MATH 2UU3 * GLOSSARY OF TERMS blue: covered so far black: to be covered

Quantitative reasoning: Applying and correct logical reasoning about math symbols, objects, calculations, algorithms, ideas in areas outside mathematics (meaning in context which is not abstract mathematics).

Prime number: A positive integer that has exactly two divisors; 1 and itself. [we will not talk about prime numbers; this was to illustrate the concept of a math definition, so no need to memorize this definition]

Definition: In math, and elsewhere (e.g., legal documents, financial agreements), it introduces something new (new term, concept, idea, object) based on previously defined terms, concepts, ideas or objects. Math definitions are precise, clear, unambiguous, and universal (do not depend on culture, political system, human condition, etc.). Once established they very rarely ever change.

Red herring: An expression which is misleading, confusing, or created to be misleading or confusing. It could also be a sentence, or a statement, which, by introducing certain sub-plot(s), draws the attention away from the actual issue or problem.

Correlation: Existence of a noticeable pattern (also called relationship, or trend, or association) between two quantities (or variables). For instance, if input/explanatory variable and output/ response variables both increase, or one increases when the other decreases (say over time), they are said to be correlated. Note: correlation does not mean causation. Just because a correlation exists does not mean that changes in one variable actually cause changes in the other variable. This might be the case, but there could also be a third variable that is causing the changes in both other variables; as well, there could be other possibilities, including coincidence.

Positive correlation: One quantity (variable) increases when the other increases. Or, one quantity (variable) decreases when the other decreases.

Negative correlation: One quantity (variable) decreases when the other increases.

Theorem: A mathematical statement that has been rigorously shown to be true (i.e., “proven true”) through logical reasoning from a set of axioms, or from previously established (proven) theorems. In other words proof is mathematically acceptable evidence. See implication.

Implication: a logical statement of the form “IF (assumption/s) THEN (conclusion/s).” This is the form of every theorem. An implication is valid (i.e., can be used) only if all assumption/s is/are true. If even one assumption fails to hold, the implication (theorem) cannot be used. However, the conclusions/s might still be true, but they do not follow (are not caused) by the implication.

Causation: outside mathematics, an implication is known as causation i.e., as a sentence that establishes a causal relationship. Instead of “A implies B,” we say “A causes B,” “B is a consequence of A,” “B is caused by A,” “B happened because of A,” “A is a reason for B,” etc.

Converse: The converse of the implication “IF A THEN B” is “IF B THEN A.” The converse of a true implication is not necessarily true.

Contrapositive: The contrapositive of the implication “IF A THEN B” is “IF NOT B THEN NOT A.” The contrapositive of a true implication is always true, and the contrapositive of a false implication is always false.

Equivalence: If A causes B and B causes A then we say that A is equivalent to B. In other words, IF A THEN B and its converse IF B THEN A are both true, then A is equivalent to B.

Negation: a logical structure; is we say that B is NOT A means that B is anything that is not A; for instance if B=NOT a cat, then B can be a dog, a chair, a planet, DNA molecule, etc.

Trend, association: see correlation.

Quantifier: Specifies elements of a given set which have some property. Universal quantifier (”for all”) signifies that all elements in a given set have that property. Existential quantifier (“there exists,” or “there is”) signifies that at least one element in a given set has that property.

Hypothesis: see conjecture

Conjecture or hypothesis: statement that is formulated based on identifying a pattern, or supported by data; however, there is no definitive proof of its validity. Once such proof is established, a conjecture becomes a theorem.

Replicability: Obtaining same, or very similar results as outcomes of a repeated experiment. In other words, an experiment is established as having a valid conclusion if its results can be replicated.

Randomized control trial: is a study/ experiment, in which participants are divided randomly into two groups: treatment group (that is under some kind of intervention, e.g., given a pill that is supposed to reduce pain) and control group (that is not under intervention, and is often given a placebo, e.g., a pill that does nothing). Based on the differences in the reactions of the two groups, and through statistical analysis, one can establish whether or not the intervention is effective.

