Mathematics · 2004-09-14 · Polya's 4 Steps to Problem Solving: 1. Understand Problem 2. Devise a...

MathematicsFOR ELEMENTARY TEACHERS

A CONTEMPORARY APPROACH

Supplimentary Text

By Courtney Pindling

Department of Mathematics- SUNY New paltz

Mathematics for Elementary Teachers by Musser, Burger, Petersonand Pharo is a key source for the content of this paper Edited 6/04/2002

Cover page

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1. Introduction - Problem Solving Process

Polya's 4 Steps toProblem Solving:

1. UnderstandProblem

2. Devise a Plan

3. Carry Out Plan

4. Look Back


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Some Problem - Solving Strategies

1. Guess and test

2. Use a variable

3. Look for a pattern

4. Make a list

5. Solve a simplerproblem

6. Draw a picture

7. Draw a diagram

8. Use directreasoning

9. Use indirectreasoning

10. Use properties ofnumbers

11. Solve anequivalent problem



12. Work backward

13. Use cases

14. Solve an equation

15. Look for aformula

16. Do a simulation

17. Use dimensionalanalysis

18. Identify subgoals

19. Use coordinates

20. Use symmetry



2.1. Introduction to Set Theory

Definitions: Set { },

Union (either A or Bor Both)

Intersection(elements in commonto both)

Complement (allelements in U not inA) Â

Difference (A - B)


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Disjointed ( ) Subset ( )


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2.2. Whole Numbers & Numeration

Math History:http://www.seanet.com/~ksbrown/ihistory.htm

Translale Egyption Numbering System: (3400 BC)http://www.psinvention.com/zoetic/tr_egypt.htm

The Egyptians had a decimal system using seven different symbols.

1 is shown by a single stroke. 10 is shown by a drawing of a hobble for cattle. 100 is represented by a coil of rope. 1,000 is a drawing of a lotus plant. 10,000 is represented by a finger. 100,000 by a tadpole or frog 1,000,000 is the figure of a god with arms raised above his head.

1 10 100 1,000 10,000 100,000 Million

Roman Numeration System: (AD 100)


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http://www.seanet.com/~ksbrown/ihistory.htm

http://www.psinvention.com/zoetic/tr_egypt.htm

System

I - 1

V - 5

X - 10

L - 50

C - 100

D - 500

M - 1000

SubstractionMethod

IV - 4

IX - 9

XL - 40

XC - 90

CD - 400

CM - 900

Examples

CCLXXX1 - 281

MCVII - 1107

MCMXLIV >

M CM XL IV >

1000+900+40+4

Babylonian Numeration System: ( 3000 - 2000 BC)



Mayan Numbering System: (AD 300 - 900)

Abacus: (500 BC - Present) Basics - Calculations areperformed by placing the

abacus flat on a

table or one's lap and manipulating the beads with the fingers of one hand. Eachbead in the upper deck has a value of five; each bead in the lower deck has avalue of one. Beads are considered counted, when moved towards the beam thatseparates the two decks. The rightmost column is the ones column; the nextadjacent to the left is the tens column; the next adjacent to the left is the hundredscolumn, and so on. After 5 beads are counted in the lower deck, the result is"carried" to the upper deck; after both beads in the upper deck are counted, theresult (10) is then carried to the leftmost adjacent column. Floating pointcalculations are performed by designating a space between 2 columns as thedecimal-point and all the rows to the right of that space represent fractionalportions while all the rows to the left represent whole number digits



Hindu-Arabic System: (AD 800)

Digits {0,1,2,3,4,5,6,7,8,9}, Base 10 (decimal system),

Place value ( ..., million, thousands,hundred,tens,ones . tenth,hundredth, ...)



5. Number Theory

5.1 Primes, Composites, and Tests for Divisibility

Counting Numbers: 1, 2, 3, 4, 5, 6, .......

Prime Numbers: Divisable by itself and 1: 2, 3, 5, 7, 11, 13 , 17, 19, ...

Composite Numbers: at least 3 factors - e.g. 60 = 2 x 2 x 3 x 5

a | b means a divides b (quotient is a whole number)

Theorem:FundamentalTheorem ofArithmetic -Each Compositenumber can be afactor of primenumbers -

e.g. 60 = 2 x 2 x 3x 5

Theorem: Test for divisibilityby 2, 5 & 10 - Number divisible by 2 if ends in 0 or even digit

Number divisible by 5 if ends in 0 or 5

Number divisible by 10 if ends in 0

Theorem:Let a, m, nbe wholenumbers - If a | m & a | n, then a |(m+n)

If a | m & a | n, then a |(m-n) for m n

If a | m, then a | km(multiple of )

