DOCSLIB.ORG

Q Skills Review Dr

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Q Skills Review Dr Q Skills Review Dr. C. Stewart Measurement 4: Scientific Notation The Decimal System We are so very familiar with our decimal notation for writing numbers that we usually take it for granted and do not think about what it actually means. For example, we know that 6 means six “ones” or “units”, but when we write 60, then the 6 indicates six tens, or sixty, of the units. The zero must be written to ensure we know that the 6 is in the tens’ place. When we write the digits in a row, we add the values, so that 64 indicates “six tens plus four” of the units. If the quantity we want to indicate is not a whole number then we sometimes use fractions, for example 6⅓ indicates “six units plus one third of a unit”, but this takes us away from the idea of simply writing digits in a row. Let us now consider only those fractions in which the denominator is a power of ten, such as 1 4 9 10 ,1000 , 1000000 . Fractions of this type are called decimal fractions. They are a continuation of the base ten place-value notation, and with this understanding we can simply continue to write down the digits of the numerators in the correct place. We keep track of the value associated with each digit by using a ‘decimal point’ to separate the digits representing whole numbers from those representing parts of a whole. For example, the number 2453.176 represents 2 thousands, 4 hundreds, 5 tens, 3 ones (or units), 1 tenth, 7 hundredths and 6 thousandths. We can expand 2453.176 to show the meaning of this notation as follows: 1 7 6 2453.176= 2000 + 400 +++++ 50 3 10 100 1000 If the numerator of a decimal fraction has more than one digit, then the fraction can be expressed as the sum of decimal fractions with single digit numerators by considering equivalent fractions, 57 50 7 5 7 0 5 7 e.g. 1000 = 1000 + 1000 = 100 + 1000 . This can then be written as 10 + 100 + 1000 to allow us to write it as 0.057 in decimal notation. The ‘0’ after the decimal point is necessary to ensure the 5 and 7 are in the correct places. By convention we very rarely write ‘0’ at the beginning of a number larger than one, and so seeing ‘0’ at the beginning of a number indicates that it has a value less than one. It is important to remember to write this ‘0’, is the convention in Canada: according to the Government of Canada’s terminology and linguistic data bank (www.btb.termiumplus.gc.ca, writing tools, The Canadian Style, section 5.09), “Normally, no number should begin or end with a decimal point. A zero is written before the decimal point of numbers smaller than 1, while in whole numbers the decimal point should either be dropped or be followed by a zero: $0 .64 not $.64; 11 or 11 .0 not 11 .”. This is also an internationally agreed convention (www.bipm.org/en/si/si_brochure, section 5.3.4). One feature of the decimal system is that each digit represents the value of the digit multiplied by a power of ten. We can use this feature to multiply or divide numbers by powers of ten very easily. For example, what is the value of 56.437× 10 2 ? If we look at the meaning of the notation 4 3 7 2 we see that 56.437 means 50++ 6 10 + 100 + 1000 , and so when we multiply by 10= 100 , we get 4 3 7 4 3 7 (50++++ 610 100 1000 ) × 100 =× 50 100 +× 6 100 + 10 ×100 + 100 × 100 +1000 × 100 7 =5000 + 600 +×+ 4 10 3 + 10 The end result is that 56.437× 102 = 5643.7 Isn’t this amazing? 2 [You’re probably wondering what the big deal is, but think about the notations used before the decimal system. How would you do this in Roman Numerals, for example? Apart from difficulties in expressing 0.437, what would 56× 100 = 560 look like? It would be written as LVI× C = DCLX . Because the decimal system is positional (the place where the digits are written indicates their value), then when we multiply (or divide) by a power of ten, we move the digits to a different place, but we use the same digits in the same order!] We often say we are moving the decimal point by the number of places indicated in the exponent of the ten. Although this is technically incorrect, since the decimal point always remains between the whole numbers and the fractional part, it relates easily to what we see written down. In 56.437× 102 = 5643.7 we feel that the decimal point has been moved 2 places to the right. 