volume of a cube of side 1/2metresis(1/2)3 =1/8 metres cubed, and the volume of a cube of side 1.4mmis1.43 =2.744 mm3. How does the volume behave under scaling? From (mx)3 = m3x3 we see that the volume of a cube whose side has been scaled by a factor of m is the scale cubed times the original volume. This is true in general: if we double the radius of a sphere, its volume increases 8-fold (since 23 = 8). If we scale each of the length, the width and the height of a rectangular box by 0.1 (i.e., make them 10 times smaller), then the volume of the scaled box (because 0.1=0.001) is thousand times smaller than the original box. In practice, we usually state this result in the following way: if a linear dimension is scaled by a factor m, then the volume changes as m3 (recall that the area changes as m2). (2) If an object is homogeneous (i.e., built of the same material), then its mass is proportional to its volume. More technically, since density = mass/volume it follows that mass = density · volume Assume that dogs are homogeneous (which, in reality, is a very reasonable assumption). Pineapples (a real dog name) is twice as long as Fido (i.e., Pineapples’ linear dimension is twice as large as Fido’s linear dimension). Thus, Pineapples’ volume is 8 times the volume od Fido, and consequently, Pineapples’ mass is 8 times larger than Fido’s mass.

Example 6.10 Could King Kong (as Envisioned by Hollywood) Exist? An averge gorilla is about 6 ft tall, and has a mass of about 150 kg. Consulting multiple King Kong movies, we assume that King Kong is about 160 ft tall, i.e., its linear dimension is 10 times larger than the liner dimension of an average gorilla.

King Kong

average gorilla

(Monkeys shown to scale. Picture source: BoredPanda https://bit.ly/2Z54cpu) Section 6 Linear and Non-linear Relationships 117

An animal 10 times taller than an average gorilla would have the volume, and thus the mass, which are 103 = 1000 times larger. Thus, a real King Kong would weigh about 150, 000 kg, or 150 tonnes; for comparison, the heaviest land animal is elephant, whose mass can reach 7-10 tonnes. The heaviest known animal is blue whale, known to weigh at most 190 tonnes. Land animals (with calcium-based skeletons) cannot weigh nearly as much, as their bones would crumble under the weight. (The buoyancy of water reduces the weight in water.) Assume, for now, that weight is not a problem. Recall the fact that the strength of bones and muscles is proportional to their cross sectional area.So,a 10-fold increase in linear dimension means a 102 = 100-fold increase in area, and thus in strength – which is no match for a 1000-fold increase in mass. Thus, to match the increase in mass, the linear dimension of the cross-sectional area (of a bone or a muscle) needs to be such that its square is 1000. Since the square root of 1000 is approximately 31.6 (i.e., 31.62 = 1000), to match the mass increase, the width of the bones and muscles has to scale by a factor of 31.6. In conclusion, to properly scale an average gorilla to become King Kong, we have to make it 10 times taller and 31.6 times wider! But as Hollywood likes thin, they disregarded the calculus of growth we just did. So Hollywood’s King Kongs are ﬁction and cannot be real.

Example 6.11 How Bones Grow Human bones and muscles must grow (and they do) in a way to support the increase in body mass (otherwise we would not be living on this planet). Example: as a baby girl grows to twice her size (so linear dimension doubles), her volume, and thus her mass, increase eightfold. Recall that the strength of a bone is proportional to its cross-sectional area. So if a bone in the baby’s body grows so that its radius doubles, the cross-sectional area would quadruple – which is not enough to match the eight-fold oncrease in her mass. What is the proper scaling? That scale must be the number whose square is 8, which is (approximately) 2.83. In other words, to support an eightfold increase in mass, brought on by the babys dimensions all doubling, her bones must increase their radius by a factor of about 2.83.

(Source: see [3] in References, page 69). Repeat this argument to show that if a body grows to three times its original size (so the length of a bone triples), the width of the bone will have to grow by a factor of about 5.20 to maintain its strength. 118 NUMERACY

Note: this is a simplified, ‘back of an envelope’ calculation, which illustrates the principles of growth and development. The actual scaling factors could be different (but not much dfferent).

Example 6.12 Hot Paws: Hard to Find a Pattern The paper Thermal Contact Burns From Streets and Highways (see [4] in Ref- erences) discusses the relationship between high air temperatures and the corresponding temperatures of paved surfaces, such as walkways, streets, and highways. Part of the data is presented in this infographics, warning us to be aware of high temperatures when we walk our pets.

(Source: CrazyRebels.com https://bit.ly/30qSuT0) It is not possible to determine a pattern, as we do not have enough data (nor is air temperature equally spaced – so we cannot meaningfully compare marginal changes). The paper lists data only, and does not underline principles based on which we could ﬁgure out a pattern. As exercise, convert the temperatures in the picture into degrees Celsius.