Theorem: Test for divisibilityby 4 & 8 - Number divisible by 4 if last 2 digits divisible by 4

Number divisible by 8 if last 3 digits divisible by 8


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Theorem:Test fordivisibilityby 3 & 9 - Number divisible by 3if sum of digitsdivisible by 3

Number divisible by 9if sum of digitsdivisible by 9

Theorem: Test for divisibilityby 11 - Number divisible by 11 if ( sum of

digits in even positions) - (sum of digits in oddpositions) divisible by 11

e.g. 909381=>(9+9+8)-(0+3+1)=22 is / 11

Theorem:Test fordivisibilityby 6 -Passes testsfor divisibility by 2 &3

Theorem:Productdivisibility -Number divisibility byboth a & b, then a & bhas 1 as commonfactor

Theorem: Prime Factor Test -Test if n is prime: see if primes up to p is divisor ofn: where

Is 299 a prime ?

So

299 is a prime

5.2 Counting factors, Greatest Common Factor (GCF)& Least Common Multiple (LCM)

Theorem: Counting Factors - If counting number expressed as product of distinctprimes:



number of factors for 144=24 x 32=> (4+1)(2+1)=15

Greatest Common Factor (GCF): The GCF of 2 or more whole numbers is the largest whole number that is a factor of both (all)

1. Prime Factor Method: The product of highest prime common to both (all):

e.g. GCF(24, 36): 24= 23 x 3 and 36=22 x 32 so GCF=> 22x3 = 12

2. GCF Theorem Method: GCF(a,b) = GCF(a-b, b) when :

3. Remainder Method: Theorem: GCF(a, b) = GCF(r, b):

If a & b are whole numbers and a >= b and a = kb + r , where r < b

Least Common Multiple (LCM): The LCM of 2 or more whole numbers is the smallest whole number that is a multiple of each (all) of the numbers. 1. Set Intersection Method: Smallest element of the intersection of multiple of the set ofeach numbers:

e.g. LCM(24, 36): 24= {24,48,72,96,120,144..} 36={36,72,108,144..}= {72, 144} So

LCM(24, 36) = 72



2. Prime Factor Method: The product of largest prime exponent in each (all):

e.g. LCM (24, 36): 24= 23x 3 and 36=22 x 32 so GCF=> 23x32 = 72

3. Buildup Method: State all prime, select prime of one number and build up to largestexponent:

e.g. LCM(42, 24): 24= 23 x 3 and 42=23 x 3 x 7 so LCM(42,24)= 23x3x7 = 168

GCF and LCM - Theorems

Theorem: GCF & LCM: GCF(a, b) x LCM (a, b) = ab

For example, find LCM (36,56) if GCF(36,56)=4

LCM x GCF = 36 x 56, So

Theorem: Infinite Number of Primes: There is an infinite number ofprimes

Algorithm for primes: Sieve of Eratosthenes

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

Directions: Skip the number 1, circle 2 and cross out evry second number after 2,Circle 3 and cross out every 3rd number after 3 (even if it had been crossed out before).Continue this procedure with 5, 7, and each succeeding number not crossed out.Circled numbers are primes and crossed out numbers are compsites.



6. Fractions

Parts of Fractions:

Definition of Fractions: a number represented by ordered pair of a wholenumber:

(Relative amount, part of a whole, numeral)

Fractions Equality : (cross product):

Theorem: Given Fraction must be written in simplest form

Improper Fractions: when numerator > denominator (mixed number):

Ordering Fractions : (Theorems)

Theorem (<) Theorem CrossMultiplication

Theorem (in betweens)

Multiplication:

Properties of Fractions Multiplication:

Meaning: (cases: whole x Fraction & Fractions x Fraction)

Properties:

6. Fractions

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Closure: Fract. XFract. = Fract.

Cumutative:

Associative: Distributive:

Identity :

Division:

Division - CommonDenominator:

Division - DifferentDenominator:

6. Fractions

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7. Decimals (base ten) Another way of representing the fractional part of a whole:

Every fraction can be represented in decimal form:

Some Observations:

A fraction can be transformed into a terminating decimal

if the prime factor if b is divisible by either 2 or 5

Order decimal from smallest to largest via its position along the number line

Addition / Substraction of decimals:

Like whole number additions (add the decimal portion first keeping true tothe dot that separates the whole number from the fractional part.

Fractional Equivalence:

Every terminating decimal has a fractional equivalence:

Converting Termination to fraction:

Divide the decimal by 1; Multiply both numerator and denominator bymultiples of 10s to remove the remove the decimal; then factor and simplify

Example convert 0.125 to fractional equivalence:

7. Decimals (base ten)

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(power of ten and decimals / common fractionalequiv.)