472.3 400 70 2 3 1 Consider the following example: 472.3÷= 10 2 = + ++× 100 100 100 100 10 100 7 2 3 =4 +++ = 4.732 10 100 1000 472.3 1 But 472.3÷= 102 = 472.3 ×= 472.310 × −2 , and so 472.3× 10−2 = 4.723 . Here it looks 102 10 2 like the decimal point has been moved 2 places to the left. In summary: When we multiply by a positive power of ten, 10 n , we see the decimal point is now n places to the right. When we divide by a positive power of ten, 10 n , or multiply by a negative power of ten, 10 −n , we see the decimal point is now n places to the left. Scientific Notation Very large or very small numbers can be difficult to read and so we sometimes use words like million or trillion to help us. Another technique is to give prefixes to the units, for example milligram (mg) or gigabyte (GB). A useful notation for writing very large or very small numbers is “scientific notation”. A number in scientific notation has the form: a ×10 n where 1≤a < 10 , and n is an integer This means that in the number a, there is exactly one digit before the decimal point (but this cannot be ‘0’), and the power of ten (10 n) is used to ensure the correct value of each digit. Here are some examples: 472.3= 4.723 × 10 2 , 0.00345= 3.45 × 10 −3 , 0.16= 1.6 × 10 −1 . How would we write the value 2.7 in scientific notation? We need to find a value for n such that 2.7= 2.7 × 10 n . We know that 2.7= 2.7 × 1 , and that 100 = 1 , and so we can write 2.7= 2.7 × 10 0 . There different ways to enter numbers in scientific notation on a calculator, for example, some have a button marked EXP or EE which you press before entering the power of ten. Also, calculators differ in the way they present numbers in scientific notation: some miss out the ‘×10’, for example, showing 3.6× 10 7 as 3.6 07 or 3.6 E07; others may use a different notation. It is very important to get to know your own calculator. 3 In addition to knowing how to enter numbers in scientific notation for use with a calculator or computer, it will help your understanding of this notation if you learn how to calculate without using a machine. Multiplying and Dividing When multiplying or dividing numbers given in scientific notation it is easiest to deal with the number part and the powers of ten separately. Example 1: (610×23) ××( 710 5) =×× 6710 235 × 10 [re-order the multiplications] (23+ 5 ) 28 =×42 10 =× 4210 [using laws of exponents] 1 28 29 =××4.2 10 10 =× 4.2 10 [write in correct scientific notation] Example 2: 17 17 2 8 ×10 2× 10 810×17 ÷× 410 − 8 = = [write the division as a fraction and cancel] ()() −8 − 8 1 4 ×10 10 (17−( − 8 )) =2 × 10 [using laws of exponents] =2 × 10 25 Adding and Subtracting When adding or subtracting numbers given in scientific notation we must first write the numbers with the same power of ten - we cannot use laws of exponents to combine the powers of ten when adding or subtracting numbers. Once the power of ten is the same we simply add or subtract the number part (and re-write in correct scientific notation if necessary). Example 3: 13 12 13 13 (a) (2.34× 10) +×( 3.110) =( 2.34 × 10) +×( 0.3110 ) [get the same power of ten – the largest] 13 =()2.34 + 0.31 × 10 [the power of ten is a common factor] 13 =2.65 × 10 OR – 13 12 12 12 (b) (2.34× 10) +×( 3.110) =( 23.4 × 10) +×( 3.110 ) [the same power of ten – the smallest] 12 =()23.4 + 3.1 × 10 [the power of ten is a common factor] 12 13 =26.5 × 10 = 2.65 × 10 [write in correct scientific notation] Method (a), writing both numbers with the largest power of ten, is slightly shorter, but some people find it easier to use the smaller power of ten, method (b). The important thing is to write both numbers in a form with the same power of ten – you can choose whichever method you find easier. Look at the following examples and decide which you find easier when there are negative powers of ten involved. 4 Example 4: −5 − 8 − 5 −− 35 (a) (7.594× 10) −×( 3.8 10) =( 7.594 × 10) −××( 3.8 10 10 ) [‘split up’ one power of ten ...] −5 − 5 =(7.594 × 10) −( 0.0038 × 10 ) [..