Chapter references [1] Messerli, F. H. (2012). Chocolate Consumption, Cognitive Function, and Nobel Laureates. New England Journal of Medicine 367:1562-1564 DOI: 10.1056/NEJ- Mon1211064 [2] Stone, A. A., Schwartz, J. E., Broderick, J. E., and Deaton, A. (2010). A snapshot of the age distribution of psychological well-being in the United States Proceedings of the National Academy of Sciences U.S.A., Jun 1;107(22):9985-90. doi: 10.1073/pnas.1003744107. Epub 2010 May 17. [3] Adler, F. R., Lovric, M. (2015). Calculus for the Life Sciences: Modelling the Dynamics of Life. Toronto: Nelson Education. [4] Berens, J. J. (1970). Thermal Contact Burns From Streets and Highways JAMA, 214(11), pp. 20252027. doi:10.1001/jama.1970.03180110035007 Section 7 Quantities Changing Exponentially 119

7 Quantities Changing Exponentially

If we start with some quantity Q and keep adding the same (fixed) amount a to it (thus creating identical marginal changes), we obtain linear growth: Q, Q + a, Q +2a, Q +3a, Q +4a,... (if we keep subtracting a, we obtain linear decrease). What happens if we start with Q and keep multiplying by the same (fixed) amount r? For example, assume that a quantity Q grows by 9 percent per year. At the end of the first year, it will grow to 1.09Q. At the end of the second year, it will grow by 9% of the first year’s value: 1.09 · (1.09Q)=1.092Q. At the end of the third year, it will grow by 9% of the second year’s value: 1.09 · (1.092Q)=1.093Q. Continuing, we obtain exponential growth or exponential increase Q, 1.091Q, 1.092Q, 1.093Q, 1.094Q, 1.095Q,... Repeated multiplication by a positive number smaller than 1 generates exponential decrease or exponential decay: 1 1 1 1 1 1 Q, Q, Q, Q, Q, Q, Q,... 2 4 8 16 32 64 Remember: (1) Accumulation by addition (subtraction) creates a linear pattern, and accumulation by multiplication (division) creates an exponential pattern. (2) If, for some quantity, next value = present value · r then the quantity grows exponentially if r>1, and decays exponentially if 0 < r<1.

Exercise 7.1 What Works Better? You are negotiating a salary for a job scheduled to start on 1 January 2020 and end on 31 January 2020, and are given two options: (a) Accept $ 10 million as a salary for the job (b) Accept the following option: on 1 January, you are paid 1 cent; on 2 January, you are paid 2 cents, on 3 January 4 cents, on 4 January 8 cents, and so on; i.e., each day you’re paid double of what you were paid the previous day. Which option would you select, hoping to maximize your salary? Answer: Let’s ﬁgure out (b): start with 1 cent on 1 January, and double on each following day:

Does not look promising: on 15 January, your salary is a bit less than $ 164, a far cry from the $ 10 million in option (a). However, there is something that suggests better news – look at the marginal changes: on 2 January you were paid only 1 120 NUMERACY

cent more than on 1 January, but on 15 January we were paid $ 163.84 − $ 81.92 = $ 81.82 more than the day before. Continue the calculation:

Note how your salary starts to increase more quickly, and the marginal change on 24 January (compared to 23 January) is $ 41,943.04. Still far from 10 million, however:

The dollar amounts are skyrocketing, and on 31 January you take home about $ 10.7 million! By the way, that’s just the amount you’re paid on the last day. Your total salary is lots more, as you need to add up all the amounts you were paid from 1 January to 31 January.

In the diagrams we show the salary for the ﬁrst 20 days, and then for the entire month.

As the accumulation is multiplicative (each entry is equal to the ﬁxed number (2) times the previous entry), the growth is exponential. Note the distinct feature: a Section 7 Quantities Changing Exponentially 121

slow initial growth is followed by a more and more rapid increase (the units of the vertical axis in the diagram on the right are ten millions; 1e7 means times 107). Similarly, a tell-tale sign of an exponential decay is a rapid initial decrease, followed by a quick slow-down in the rate of decrease. The diagram shows the values of a quantity which loses 7 percent of its value per day.

Going back to the salary exercise, we would like to introduce a formula which tells us how to add the powers of a number: rn+1 − 1 1+r + r2 + r3 + ···+ rn = r − 1 For instance, when r =2andn =30 231 − 1 1+2+22 +23 + ···+230 = =231 − 1=2, 147, 483, 647 2 − 1 i.e., about 2.1 billion. Note that this is exactly the total amount (in cents) that we would receive under option (b) in Exercise 7.1. Converting, it gives a bit over 21.47 million dollars.

Note: it’s good to remember the values of there powers of 2: 210 =1, 024 is about a thousand (kilo) 220 =1, 048, 576 is about a million (mega) 230 =1, 073, 741, 824 is about a billion (giga).