Meta-analysis: a summary, or a critical analysis, of multiple studies.

Axiom: A mathematical fact (or a fact in general) that we take for granted, to start a math theory (or other scientific theory); thus, we cannot prove that an axiom is true (e.g. of an axiom from calculus: x*y=y*x for any two real numbers x and y)

Algorithm: A step by step process or set of rules for accomplishing some task. An example was given in class: Luhn algorithm that verifies whether or not a given number is a legitimate credit card number.

Estimate: A calculation or an algorithm that involves making assumptions about something that cannot be directly measured (or is hard or impractical to measure), or when exact value of some data is not (or could not have been) known. When making an estimate, we need to write down all assumptions that we made, and list the sources of these assumptions.

Approximation: A value that is nearly exact, or close to a true value (which might, or might not be known, or knowable).

Scientific notation: A method of writing concisely by using powers of ten (e.g. 0.00278 = 2.87*10^-3, 10000 = 1.0*10^4). Convenient for very small and very large numbers. To write a number in scientific notation means to write it in the form D.dddd * 10 ^ exponent, where D is a non-zero digit, dddd are decimals and the exponent could be positive, negative, or zero.

Order of magnitude: Very rough way to compare numbers. It refers to the powers of ten in scientific notation. For example, 4.7*10^7 has order of magnitude 7, and is one order of magnitude bigger than, say, 3.9*10^6. The number 4.3*10^-5 has order of magnitude -5.

Nanoscale technology: manufacturing objects (or structures) whose size is about a nanometre, i.e., one billionth of a metre.

Absolute number: A number quoted as is, without reference to anything else (for example: I paid $2.45 for bananas; there are about 500 thousand people in Hamilton)

Relative number: A number related to something else (for instance price per unit: bananas cost $0.69 per pound; or comparison: by population size, Hamilton is 10th largest city in Canada)

Absolute change: A direct comparison of the values of two quantities, computed by subtraction (by how much is one quantity larger than the other quantity?) In particular, if some quantity changes from A to B, then the absolute change is B-A.

Relative change: A comparison of two quantities by expressing one as proportion (or percent) of the other. In particular, if some quantity changes from A to B, then the relative change (relative with respect to the initial value) is (B-A)/A.

Positional number system: number system where the position of a digit determines its value; for instance, 4 in 3456 represents 4 hundred, whereas 4 in 2249 is forty (four tens). Example of a non-positional system are roman numerals, where the same symbol always has the same value. For Instance, all Cs in MCCCXIX and all C’s in DCCV have the same value of 100.

Decimal number system: Number system that we use daily, based on powers of 10, which uses Hindu-Arabic numerals 0, 1, 2, 3, …, 8, 9.

Binary number system: Number system based on powers of 2; uses digits 0 and 1; use in computers, data transmission, etc.

Hexadecimal number system: : Number system based on powers of 16; uses digits 0, 1, 2, ..., 9, and letter equivalents A, B, C, D, E, F for 10, 11, 12, 13, 14, 15 respectively; use in computers, data transmission, etc.

Vigesimal number system: Number system based on powers of 20; uses pictorial representations for digits 0, 1, 2, 3, …, 19. Still, in some forms, in sporadic use.

Sexagesimal number system: Number system based on powers of 60; uses pictorial representations for digits 0, 1, 2, 3, …, 59. This is why an hour has 60 minutes and a minute has 60 seconds. (and also why the right angle has 90 degrees).

Buy rate: In currency exchange, a buy rate (common meaning) is the rate at which a bank buys currency from individuals. Sometimes, however, it’s the rate at which an individual buys a currency from a bank. To figure which of the rates posted is a buy rate, we keep in mind is that it is the one that works against us.