Load more

Recommended publications
  • Stage 1: the Number Sequence
    1 stage 1: the number sequence Big Idea: Quantity Quantity is the big idea that describes amounts, or sizes. It is a fundamental idea that refers to quantitative properties; the size of things (magnitude), and the number of things (multitude). Why is Quantity Important? Quantity means that numbers represent amounts. If students do not possess an understanding of Quantity, their knowledge of foundational mathematics will be undermined. Understanding Quantity helps students develop number conceptualization. In or- der for children to understand quantity, they need foundational experiences with counting, identifying numbers, sequencing, and comparing. Counting, and using numerals to quantify collections, form the developmental progression of experiences in Stage 1. Children who understand number concepts know that numbers are used to describe quantities and relationships’ between quantities. For example, the sequence of numbers is determined by each number’s magnitude, a concept that not all children understand. Without this underpinning of understanding, a child may perform rote responses, which will not stand the test of further, rigorous application. The developmental progression of experiences in Stage 1 help students actively grow a strong number knowledge base. Stage 1 Learning Progression Concept Standard Example Description Children complete short sequences of visual displays of quantities beginning with 1. Subsequently, the sequence shows gaps which the students need to fill in. The sequencing 1.1: Sequencing K.CC.2 1, 2, 3, 4, ? tasks ask students to show that they have quantity and number names in order of magnitude and to associate quantities with numerals. 1.2: Identifying Find the Students see the visual tool with a numeral beneath it.
    [Show full text]
  • Using Concrete Scales: a Practical Framework for Effective Visual Depiction of Complex Measures Fanny Chevalier, Romain Vuillemot, Guia Gali
    Using Concrete Scales: A Practical Framework for Effective Visual Depiction of Complex Measures Fanny Chevalier, Romain Vuillemot, Guia Gali To cite this version: Fanny Chevalier, Romain Vuillemot, Guia Gali. Using Concrete Scales: A Practical Framework for Effective Visual Depiction of Complex Measures. IEEE Transactions on Visualization and Computer Graphics, Institute of Electrical and Electronics Engineers, 2013, 19 (12), pp.2426-2435. 10.1109/TVCG.2013.210. hal-00851733v1 HAL Id: hal-00851733 https://hal.inria.fr/hal-00851733v1 Submitted on 8 Jan 2014 (v1), last revised 8 Jan 2014 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Using Concrete Scales: A Practical Framework for Effective Visual Depiction of Complex Measures Fanny Chevalier, Romain Vuillemot, and Guia Gali a b c Fig. 1. Illustrates popular representations of complex measures: (a) US Debt (Oto Godfrey, Demonocracy.info, 2011) explains the gravity of a 115 trillion dollar debt by progressively stacking 100 dollar bills next to familiar objects like an average-sized human, sports fields, or iconic New York city buildings [15] (b) Sugar stacks (adapted from SugarStacks.com) compares caloric counts contained in various foods and drinks using sugar cubes [32] and (c) How much water is on Earth? (Jack Cook, Woods Hole Oceanographic Institution and Howard Perlman, USGS, 2010) shows the volume of oceans and rivers as spheres whose sizes can be compared to that of Earth [38].