Example 7.2 Fun With Exponential Growth and Decay (1) Read about the Wheat and chessboard problem, related to the powers of 2, on Wikipedia https://bit.ly/1OaaCXe. (2) Take a sheet of paper and fold it in half 20 times (in theory, because in practice you will not be able to do it). Assuming that a stack of 100 sheets is about 0.5 cm in height, how tall would the folded paper be? How tall would it be if you fold it 30 times? (3) Take an apple (of, say, diameter equal to 7 cm) cut it in half, then take one of the halves and cut it in half, and so on. What happens after you repeat this routine 10 times? 20 times? 30 times? Compare to the approximate diameter of a 122 NUMERACY

water molecule (which is very small in relation to other molecules), which is about 0.275 nanometers.

How to recognize a quantity which changes exponentially? One way is to remember that it’s a multiplicative accumulation: so the next value must be equal to a fixed number times the previous value (or equal to the previous value divided by a fixed number, whichever is more convenient). Keep in mind that this works, as in the case of marginal changes, under the assumption that the values we are looking at correspond to equally spaced values of the variable with respect to which the quantity is changing. Another way is to look at the marginal changes. The quantity in the table below is an exponentially increasing quantity, where the fixed number we keep multiplying by is 2.

Note that marginal changes show the pattern identical to the pattern that deﬁnes the quantity: to advance to the next marginal change, multiply the previous one by 2. In the following example the quantity is exponential (next value = 3 times the previous value), and its marginal changes follow the identical pattern:

Finally, the table below shows an exponentially decreasing quantity (note that the next value = 0.2 times the previous value, or the previous value divided by 5). Again, marginal changes follow the same pattern.

The quantities that we study in this course, in most cases, change with time (hence the references to time as the input/ independent variable). Exponentially growing quantity is characterized by the doubling time:itis the time needed for the quantity to grow to twice its original size. Exponentially Section 7 Quantities Changing Exponentially 123

decaying quantity is characterized by the half-life: it is the time needed for the quantity to decay to one half of its original amount. Assume that the doubling time of some quantity is T (in some time units, say hours). If its initial amount was q,thenafterT hours it will grow to 2q; after another T hours expire this new amount will double to 4q.Afterthreetime intervals T have expired (i.e., three doubling times), the quantity will reach 8q. In general, after n doubling times have expired, the quantity will reach 2nq. Likewise, assume that the half-life of some quantity is T. It will decay according to the following schedule:

Number of half lives Amount left zero q 1 one 2 q =0.5q 1 two 22 q =0.25q 1 three 23 q =0.125q 1 four 24 q =0.0625q 1 five 25 q =0.03125q 1 six 26 q =0.15625q ... 1 n 2n q Thus, after four half-lives, a bit over 6% of the original amount is left, and after five half-lives, the quantity is down to about 3% of its initial amount. Most common drugs (such as pain killers or caffeine) become ineffective in small amounts, and it’s generally viewed that after 4-5 half-lives they become ineffective. However, not all drugs decay according to exponential decay (for instance, the way our body absorbs alcohol follows a different non-linear regime). Below are half-lives of several drugs:

Comment on advantages and disadvantages of taking Zalkeplon vs. Doxylamine, or Aleve vs Advil. Radioactive decay is an example of exponential decay. The half-life schedule above shows why certain areas contaminated by radioactive fallout (such as Fukushima in Japan or Chernobil in Ukraine, following major disasters in nuclear power plants) are still uninhabitable (for humans): some radioactive substances have a long half-life (years, or decades); as well, some which decay more quickly are dangerous even in very small quantities.

Case Study 7.3 Breast Cancer: Clinical Exam vs. Mammogram In the abstract of the article Is clinical breast examination an acceptable alternative to mammographic screening? published in British Medical Journal (see [2] 124 NUMERACY

in References at the end of this section) we read “Breast cancer screening and mammography have almost become synonymous in the public perception, yet this should not necessarily be the case. Ideally, a screening tool for breast cancer would reduce mortality from breast cancer while having a low false alarm rate and being relatively cheap. Screening should not be at the expense of the symptomatic services nor inappropriately divert scarce resources away from equally deserving areas of the NHS [= National Health Services, UK equivalent of OHIP] that are less politically sensitive. An ideal screening test would be simple, inexpensive, and effective. Of the three modalities of breast cancer screening - breast self examination, clinical breast examination, and mammography - breast self examination fulfils the first two criteria, but early results of two randomised trials conducted in Russia and China suggest that it would not be effective in reducing mortality from breast cancer. Clinical breast examination is also relatively simple and inexpensive, but its ef- fectiveness in reducing mortality from breast cancer has not been directly tested in a randomised trial. Mammography is complex, expensive, and only partially effective. We believe that there is sufficient circumstantial evidence to suggest that clinical breast examination is as effective as mammography in reducing mortality from breast cancer and that the time has come to compare these two screening methods directly in a randomised trial.” Like other cancers, breast cancer begins with certain (not fully understood) changes within one cell, which then propagate as the cell grows and multiplies (by doubling, i.e., by producing two daughter cells). The doubling times depend on patient’s age, and possibly other factors. Here is information from [3]:

Age Median doubling time and interval (days) less than 50 80 (44-147) 50 − 70 157 (121-204) over 70 188 (120-295)

The doubling times vary quite a bit: for instance, for women aged 50-70, the shortest doubling time recorded was 121 days (about 4 months), and the longest was 204 days (almost 7 months). For one half of the women, the doubling time is below 157 days, and for the other half it is above 157 days (that’s what the median tell us). Going back to the article, we read: “[...] if you consider the exponential growth rate and doubling time of breast cancer you find that a single breast cancer cell has to undergo 30 doublings to reach a size of 1 cm, when it will contain 109 cells and be clinically palpable. Since the average size of a non-palpable, mammographically detected cancer can be assumed to be about 0.5 cm, the lead time gained by mammography over clinical breast examination would be of the order of only one doubling. Whether this lead time equivalent of one doubling in the natural course of 30 doublings would lead to a significantly greater reduction in mortality is questionable.” Let’s try to understand what the authors are saying. After 30 doublings, the cancer grows (ideally, assuming that no cells die) to 230 cells, which is about one billion (i.e., 109). So we verified the first statement. The term ’size’ in this context means a linear dimension (i.e., the diameter of the cancer as measured in a mammogram with a measuring tape/ruler). The doubling time is based on the cell count, so during one doubling time, the number of cells doubles. Thus, during one doubling the volume of the cancer doubles. Section 7 Quantities Changing Exponentially 125

How long does it take for a cancer to grow from the diamemeter of 0.5 cm to the diameter of 1 cm? It cannot be one doubling as claimed in the paper! If the diameter (linear dimension) doubles, then the volume would increase 8-fold, and not double. So, something is not right. We know that a volume depends on the third power of a liner dimension, and we write V = k · a3, where k is some constant and a is a linear dimension (k =1if a is the length of a side of a cube; k =4π/3ifr is the radius of a ball). If we want the volume to double, i.e., V =2k · a3, then the new linear dimension A must be 2k · a3 = k · A3, i.e., A3 =2a3, and A ≈ 1.26a. Thus, an increase of approximately 26% in the linear dimension guarantees that the volume doubles. With this in mind, we work backward: ♦ 30th doubling: 230 ≈ 1.07 billion cells, linear dimension = 1 cm ♦ 29th doubling: 229 ≈ 536 million cells, linear dimension = 1/1.26 = 0.79 cm ♦ 28th doubling: 228 ≈ 268 million cells, linear dimension = 0.79/1.26 = 0.63 cm ♦ 27th doubling: 227 ≈ 134 million cells, linear dimension = 0.63/1.26 = 0.50 cm! Thus, it takes three doublings for the cancer to grow from the size (linear dimension) of 0.5 cm (when it is detectable on a mammogram) to 1 cm (which is large enough to be detected in a clinical breast examination).

We conclude that the claim made in the paper: “the lead time gained by mammography over clinical breast examination would be of the order of only one doubling” is incorrect! It should be three doublings, which is a lot more signiﬁcant as a lead time. The source of error is the ambiguous vague term ’size,’ which the authors of the paper interpreted incorrectly as both linear dimension and volume. Epilogue: although the journal and the authors have been warned about the error, no one has decided to correct the paper, nor to publish a warning that one of its major statements is wrong.

If a quantity grows by R percent in a given time interval, when will it double? If it decays by R percent in a given time interval, what is its half-life? Note that here R represents a percent, so 17% means that R = 17 (and not 0.17). People working in ﬁnance very often do mental estimates of doubling time and half-life, using either the rule of 70 or the rule of 72. The rule of 70 (divide 70 by the rate R) gives a closer approximation for the half-life, and for the doubling time with the growth rate up to about 5%. The rule 126 NUMERACY

of 72 (divide 72 by the rate R) gives a better approximation of the doubling time when the growth rate is 5% or more. As the differences in estimates are not large, in practice either rule is used (often depending on convenience). The rule of 72 is sometimes easier for mental calculations, as 72 is divisible by many numbers: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. Likewise, we can use the rule of 70 what it’s easier to divide 70 by the given rate. Here is a sample of mental calculations one can do: if we invest $ 1,000 at a rate of 8% per year, our investment will double in approximately (use the rule of 72) 72/8 = 9 years. If we invest at a rate of 3.5% per year, our investment will double in approximately (use the rule of 70) 70/3.5 = 20 years. Inflation decreases money’s buying power (i.e., money is worth less over time). For instance, at a constant rate of inflation of 4% per year, it would take about 72/4 = 18 years for any amount of money (not invested, just kept, say in a sock) to halve. Under a 14% per year inflation, it would take about 70/14 = 5 years for any amount of money (not invested) to halve. In the table below we compare the exact values for the doubling time and the half-life with the approximations obtained from the rule of 70 and the rule of 72.