7.2 Decimals Operations:

Multiplication: number of decimal place is expanded:

e.g 437.09 x 3.8 = 1600.942

Significant figure: By Example:

e.g. 437.0923 = 437.09 (since value of place after 9 is 2 < 5unchanged )

437.0961 = 437.10 (since value of place after 9 is >= 5 round up)

Division: number of decimal place is reduced to significance of smallestdecimal place: (practice)

e.g.. 437.09 / 3.8 = 115.0

Decimals with repeating series; repeating values called repetend

Number of values that repeats called period

For example:

Long Division Algorithm: preserve decimal place or introduce it:

e.g..



Theorem: A repeating decimal doesnot terminate (NonterminatingDecimal Representation):

iff its fractional equivalent has primefactor other than 2 or 5 for itsdenominator

Theorem: Everyfraction has arepeating decimaland everyrepeating decimalhas a fractionalrepresentation

Fraction <==>RepeatingDecimal

If p is the period of a repetend:

Then with any given repeating decimal, n

(subtract n from both sides):

99n = 34, so (fractional equivalence)

In General to convert a repeating Decimal tofraction.

Introduce: Scientific Notation for decimals: 10-n

7.3 Ratio and Proportion

Ratio: a : b, with b not equal to 0 = denote relative size, comparison, rate,percent: a relative to b

Equality of Ratios: 2 ratios are equal if given:



Proportion: a statement that 2 ratios are equal: e.g.

7.4 Percent

Another way of representing fractions or decimal:

(Number per hundred)

Cases:

Percent to Decimal

Percent to Fraction

Decimal to Percent

Fraction to Percent

Common Percent / Fraction Equivalence (Appendix B)

Approaches to solving problems with percent:

1. 10 x 10 Grid

2. Properties of proportion / ratios



3. Solve equation



8. Integers Negative representations: (discuss historic and present international)

Integers: set of numbers: {..,-3,-2,-1,0,1,2,3,..} Positive Integers, Zero, Negative Integers (Set View via models or Measurement view via the number line); concept of negative being opposite of positive across pivot point at Zero)

8.1 Addition & Subtraction:

Set Model: Cancel effect: 4 positives + 3 negatives [i.e. 3 (-) cancels 3 (+) leaving1 (+)]

Number Line Model: a positives + b negatives: move from Zero a units right and then b units left from new position

Addition Properties: (if a, b, c are integers)

Closure: a + b is an integer Cumutative:

a + b = b + a

Associative:

(a + b) + c = a + (b + c)

Additive Inverse:

a + (-a) = 0

Identity :

a + 0 = a = 0 + a for all a

Theorem - Additive Cancellation forIntegers:

If a + c = b + c, then a = b

Theorem - Inverse of opposite:

- ( - a ) = a

8.2 Multiplication, Division, and Ordering Integers:

8. Integers

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If a and b are integers:

1.

2.

3.

Multiplication Properties: (if a, b, c are integers)

Closure: ab is an integer Cumutative:

a x b = b x a

Associative:

(ab)c = a(bc)

Identity:

a x 1 = a

Distribution: (Multipilcation overaddition):

a( b + c ) = ab + ac

Multiplication Cancellation:

Ac = bc, then a = b

Zero Divisors:

ab = 0, iff a = 0 or b = 0 or both = 0

Theorem - Multiplication by -1:

a (-1) = - a

Theorem - Multiplication of (-):

Case 1: (-a)b = -(ab)

Case 2: (-a)(-b) = ab

Scientific Notation: An exponential representation of numbers in the form:

Where a is called the mantissa and n the characteristic of exponent

8. Integers


Ordering Integers Properties: (if a, b, c are integers)

Transitive Properties:

If a < b and b < c, then a < c

< addition:

If a < b, then a + c < b + c

< Multiplication by (+):

If a < b, then ac < bc

< Multiplication by (-):

If a < b, then a(-c) > a(-c)

Use number line to order integers

8. Integers


9.1 Rational Numbers

Real Number Line

-4 -3 -2 -1 0 ½ 1 2 3 4

_________________________________________________________________Negative real numbers Zero(neither + or -) Positive realnumbers

Set of Rational Numbers

Real numbers {Rational Numbers{Fraction / Integers{Whole numbers{Counting}, 0}} }

Real numbers {Irrational numbers}

Definition Rational Numbers: {fractions, whole numbers ( ), integers}

The set of rational numbers is : Q={

Equality of Rationals:Definition

Equality Theorem: n =nonzero integer

(smiplest form:

lowest term)

Addition of Rationals:Definition

Additive InverseTheorem:

(note -b

hard to interpret)

Properties (Rational Numbers Addition):

Closure: Fract. X Fract.= Fract.