    [Show full text]
  • Basic Statistics Range, Mean, Median, Mode, Variance, Standard Deviation
    Kirkup Chapter 1 Lecture Notes SEF Workshop presentation = Science Fair Data Analysis_Sohl.pptx EDA = Exploratory Data Analysis Aim of an Experiment = Experiments should have a goal. Make note of interesting issues for later. Be careful not to get side-tracked. Designing an Experiment = Put effort where it makes sense, do preliminary fractional error analysis to determine where you need to improve. Units analysis and equation checking Units website Writing down values: scientific notation and SI prefixes http://en.wikipedia.org/wiki/Metric_prefix Significant Figures Histograms in Excel Bin size – reasonable round numbers 15-16 not 14.8-16.1 Bin size – reasonable quantity: Rule of Thumb = # bins = √ where n is the number of data points. Basic Statistics range, mean, median, mode, variance, standard deviation Population, Sample Cool trick for small sample sizes. Estimate this works for n<12 or so. √ Random Error vs. Systematic Error Metric prefixes m n [n 1] Prefix Symbol 1000 10 Decimal Short scale Long scale Since 8 24 yotta Y 1000 10 1000000000000000000000000septillion quadrillion 1991 7 21 zetta Z 1000 10 1000000000000000000000sextillion trilliard 1991 6 18 exa E 1000 10 1000000000000000000quintillion trillion 1975 5 15 peta P 1000 10 1000000000000000quadrillion billiard 1975 4 12 tera T 1000 10 1000000000000trillion billion 1960 3 9 giga G 1000 10 1000000000billion milliard 1960 2 6 mega M 1000 10 1000000 million 1960 1 3 kilo k 1000 10 1000 thousand 1795 2/3 2 hecto h 1000 10 100 hundred 1795 1/3 1 deca da 1000 10 10 ten 1795 0 0 1000
    [Show full text]
  • Common Units Facilitator Instructions
    Common Units Facilitator Instructions Activities Overview Skill: Estimating and verifying the size of Participants discuss, read, and practice using one or more units commonly-used units, and developing personal of measurement found in environmental science. Includes a fact or familiar analogies to describe those units. sheet, and facilitator guide for each of the following units: Time: 10 minutes (without activity) 20-30 minutes (with activity) • Order of magnitude • Metric prefixes (like kilo-, mega-, milli-, micro-) [fact sheet only] Preparation 3 • Cubic meters (m ) Choose which units will be covered. • Liters (L), milliliters (mL), and deciliters (dL) • Kilograms (kg), grams (g), milligrams (mg), and micrograms (µg) Review the Fact Sheet and Facilitator Supplement for each unit chosen. • Acres and Hectares • Tons and Tonnes If doing an activity, note any extra materials • Watts (W), kilowatts (kW), megawatt (MW), kilowatt-hours needed. (kWh), megawatt-hours (mWh), and million-megawatt-hours Materials Needed (MMWh) [fact sheet only] • Parts per million (ppm) and parts per billion (ppb) Fact Sheets (1 per participant per unit) Facilitator Supplement (1 per facilitator per unit) When to Use Them Any additional materials needed for the activity Before (or along with) a reading of technical documents or reports that refer to any of these units. Choose only the unit or units related to your community; don’t use all the fact sheets. You can use the fact sheets by themselves as handouts to supple- ment a meeting or other activity. For a better understanding and practice using the units, you can also frame each fact sheet with the questions and/or activities on the corresponding facilitator supplement.
    [Show full text]
  • Order-Of-Magnitude Physics: Understanding the World with Dimensional Analysis, Educated Guesswork, and White Lies Peter Goldreic
    Order-of-Magnitude Physics: Understanding the World with Dimensional Analysis, Educated Guesswork, and White Lies Peter Goldreich, California Institute of Technology Sanjoy Mahajan, University of Cambridge Sterl Phinney, California Institute of Technology Draft of 1 August 1999 c 1999 Send comments to [email protected] ii Contents 1 Wetting Your Feet 1 1.1 Warmup problems 1 1.2 Scaling analyses 13 1.3 What you have learned 21 2 Dimensional Analysis 23 2.1 Newton’s law 23 2.2 Pendula 27 2.3 Drag in fluids 31 2.4 What you have learned 41 3 Materials I 43 3.1 Sizes 43 3.2 Energies 51 3.3 Elastic properties 53 3.4 Application to white dwarfs 58 3.5 What you have learned 62 4 Materials II 63 4.1 Thermal expansion 63 4.2 Phase changes 65 4.3 Specific heat 73 4.4 Thermal diffusivity of liquids and solids 77 4.5 Diffusivity and viscosity of gases 79 4.6 Thermal conductivity 80 4.7 What you have learned 83 5 Waves 85 5.1 Dispersion relations 85 5.2 Deep water 88 5.3 Shallow water 106 5.4 Combining deep- and shallow-water gravity waves 108 5.5 Combining deep- and shallow-water ripples 108 5.6 Combining all the analyses 109 5.7 What we did 109 Bibliography 110 1 1 Wetting Your Feet Most technical education emphasizes exact answers. If you are a physicist, you solve for the energy levels of the hydrogen atom to six decimal places. If you are a chemist, you measure reaction rates and concentrations to two or three decimal places.