The formulas for the exact values are derived in the following example.

Example 7.4 Calculus: Computing the Doubling Time and the Half-Life A quantity which grows or decays exponentially can be described using the formula t q = q0(1 + r)

where q0 is the initial quantity. When r>0, then 1 + r>1, and q is increasing; when r<0, then 1+r<1, and q is decreasing. Here r is the decimal interpretation of the rate, so 17% is expressed as r =0.17. t To calculate the doubling time, we need to ﬁnd t so that 2q0 = q0(1 + r) , i.e., (1 + r)t =2. Using the natural logarithm, ln 2 t = ln(1 + r)

t t For the half-life, we need to ﬁnd t so that q0/2=q0(1+r) , i.e., (1+r) =1/2. Using the natural logarithm, ln(1/2) t = ln(1 + r)

As an example, assume the annual growth of 7%. The doubling time is ln 2 ln 2 t = = ≈ 10.24 years ln(1 + r) ln 1.07 Compare this exact value with the two approximations: the rule of 70 gives 70/7= 10 years, and the rule of 72 gives 72/7 ≈ 10.29 years. Assume the yearly decay of 7%. The half-life is ln(1/2) ln(1/2) t = = ≈ 9.55 years ln(1 + r) ln 0.93 Section 7 Quantities Changing Exponentially 127

Compare this exact value with the two approximations: the rule of 70 gives 70/7= 10 years, and the rule of 72 gives 72/7 ≈ 10.29 years.

Example 7.5 Quantities Changing Exponentially (1) The global consumption of plastic has been increasing at alarming rates, creating huge amounts of plastic waste. The diagram (taken from [4], see References), shows the trend in the total production of plastic since 1950s, when it emerged as a major material (the ﬁrst fully synthetic plastic was invented in 1907).

The units on the vertical axis are megatonnes (Mtones); recall that 1 megatonne is one million tonnes. Graph source: https://bit.ly/2koqm2Q. The authors of the paper [4] encourage recycling as “the best treatment for plastic waste since it cannot only reduce the waste but also reduce the consumption of oil for producing new virgin plastic.” The European Commission in-depth report Plastic Waste: Ecological and Hu- man Health Impacts (see [5]) shows an exponential increase in plastic production that ends around 2007, and is replaced by a downward trend:

The units are the same in both diagrams. Reading the reports, we realize that the differences in the two diagrams are due to their authors using different definitions of what plastic is. The European Commission report states an alarming statistics that we should all be aware of: “About 50 per cent of plastic is used for single-use disposable applications, such as packaging, agricultural films and disposable consumer items.” 128 NUMERACY

(2) The Economist on 15 May 2019 published a piece America’s avocado supply is set to tighten (https://econ.st/2JnJ3P2) in which they claim that “the market for avocados has undergone a transformation unlike that of any other fruit (yes, fruit) over the past few decades – from chic canape of the 1970s to millennial staple today. In 2018 Americans consumed 3.5kg (7.7lbs) of avocado per person – nearly four times the level in 2000. Even McDonalds is serving up guacamole. Yet the outlook for American avocado lovers is a concern.” The graph below shows the trend in avocado consumption. As the last sentence in the quoted text suggests, the (almost) exponential growth pattern cannot continue for much longer.

(3) The richer the people are, the fewer children they have. This statement is given evidence in the following graph (taken from The Economist, August 30 2019, at https://econ.st/2jRfVoe) which relates fertility (the number of children per woman) with the GDP (gross domestic product), which is an indicator of a country’s wealth.

The graph suggests an exponential decay. Section 7 Quantities Changing Exponentially 129

(4) Writer and blogger Darrin Qualman describes himself as “a civilizational critic, a researcher and data analyst, and an avid observer of the big picture.” In his blog post Another trillion tonnes: 250 years of global material use data (https://bit.ly/2kr0l2S) he writes “Our cars, homes, phones, foods, fuels, clothes, and all the other products we consume or aspire to are made out of stuﬀ – out of materials, out of wood, iron, cotton, etc. And our economies consume enormous quantities of those materialstens-of-billions of tonnes per year.”