Sell rate: In currency exchange, a sell rate (common meaning) is the rate at which a bank sells currency to individuals. Sometimes, however, it’s the rate at which an individual sells a currency to a bank. To figure which of the rates is a buy rate, we keep in mind is that it is the one that works against us.

Proportional relationship: Quantities A and B are called proportional if one is a multiple of the other, i.e. A = some number * B. Examples of proportional relationships: proportional scaling, percent, currency exchange, conversion of units.

One plus rule: This rule makes understanding phrases like “increased by 26%” easier, by representing the change as 1 plus the percent increase. For example, 126% = 1.26 = 1+0.26 = 1 +26%, i.e., the original quantity has increased by 26%. Another example: a quantity that is tripled (e.g. 1 becomes 3) has increased by 200%, and can be written as 3 = 1 + 2 using the “one plus” rule.

One minus rule: This rule makes understanding phrases like “decreased by 15%” easier, by representing the change as 1 minus the percent decrease. For example, 0.85= 1-0.15 = 1-15%, i.e., the original quantity has decreased by 15%.

Outlier: Refers to data points on a plot that fall outside of the trend. For example, in a data set with a clear upward trend, a large value of the independent variable that corresponds to a small value of the dependent variable would be an outlier since it is not explained by the upward trend.

Scatter plot: Visual representation of a data set in a graph/coordinate system. Data represented this way can be difficult to make sense of (just by looking at it), and usually requires additional calculations or processes to identify possible trends and draw conclusions.

Line: Graph of a function of the form y=A+Bx, where B is a constant that represents the slope of the line; slope says how y changes for a unit change in x. Other formulas are also used, such as y=mx+b.

Function: Mathematical relationship between two quantities x and y, expressed as y = f(x), where x is an independent variable, and y is a dependent variable. What is special for functions is that the value for y is uniquely determined from the given value of x.

Independent variable: Also called explanatory variable, input, or cause. In the notation y=f(x), x is the independent variable. The independent variable on a graph represents some information or data that is selected in order to observe the resulting output. In a coordinate system, it is represented on the horizontal axis.

Dependent variable: Also called response (or reaction) variable, output, or effect. In the notation y=f(x), y is the dependent variable. The value of the dependent variable is determined uniquely by selecting the value of the independent variable. In a coordinate system, it is represented on the vertical axis.

Linear function: Function of the form y=A+Bx, where A and B are constants. The graph of a linear function is a line of slope B, which crosses the vertical axis at A. If A=0, then the linear function y=Bx represents a proportional relationship.

Range: Range refers to an interval on the horizontal axis (i.e. an interval in the independent variable) from a smallest/earliest data point to a largest/latest data point). It can also refer to an interval on the vertical axis (again, from the smallest to the largest values). For instance, we can say “the values of the variable are in the range from 10 to 90,” or “the output is in the range from 200 to 300”).

Interpolation: A method of interpreting data by using the trend (i.e., the graph derived from the data, for instance by regression) to draw conclusions about data points inside the data range.

Extrapolation: A method of interpreting data by using the trend (i.e., the graph derived from the data, for instance by regression) to draw conclusions about data points outside the data range.

Marginal change: Marginal change is the change in the dependent variable which occurs as a reaction to a unit change (or small change) in the independent variable. For a straight line, the marginal changes are all the same, and equal to the slope of the line (i.e. for a line with slope 3, the y-value increases by 3 units each time the x-value increases by 1 unit). In a parabola y=x^2 the marginal change increases (e.g. the change from 3^2=9 to 4^2=16 is smaller than the change from 4^2=16 to 5^2=25, and so on). A function with positive marginal changes is said to be increasing, while a function with negative marginal changes is said to be decreasing.

Correlation coefficient r: The strength of the correlation is denoted by r, which takes a value between -1 (strongest negative correlation) and 1 (strongest positive correlation). The correlation is strong if it satisfies r>0.7 (strong positive correlation) or r<-0.7 (strong negative correlation).