    [Show full text]
  • Metric Prefixes and Exponents That You Need to Know
    Remember, the metric system is built upon base units (grams, liters, meters, etc.) and prefixes which represent orders of magnitude (powers of 10) of those base units. In other words, a single “prefix unit” (such as milligram) is equal to some order of magnitude of the base unit (such as gram). Below are the metric prefixes that you are responsible for and examples of unit equations and conversion factors of each. It does not matter what metric base unit the prefixes are being applied to, they always act in the same way. Metric prefixes and exponents that you need to know. Each of these 1 Tfl 1×1012 fl tera T = 1x1012 1 Tfl = 1x1012 fl or 1x1012 fl = 1 Tfl or have positive 1×1012 fl 1 Tfl exponents because each prefix is larger than the 1 GB 110× 9 B giga G = 1x109 1 GB = 1x109 B or 1x109 B = 1 GB or base unit 1×109 B 1 GB (increasing orders of magnitude). 6 Remember, the 1 MΩ 1×10 Ω mega M = 1x106 1 MΩ = 1x106 Ω or 1x106 Ω = 1 MΩ or power of 10 is 1×106 Ω 1 MΩ always written with the base unit 1 kJ 110× 3 J whether it is on kilo k = 1x103 1 kJ = 1x103 J or 1x103 J = 1 kJ or the top or the 3 1×10 J 1 kJ bottom of the Larger than the base unit base the than Larger ratio. 1 daV 1×101 V deka da = 1x101 1 daV = 1x101 V or 1x101 V = 1 daV or 1×101 V 1 daV -- BASE UNIT -- (meter, liter, gram, second, etc.) 1 dL 110× -1 L deci d = 1x10-1 1 dL = 1x10-1 L or 1x10-1 L = 1 dL or Each of these 1×10-1 L 1 dL have negative exponents because 1 cPa 1×10-2 Pa each prefix is centi c = 1x10-2 1 cPa = 1x10-2 Pa or 1x10-2 Pa = 1 cPa or smaller than the -2 1×10 Pa 1 cPa base unit (decreasing orders 1 mA 110× -3 A of magnitude).
    [Show full text]
  • Lesson 2: Scale of Objects Student Materials
    Lesson 2: Scale of Objects Student Materials Contents • Visualizing the Nanoscale: Student Reading • Scale Diagram: Dominant Objects, Tools, Models, and Forces at Various Different Scales • Number Line/Card Sort Activity: Student Instructions & Worksheet • Cards for Number Line/Card Sort Activity: Objects & Units • Cutting it Down Activity: Student Instructions & Worksheet • Scale of Objects Activity: Student Instructions & Worksheet • Scale of Small Objects: Student Quiz 2-S1 Visualizing the Nanoscale: Student Reading How Small is a Nanometer? The meter (m) is the basic unit of length in the metric system, and a nanometer is one billionth of a meter. It's easy for us to visualize a meter; that’s about 3 feet. But a billionth of that? It’s a scale so different from what we're used to that it's difficult to imagine. What Are Common Size Units, and Where is the Nanoscale Relative to Them? Table 1 below shows some common size units and their various notations (exponential, number, English) and examples of objects that illustrate about how big each unit is. Table 1. Common size units and examples. Unit Magnitude as an Magnitude as a English About how exponent (m) number (m) Expression big? Meter 100 1 One A bit bigger than a yardstick Centimeter 10-2 0.01 One Hundredth Width of a fingernail Millimeter 10-3 0.001 One Thickness of a Thousandth dime Micrometer 10-6 0.000001 One Millionth A single cell Nanometer 10-9 0.000000001 One Billionth 10 hydrogen atoms lined up Angstrom 10-10 0.0000000001 A large atom Nanoscience is the study and development of materials and structures in the range of 1 nm (10-9 m) to 100 nanometers (100 x 10-9 = 10-7 m) and the unique properties that arise at that scale.