The legend points to different sources of information for the graph. Qualman continues: “The graph above shows 250 years of actual and projected material flows through our global economy. The graph may initially appear complicated, because it brings together seven different sources and datasets and includes a projection to the year 2100. But the details of the graph arent important. What is important is the overall shape: the ever-steepening upward trendline – the exponential growth.” (5) On its web page https://bit.ly/32ftWwA, Monex (Australian securities company) tracks worldwide profits of Netflix (media service provider and production company):

In the commentary, we read: “Netﬂix increased annual revenue 35% to $ 16 billion in 2018 and nearly doubled operating income to $ 1.6 billion with an operating proﬁt margin of 10%” 130 NUMERACY

Example 7.6 7 Percent per Year Growth By the rule of 70, a quantity which grows at 7% per year will double in about 70/7 = 10 years. It is amazing how many diﬀerent quantities show this trend. Here we explore a few of them. (1) The UK Newspaper telegraph report Cost of university accommodation doubles in 10 years (at https://bit.ly/2Lj6Ka3) states that “The price of a room in university halls of residence has doubled in just 10 years amid rising fears over student debt levels, a major report has found.” (2) Statistics New Zealand (https://bit.ly/2zDX58t) presents data which proves the claim made in the title of their report Kaumatua population [in New Zealand] doubles in 10 years (Kamatua is respected elder of either gender in Maori community. In this context, it refers to people aged 80 or older in general.)

Compare 2002 and 2012 data for both males and females to check the claim. (3) On 7 September 2016, BBC News published a report titled Number of children who are refugees doubles in 10 years (report and photo: https://bbc.in/2PyD123)

We read: “The number of children who are refugees has doubled in the last 10 years says charity UNICEF. Many of the children have escaped from danger and Section 7 Quantities Changing Exponentially 131

wars in countries like Syria, Afghanistan, Eritrea or Iraq. The report by UNICEF says that more than half of all refugees in the world are children.” (4) In Saskatchewan diabetes more than doubles in 10 years, part of worldwide trend, CBC News (6 April 2016, https://bit.ly/1T0q4G6) reports that “over a quarter of province’s population has been diagnosed with diabetes or pre-diabetes; An estimated 97,000 people in Saskatchewan have diabetes, according to the Cana- dian Diabetes Association. That’s a 59 per cent increase from 10 years ago, and part of an international trend.” (5) In the video report Digital: NY Student Loan Debt Doubles in 10 Years NCC News (21 September 2016, https://bit.ly/2MKaFQp) we hear that “the average New Yorker with college loads owed $ 22,200 in 2015 [...] Student load debt has more than doubled in NY in the last decade.” (6) In the article World heritage tourists destroying the sites they love New Zealand Herald (11 July 2018, https://bit.ly/2AJYptI)iscriticalofthewayhow,evenwith best intentions, tourists are bad news (so much so that local authorities worldwide started limiting the number of people visiting certain locations). As well, we ﬁnd this statement: “The number of people travelling by air internationally has increased by an average of around 7 per cent a year since 2009. This growth is expected to continue at a similar rate.”

Example 7.7 Exponential Model is Sensitive With Respect to its Exponent As we have seen, after some initial period, a quantity growing exponentially picks up the pace and increases at larger and larger rates. By comparing two exponential growth patterns, in our case 1.1x and 1.11x, we discover that, on top of rapid increases, the two patterns grow apart from each other as well (even though there is a diﬀerence of one hundredth between the two numbers 1.1and1.11), and that pattern is also exponential:

When x = 100, the two quantities are of the same order of magnitude. How- ever, when x =1, 000, 1.11x is 4 orders of magnitude larger than 1.1x, and when x =10, 000, it is 40 orders of magnitude larger than 1.1x.

Note how large the numbers in the table have become. There is absolutely nothing in universe that we could relate these numbers to. For instance, the total number of elementary particles in universe is believed to be somewhere between 1078 and 1082 atoms. Even if we wish to measure the diameter of the known universe by using a stick whose length is the diameter of an electron we would not reach nearly these numbers. Hence an important conclusion: an exponential model, no matter what the quantity involved, becomes unrealistic after some time. In other words, a quantity growing exponentially, sooner or later, will change its growth pattern. (This is not true just for exponential growth, but for ant model which assumes unlimited growth, such as a line with positive slope.) 132 NUMERACY

Exponential Regression

Of all exponential growth patterns (functions), there is one that we view as special, and call it a natural exponential function: y = ex, where e =2.71828 ... is a decimal number which keeps going (i.e., has infinitely many decimals), in such a way that there is no periodicity in its decimals (such numbers are called irrational). Why someone would choose e to call “natural” (as opposed to, say, 2, or 3, or 10) is clarified in calculus courses. In real life situations, of course, we use the one that’s most convenient. One reason why we need natural exponential functions is for exponential regression, as many online calculators work with it. Recall that regression is an attempt at describing a data set by an object which is easier to work with, such as a curve (in the case of linear regression, it’s a line). Knowing linear regression does not suffice, as there are phenomena (data sets) which cannot be described well by a line. For instance, the data set below represents cranial capacity (brain volume) of humans dating about 3 million years (the diagram has been drawn based on data from [1]):

In this case, as suggested, an exponential function could represent (we can also say – summarize) the data.