Line of regression: Also called a trend line. A line of regression is a line that is constructed from the given data set to represent the trend in it (if it exists). The line of regression can then be used to extrapolate the trend to data points outside the range. How well the line of regression represents the data is measured by the correlation coefficient.

(Mathematical) model: Construct which is used to describe something in real life (which, dues to complexity of real life, is only an approximation). Regression lines and regression curves are examples of models.

Logistic curve: A logistic curve is an S-shaped curve which is increasing, and initially has increasing marginal changes and then switches to decreasing marginal changes. The point where the marginal changes switch from increasing to decreasing is called an inflection point. Logistic curves have been used to model limited population growth, where the population initially grows exponentially, and then slows down due to limited resources available. Logistic functions for population growth include a carrying capacity, which is a theoretical maximum number that the population can reach given available resources. The value of the logistic function at the inflection point is half of the carrying capacity.

Inference (statistical inference): An inference is a conclusion that something is true of a certain population based on that thing being true of a sample of that population. For example, researchers might discover a certain gene present in a sample of 100 people, and use that information to infer that the gene is present in a larger population from which the sample was taken.

Parabola: Graph of the function y=x^2, or its modifications (shifts, scales or reflections). In practice, it’s often referred to as the “U shaped curve” (opens upward; has a minimum value) or “inverted U shaped curve” (opens downward, has a maximum value).

Curve fitting: A method of finding (“forcing”) a curve to fit a set of data, without being guided by some principle. Curve fitting is commonly used, but often leads to erroneous extrapolation predictions.

Flops: abbreviation for “floating point operations per second” (floating point means real numbers). It is used to measure how fast a computer can process information.

Weighted average: The usual average applies equal weights to each item. A weighted average assigns higher weights to some items and lower weights to others, with the restriction that the weights must be positive and add up to 1. Higher weights are assigned to more important (more relevant) items, i.e., those which are supposed to have a larger impact.

Population: The entire set of individuals about which we would like to claim something (numeric).

Sample: A subset of the entire population from which we collect data.

Exponential function: A function of the form y = ax, where ‘a’ is the base and ‘x’ is the exponent. For example, y=1.12x and y=0.9x are exponential functions.

Exponential model: A function of the form y=A*e^B, where e^B is the natural exponential function. In certain situations, it is more convenient to use y=A*10^B is also used.

Doubling time: time needed for an exponentially increasing quantity to double. There are several rules of thumb to figure out doubling time: if it’s exponential growth (modelled by e^rt) then the doubling time is 69.3/r or 70/r, where r is the rate expressed as percent. So if r=2% then the doubling time id 70/2=35 time units. For interest compounded once a year (modelled by (1+r)^t), the doubling time is roughly 72/r.

Half-life: In exponential decay, half-life refers to the time needed for a substance to be reduced to half of its original amount.

Logarithm: Taking the logarithm of a number x results in the exponent y that would satisfy the equation 10y = x. In other words, by taking the logarithm of a number, you are finding the power that you would have to raise 10 to in order to get that number. For example, log(100) = 2, because you would have to raise 10 to the power of 2 in order to get 100. Common logarithm is with base 10 (meaning that 10 is the number you are raising to a certain power); however, logarithms can have any base. Logarithms are used to take very large numbers and reduce them to smaller numbers. If you’re measuring something that reaches very high values it can be useful to use a logarithmic scale where values increase by a factor of 10, rather than a linear scale. This allows you to fit large quantities onto a smaller scale.

Interest: Interest is a particular example of growth that is used to model changes in monetary values like the amount owed on a loan or the return on an investment. Simple interest is represented by linear growth, where the value increases by a fixed amount for each specified time interval. Compound interest is represented by exponential growth, where the amount increases by a certain percentage of the previous amount after each interval. An example of application of compound interest applied to loans and mortgages.

Simple interest and compound interest: see interest.

Depreciation: Depreciation refers to the decrease in value of something over a certain period of time and is often modeled by exponential decay. For instance, the value of a car decreases with time after it is purchased, so the car is said to depreciate in value.