    [Show full text]
  • Redalyc.Recent Results in Cosmic Ray Physics and Their Interpretation
    Brazilian Journal of Physics ISSN: 0103-9733 [email protected] Sociedade Brasileira de Física Brasil Blasi, Pasquale Recent Results in Cosmic Ray Physics and Their Interpretation Brazilian Journal of Physics, vol. 44, núm. 5, 2014, pp. 426-440 Sociedade Brasileira de Física Sâo Paulo, Brasil Available in: http://www.redalyc.org/articulo.oa?id=46432476002 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Braz J Phys (2014) 44:426–440 DOI 10.1007/s13538-014-0223-9 PARTICLES AND FIELDS Recent Results in Cosmic Ray Physics and Their Interpretation Pasquale Blasi Received: 28 April 2014 / Published online: 12 June 2014 © Sociedade Brasileira de F´ısica 2014 Abstract The last decade has been dense with new devel- The flux of all nuclear components present in CRs (the opments in the search for the sources of Galactic cosmic so-called all-particle spectrum) is shown in Fig. 1.Atlow rays. Some of these developments have confirmed the energies (below ∼ 30 GeV), the spectral shape bends down, tight connection between cosmic rays and supernovae in as a result of the modulation imposed by the presence of a our Galaxy, through the detection of gamma rays and the magnetized wind originated from our Sun, which inhibits observation of thin non-thermal X-ray rims in supernova very low energy particles from reaching the inner solar sys- remnants. Some others, such as the detection of features tem.
    [Show full text]
  • Our Cosmic Origins
    Chapter 5 Our cosmic origins “In the beginning, the Universe was created. This has made a lot of people very angry and has been widely regarded as a bad move”. Douglas Adams, in The Restaurant at the End of the Universe “Oh no: he’s falling asleep!” It’s 1997, I’m giving a talk at Tufts University, and the legendary Alan Guth has come over from MIT to listen. I’d never met him before, and having such a luminary in the audience made me feel both honored and nervous. Especially nervous. Especially when his head started slumping toward his chest, and his gaze began going blank. In an act of des- peration, I tried speaking more enthusiastically and shifting my tone of voice. He jolted back up a few times, but soon my fiasco was complete: he was o↵in dreamland, and didn’t return until my talk was over. I felt deflated. Only much later, when we became MIT colleagues, did I realize that Alan falls asleep during all talks (except his own). In fact, my grad student Adrian Liu pointed out that I’ve started doing the same myself. And that I’ve never noticed that he does too because we always go in the same order. If Alan, I and Adrian sit next to each other in that order, we’ll infallibly replicate a somnolent version of “the wave” that’s so popular with soccer spectators. I’ve come to really like Alan, who’s as warm as he’s smart. Tidiness isn’t his forte, however: the first time I visited his office, I found most of the floor covered with a thick layer of unopened mail.