To ﬁgure our exponential regression we use an online calculator, as we did for the linear regression. The link is https://bit.ly/31dfxAj As illustration, we consider Stats Canada information (or Wikipedia, under Pop- ulation of Canada, Census data) on the population of Canada:

Based on the 1966-2006 data, we will obtain an exponential regression model, which we will use to predict the population in 2016, and compare with the actual Section 7 Quantities Changing Exponentially 133

2016 Census data. As in the case of linear regression, we enter the data into the calculator, relabelling the years so that the earliest data (1966) is represented by zero. (Thus, 1976 is 10, 1986 is 20, and so on.)

The top line tells us that we are looking for the model y = AeBx. Hitting ’Execute,’ we obtain the following output:

The correlation coeﬃcient r ≈ 0.998 indicates a strong correlation; thus, the population data we have can be well described/ explained by an exponential model (note how the data points lie very close to the exponential curve). This exponential model is given by y =20, 236.058e0.01141x where y represents the population (in thousands) and x is the tme, measured from 1966. We rounded oﬀ A to three decimal places, and B to 5 decimal places (recall that exponential functions are sensitive to their exponents, as illustrated with the 1.1x vs. 1.11x example). This model extrapolates that, in 2016, the population of Canada is y =20, 236.058e0.01141(50) =35, 800.65 i.e., about 35.8 million. Compared to actual data, we see that the model gives an overestimate; in other words, the population grew slower than the exponential pattern based on the given data. We will look at further examples of exponential regression in our study of climate change and human population.

Limited Growth

Resources (food, water, energy, etc.) are essential for human (and animal, plant) survival on our planet. Nature, excluding humans, is able to self-regulate and con- 134 NUMERACY

trol the balance between the demand for resources and their availability. However, humans are the problem. As human population grows, so does the demand for resources. The simplest, but accurate model of this dynamics is shown in the diagram below. The resources we need to live are growing, but in a linear fashion (for instance, we build another power plant, convert 10,000 hectares of desert into farmland, etc.). However, the demand, based on the exponential growth of human population, is exponential.

demand for resources

resources resources

time when?

Assuming that these trends continue, there will be a time when the demand for resources will meet, and then overcome the resources available – which is going to be a huge problem. When, or if, this will happen is not clear. As our planet is a limited environment, all resources are limited. So neither linear nor exponential growth models can predict what will happen in a more distant future. To understand things, we need to do is to develop limited growth models. One of the most commonly used models of limited growth is called a logistic model, visualized by the logistic or S-shaped curve.

carrying capacity growth rate approaches zero

growth rate slows down

output / response inflection

rapid initial growth rate

input / explanatory

Logistic model is characterized by an initial period of rapid growth, that mimics exponential growth. For instance, when an infectious disease emerges, its initial spread within a population is exponential (as there is a large “pool” of people who could contract it). After some time, the growth slows down (it’s getting more difficult for a disease to find people who are not infected); so although the quantity we study (say, the number of infected people) still increases, it does so at a decreasing rate. The value where this happens is called an inflection. Finally, in the long run, the growth rate approaches zero, i.e., the quantity stabilizes. We also say that it reaches its carrying capacity, or that it plateaus. This is one possible way to model limited growth. Note that although this model is more appropriate in many circumstances, it assumes that the quantity remains at its plateau. Sometimes, that could be the case (for instance, an animal population achieving ecological balance, where its size matches the resources Section 7 Quantities Changing Exponentially 135

available), but sometimes not (an infectious disease will eventually die out, and the number of infected people will be decreasing). We can explain logistic growth pattern in terms of its marginal changes.

As they describe a growing pattern, all marginal changes are positive. Initially, they increase (drawn in red), until the growth reaches the point where the marginal changes start decreasing (green). The moment when the pattern changes (red- green rectangle) is the inflection. For quick estimates, it is good to remember a mathematical fact that the value of the quantity (which increases according to the logistic model) at inflection is one half of its carrying capacity (plateau). In reality, all growth is limited, but its pattern, of course, could be different from logistic.

Example 7.8 Limited Growth, Logistic Curve, Plateau (1) The diagram below shows airspeed records (in km/h) in the last century (in other words, how fast was the fastest airplane at the time). We recognize the tell-tale signs of an (initial) exponential pattern: slow initial increase followed by a stronger and stronger growth. However, after some time, the growth shows the signs of slowing down, thus suggesting logistic growth. (Source: Wikipedia Flight airspeed record https://bit.ly/32AJc70.)