Mortgage: A mortgage is a special kind of loan taken for the purpose of buying property. Mortgages function by dividing the total cost of the property, plus interest, into smaller installments that are paid over a mutually agreed longer period of time rather than all at once. This process of spreading the payments over a longer time is called amortization.

Inflation: The rate at which goods and services increase in price. As a result of inflation, the value of money decreases over time. Inflation is measured using Consumer Price Index (CPI), which is a weighted average cost of a fixed basket of goods and services. For instance, inflation for 2015 is computed as the CPI for 2105 divided by the CPI for 2014.

Consumer Price Index (CPI) is a weighted average price of a fixed basket of goods and services (such as food, shelter, transportation, alcohol and tobacco, clothing and footwear, health and personal care, etc.)

Dow 30, formerly Dow Jones Industrial Average (DJIA) is a numeric indicator showing how thirty companies in the U.S. trade on a particular day. Dow30 is computed very frequently (every 15 seconds, or even more often), according to the following formula: sum of the values of the 30 companies’ individual stocks divided by the so-called Dow divisor, whose value is adjusted as needed (for instance when a stock is split).

Standard and Poor’s 500 index or S&P500 is an indicator of the movement of stock prices of 500 selected companies, which (unlike Dow30) takes into account the volume (i.e., the number of stocks available for trading). For each company, the price of its stock is multiplied by the number of shares publicly available for trading; these are added for all 50 companies, and the sum is divided by a divisor (whose exact value is a trade secret) to compute the S&P500.

Gross Domestic Product (GDP) is a money value of all goods and services related to a specific geographic area (such as a country, or a region within a country). GDP measures the size of the economy. It is reported in absolute terms (for Canada, it was about 2.13 trillion Canadian dollars in 2016), as well as in relative terms (for instance, with respect to a previous year). Per capita GDP (i.e., GDP divided by the number of people) is a measure of living standard.

Gross National Product (GNP), also known as Gross National Income (GNI) is the total of domestic and foreign goods and services claimed by residents of a country; i.e., GNI = GDP plus income earned by residents abroad, minus income earned in the domestic economy by non- residents.

Human development index (HDI) is a composite of life expectancy (LEI), education (EI), and per capita income (II) indices, which are used to rank countries into tiers of human development. HDI is computed as the cube root of the product of LEI, EI and II (this is also known as the geometric mean).

Life expectancy index (LEI) is the ratio (life expectancy –20) / (85–20).

Education index (EI) is the average of two quantities: average number of completed years of education by people 25 years or older, divided by 15; and the number of years a child can expect to spend in school, divided by 18.

Income index (II) is the ratio (log(GNI per capita) – log(100)) / (log(75,000) – log(100)).

Gini Index is a measure of dispersion of wealth, measured on the scale from 0 (equal distribution of wealth) to 1 (maximum inequality; wealth owned by one person or entity). Gini index is the quotient: sum of absolute values of all differences between incomes is divided by 2, by the number of people, and (to normalize) by the sum of all incomes.

Lorenz curve expresses the cumulative percent of total income as a function of the cumulative percent of people from poorest to richest (e.g., “30 percent of poorest people own 10 percent of the total wealth”). The area of the region between the Lorenz curve and the diagonal, multiplied by 2, is the Gini index.

Purchasing power parity (PPP) is a way of comparing different countries’ currencies based on calculating the price of a fixed basket of goods and services (so similar to CPI) in these countries.

Average is a vague term, and it could mean any of the mean, the median, and the mode. Unless absolutely clear or irrelevant, we need to specify which exact meaning does the word “average” represent.

Mean of a set of n numbers is their sum divided by n.