    [Show full text]
  • On Number Numbness
    6 On Num1:>er Numbness May, 1982 THE renowned cosmogonist Professor Bignumska, lecturing on the future ofthe universe, hadjust stated that in about a billion years, according to her calculations, the earth would fall into the sun in a fiery death. In the back ofthe auditorium a tremulous voice piped up: "Excuse me, Professor, but h-h-how long did you say it would be?" Professor Bignumska calmly replied, "About a billion years." A sigh ofreliefwas heard. "Whew! For a minute there, I thought you'd said a million years." John F. Kennedy enjoyed relating the following anecdote about a famous French soldier, Marshal Lyautey. One day the marshal asked his gardener to plant a row of trees of a certain rare variety in his garden the next morning. The gardener said he would gladly do so, but he cautioned the marshal that trees of this size take a century to grow to full size. "In that case," replied Lyautey, "plant them this afternoon." In both of these stories, a time in the distant future is related to a time closer at hand in a startling manner. In the second story, we think to ourselves: Over a century, what possible difference could a day make? And yet we are charmed by the marshal's sense ofurgency. Every day counts, he seems to be saying, and particularly so when there are thousands and thousands of them. I have always loved this story, but the other one, when I first heard it a few thousand days ago, struck me as uproarious. The idea that one could take such large numbers so personally, that one could sense doomsday so much more clearly ifit were a mere million years away rather than a far-off billion years-hilarious! Who could possibly have such a gut-level reaction to the difference between two huge numbers? Recently, though, there have been some even funnier big-number "jokes" in newspaper headlines-jokes such as "Defense spending over the next four years will be $1 trillion" or "Defense Department overrun over the next four years estimated at $750 billion".
    [Show full text]
  • Chapter 3: Numbers in the Real World Lecture Notes Math 1030 Section B
    Chapter 3: Numbers in the Real World Lecture notes Math 1030 Section B Section B.1: Writing Large and Small Numbers Large and small numbers Working with large and small numbers is much easier when we write them in a special format called scientific notation. Scientific notation Scientific notation is a format in which a number is expressed as a number between 1 and 10 multiplied by a power of 10. Ex.1 one billion = 109 (ten to the ninth power) 6 billion = 6 × 109 420 = 4.2 × 102 0.5 = 5 × 10−1 Ex.2 Numbers in Scientific Notation. Rewrite each of the following statement using scientific notation. (1) The U.S. federal debt is about $9, 100, 000, 000, 000. (2) The diameter of a hydrogen nucleus is about 0.000000000000001 meter. Approximations with Scientific Notation Approximations with scientific notation We can use scientific notation to approximate answers without a calculator. Ex.3 Approximate 5795 × 326. 1 Chapter 3: Numbers in the Real World Lecture notes Math 1030 Section B Ex.4 Checking Answers with Approximations. You and a friend are doing a rough calculation of how much garbage New York City residents produce every day. You estimate that, on average, each of the 8 million residents produces 1.8 pounds or 0.0009 ton of garbage each day. Thus the total amount of garbage is 8, 000, 000 person × 0.0009 ton. Your friend quickly presses the calculator buttons and tells you that the answer is 225 tons. Without using your calculator, determine whether this answer is reasonable.
    [Show full text]
  • Orders of Magnitude (Length) - Wikipedia
    03/08/2018 Orders of magnitude (length) - Wikipedia Orders of magnitude (length) The following are examples of orders of magnitude for different lengths. Contents Overview Detailed list Subatomic Atomic to cellular Cellular to human scale Human to astronomical scale Astronomical less than 10 yoctometres 10 yoctometres 100 yoctometres 1 zeptometre 10 zeptometres 100 zeptometres 1 attometre 10 attometres 100 attometres 1 femtometre 10 femtometres 100 femtometres 1 picometre 10 picometres 100 picometres 1 nanometre 10 nanometres 100 nanometres 1 micrometre 10 micrometres 100 micrometres 1 millimetre 1 centimetre 1 decimetre Conversions Wavelengths Human-defined scales and structures Nature Astronomical 1 metre Conversions https://en.wikipedia.org/wiki/Orders_of_magnitude_(length) 1/44 03/08/2018 Orders of magnitude (length) - Wikipedia Human-defined scales and structures Sports Nature Astronomical 1 decametre Conversions Human-defined scales and structures Sports Nature Astronomical 1 hectometre Conversions Human-defined scales and structures Sports Nature Astronomical 1 kilometre Conversions Human-defined scales and structures Geographical Astronomical 10 kilometres Conversions Sports Human-defined scales and structures Geographical Astronomical 100 kilometres Conversions Human-defined scales and structures Geographical Astronomical 1 megametre Conversions Human-defined scales and structures Sports Geographical Astronomical 10 megametres Conversions Human-defined scales and structures Geographical Astronomical 100 megametres 1 gigametre
    [Show full text]