(2) The Rockefeller University web publication Bi-Logistic Growth discusses logistic growth model and suggests an improvement, named bi-logistic growth, which combines two logistic curves (https://bit.ly/34PzQGe). We read “Many processes in biology and other ﬁelds exhibit S-shaped growth. Often the curves are well modeled by the simple logistic growth function, ﬁrst introduced by Verhulst in 1845. Although the logistic curve has often been criticized for being applied to 136 NUMERACY

systems where it is not appropriate, it has proved useful in modeling a wide range of phenomena.” The following diagram, taken from the publication, shows the growth of a sunﬂower (height vs age):

(3) The number of mobile phone subscriptions shown in the following diagram has been taken from Our World in Data (https://bit.ly/2Q33KCx)

Note that the vertical axis represents the number of cellular subscriptions per 100 people. ITU (International Telecommunications Union) estimates that by the end of 2018 there were 8.2 billion mobile subscribers worldwide, which averages 107 mobile phones for 100 people.

(4) In his blog bit-player, the science writer Brian Hayes discusses (among other things) trends in world economics (https://bit.ly/2pOsHai). He says: “Writing in Section 7 Quantities Changing Exponentially 137

The New York Times, the business columnist Eduardo Porter quotes Paul Ehrlich quoting Kenneth Boulding: ‘Anyone who believes exponential growth can go on forever in a ﬁnite world is either a madman or an economist.’ Then Porter, siding with the economists if not the madmen, proclaims that economic growth must continue if ‘civilization as we know it’ is to survive. (He doesnt explicitly say the growth needs to be exponential.)” Then Brian Hayes presents his vision of historic and future economic growth:

He explains: “For hundreds or thousands of years before the modern era, average wealth and economic output were low, and they grew only very slowly. Life was solitary, poor, nasty, brutish, and short. Today we have vigorous economic growth, and the world is full of wonders. Life is sweet, for now. If growth comes to an end, however, civilization collapses and we are at the mercy of new barbarian hordes (equipped with a diﬀerent kind of horsepower).”

(5) Godwin’s law, when originally formulated, referred to newsgroup discussions. These days it is more broadly used, but more or less in the same context. Accord- ing to Wikipedia https://bit.ly/2q1kn75 “As an online discussion grows longer, the probability of a comparison involving Nazis or Hitler approaches 1. If an online discussion (regardless of topic or scope) goes on long enough, sooner or later someone will compare someone or something to Adolf Hitler or his deeds, the point at which eﬀectively the discussion or thread often ends.”

1 to Hitler of Nazis probability of reference length of online discussion

Note: In linguistics, the term “inflection” represents the change of form that words undergo to accommodate for number (egg vs. eggs), tense (go vs. went), gender, person, case, etc. Inflection is also used to signify change, or a particular moment in the process of change. For instance, in the report Hong Kong airport shuts down amid pro- democracy protest by CP24 News (12 August 2019, at https://bit.ly/2CvsSd6)we read that “One of the world’s busiest airports cancelled all flights after thousands of pro-democracy demonstrators crowded into Hong Kong’s main terminal ...” Characterizing thousands of people fighting for freedom as terrorists, the regime in Beijing stated that “‘One must take resolute action toward this violent criminality, showing no leniency or mercy,’ said the statement, attributed to spokesman Yang Guang. ‘Hong Kong has reached an inflection point where all those who are concerned about Hong Kong’s future must say ‘no,’ to law breakers and ‘no’ to those engaged in violence’.” 138 NUMERACY

The title of the article A Historic Inﬂection Point In Capitalism’s Battle Against Climate Change published in the Forbes Magazine on 26 April 2019 (https://bit.ly/2WYg1to) announces that the authors are reporting on some major change (in this case, an advancement in direct air capture technology).

Logarithms

The paper Hopes for the Future: Restoration Ecology and Conservation Biology (see [6] at the end of this chapter) discusses restoration ecology and its importance for the preservation of our nature and natural resources. In the abstract, the authors write: “Conversion of natural habitats into agricultural and industrial landscapes, and ultimately into degraded land, is the major impact of humans on the natural environment, posing a great threat to biodiversity. The emerging discipline of restoration ecology provides a powerful suite of tools for speeding the recovery of degraded lands. In doing so, restoration ecology provides a crucial complement to the establishment of nature reserves as a way of increasing land for the preservation of biodiversity. An integrated understanding of how human population growth and changes in agricultural practice interact with natural recovery processes and restoration ecology provides some hope for the future of the environment.” Later in the article we ﬁnd the following diagram depicting the relation between the spatial scale of natural (ovals) and human caused disasters (rectangles) and the recovery time.

Note that the scales (tickmarks) on the axes are not linear: the same amount of space is used for the numbers between 1 and 10 as between 10 and 100 or between 100 and 1000. (Recall that on a linear scale, the numbers (labels) 1, 2, 3, 4, etc. are equally spaced). What is this scale, why was it used, and how was it done? Some disasters aﬀect small areas: for instance a lightning strike aﬀects about one hundredth (10−2) of a square kilometre; a land slide could destroy about one square kilometre area (hundred times larger than a lightning strike). Industrial