Median is the midpoint of the set of numbers. To compute it, we arrange the numbers from the smallest to the largest. If there is an odd number of numbers, the median is the middle value (e.g., if there are 11 numbers, the median is the sixth number counting from either end). If there is an even number of numbers, the median is the mean of the two middle numbers (e.g., if there are 10 values, the median is ½ * sum(fifth and sixth numbers, counted form either end).

Mode is the value/values which appears/appear most often. In a bi-modal data set, two values appear equally often.

Percentile: A percentile, p, is the value in a data set that is greater than or equal to p% of the data (the data must be ordered from the smallest to the largest values). For example, a value in the 18th percentile is larger than the smallest 18% of the data. In statistics, the first quartile refers to the 25th percentile, and the third quartile refers to the 75th percentile. The median is the 50th percentile. The interquartile range (IQR) is defined as IQR = third quartile – first quartile.

Box (or box and whisker) plot is a visual representation of a data set that shows the minimum value, first quartile, median, third quartile, and maximum value.

Outliers are extreme values in a data set, defined as lying farther than a reasonable distance from the rest of the data. More precisely, an outlier is a value is smaller than (1st quartile – 1.5*IQR) or larger than (3rd quartile + 1.5*IQR).

Regression to the mean refers to the phenomenon that if some variable is extreme on its first occurrence/ measurement, it will be less so on the second occurrence/ measurement.

Variance is a measure of the spread of a data set; it is defined as the sum of the terms of the form (data point – mean)2 divided by the total number of elements in the data set.

Standard deviation is the most common measure of the spread of the data. It is computed as the square root of the variance.

Any number which describes (or which is derived from) a given set of data in some way is called a statistic (statistics is plural). Some examples we have seen so far are mean, median, mode, percentiles, variance and standard deviation. Statistics are useful ways of describing large data sets, as they are reduced to a very few numbers.

Random experiment is an experiment whose outcome is determined by chance, i.e., whose outcome cannot be predicted with certainty.

Simple event is a single outcome of a random experiment (e.g., head-tail-head as outcome of flipping a coin three times). An event is a collection of simple events (e.g., flipping a coin three times and obtaining two heads).

Sample space is the set of all outcomes of a random experiment.

Probability of an event is the measure of the likelihood of that event occurring. It is a number between 0 and 1; the probability of 0 is assigned to an impossible event, and the probability of 1 to a certain event.

Two events are called independent if the knowledge of one event does not tell us anything about the probability of the of the other event occurring. If A and B are independent events. Then the probability of A and B occurring is the product of the probability of A occurring and the probability of B occurring.

Complementary events are two events with the property that they are the only outcomes. If A denotes “black,” then the complement of A is “not black.” If p is the probability that A occurs, then the probability of the complement of A (i.e., the probability that A does not occur) is 1-p.

Contingency table is a table showing the distribution of one variable in rows and another in columns (e.g., recall the undergraduate/graduate, male/female example discussed in class). Contingency tables are used to study associations between the two variables.

False positive/false negative: Suppose someone is tested for the presence of a disease. If the test yields a positive result (i.e. that they have the disease) but in reality they do not, then the test has given a false positive. If the test yields a negative result but in reality they have the disease, then the test has given a false negative. In general, false negatives are less desirable than false positives (it’s better to treat someone for a disease they don’t have than to fail to treat them for a disease they do have).

Law of total probability expresses the probability of an event which can be realized via several mutually exclusive events (which we usually visualize as a tree). Mutually exclusive events are events that have nothing in common.

Quantitative argument: An argument (usually narrative) that is backed by quantitative data, based on logical reasoning and supported by evidence derived from the data (for instance, through statistical inference).

Algebraic argument: A quantitative argument using numbers and a known formula in order to make some claim about the world. For example, know the formula for a car’s stopping distance given its velocity, the coefficient of friction on the driving surface, and the acceleration due to gravity, then we can make an algebraic argument about how much stopping distance a driver should leave when driving in certain road conditions.

Geometric argument: A quantitative argument based on information derived from a visual interpretation of data (such as a scatter plot or a trend line).