Cumutative:


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Associative: Additive Inverse:

Identity:

Theorem:Additive cancellation OppositeofOpposite:

Subtraction:Adding Opp.

(common / uncommondenominators)

Multiplcation of Rational Numbers

Properties (Rational Numbers Multiplication):

Closure: Fract. X Fract. =Fract.

Cumutative:

Distributive of Multiplication /Addition:

MultiplicationInverse: (Theorem)

Every ratitionals has a

unique rationals such

that: (reciprocal)



Identity: Associative:

Division of Rational Numbers

Division of Rationals: Theorem

1.

2.

3.

Ordering of Rationals:

Number line approach

Common-positive denominator a/b > c/d ifi a > c

Additive approach

Cross-Multiplication Theorem: (for b > 0 and d > 0)



9.3 Functions and Their Graphs

xy-coordinates system Linear function

Step Function: Quadratic Max.


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More FunctionsQuadratic Min. Exponential Growth:



Exponential Decay: Cubic Function:



Appendix A - Multiplication Table (12 x 12)

1 2 3 4 5 6 7 8 9 10 11 12

2 2 6 8 10 12 14 16 18 20 22 24

3 6 9 12 15 18 21 24 27 30 33 36

4 8 12 16 20 24 28 32 36 40 44 48

5 10 15 20 25 30 35 40 45 50 55 60

6 12 18 24 30 36 42 48 54 60 66 72

7 14 21 28 35 42 49 56 63 70 77 84

8 16 24 32 40 48 56 64 72 80 88 96

9 18 27 36 45 54 63 72 81 90 99 108

10 20 30 40 50 60 70 80 90 100 110 120

11 22 33 44 55 66 77 88 99 110 121 132

12 24 36 48 60 72 84 96 108 120 132 144

Remembering 9's

What's 9 x 7 ? Use the 9-method! Hold out all 10 fingers, and lower the 7th finger.There are 6 fingers to the left and 3 fingers on the right.

The answer is 63!

Appendix A - Multiplication Table (12 x 12)

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Appendix B - Fraction to Decimal Comparison Table

Fraction Decimal Fraction Decimal

0.05 0.5

0.1 0.6

0.125

0.2 0.75

0.25 0.8

0.875

0.4 1.0

Need to convert arepeating decimal to afraction?

Follow these examples:

Note the followingpattern for repeatingdecimals:

0.22222222... =

0.54545454... =

0.298298298... =

Division by 9's causesthe repeating pattern.

Note the pattern if zeros

To convert a decimal that begins with a non-repeatingpart, such as 0.21456456456456456..., to a fraction, write it asthesum of the non-repeating part and the repeating part.

0.21 + 0.00456456456456456...

Next, convert each of these decimals to fractions.The first decimal has a divisor of power ten. The seconddecimal (which repeats) is convirted according to thepattern given above.


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preceed the repeatingdecimal:

0.022222222... = 2/90

0.00054545454... =54/99000

0.00298298298... =298/99900

Adding zero's to thedenominator addszero's before the repeatingdecimal.

21/100 + 456/99900

Now add these fraction by expressing both witha common divisor


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Appendix C - First 200 prime numbers

2 73 181 613 743 1231 1399 xxxx 2063 2221

3 79 191 617 751 1237 1409 1531 2069 2237

5 83 193 619 757 1249 1423 1543 2081 2243

7 89 197 631 761 1259 1427 1549 2083 2251

11 97 199 641 769 1277 1429 1553 2087 2267

13 101 211 643 773 1279 1433 1559 2089 2269

17 103 223 647 787 1283 1439 1567 2099 2281

19 107 227 653 797 1289 1447 1571 2111 2287

23 109 229 659 809 1291 1451 1579 2113 2293

29 113 547 661 811 1297 1453 1583 2129 2297

31 127 557 673 821 1301 1459 1993 2131 2309

37 131 563 677 823 1303 1471 1997 2137 2311

41 137 569 683 827 1307 1481 1999 2141 2333

43 139 571 691 829 1319 1483 2003 2143 2339

53 151 587 709 853 1327 1489 2017 2161 2347

59 157 593 719 857 1361 1493 2027 2179 2351

61 163 599 727 859 1367 1499 2029 2203 2357

67 167 601 733 863 1373 1511 2039 2207 2749

71 173

179

607 739 1229 1381 1523 2053 2213 2753

Appendix C - First 200 prime numbers

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Appendix D: Pascal's Triangle to Row 191

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 330 165 55 11 1

1 12 66 220 495 792 924 792 495 220 66 12 1

1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1

1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1

1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1

1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1

1 17 136 680 2380 6188 12376 19448 24310 24310 19448 12376 6188 2380 680 136 17 1

1 18 153 816 3060 8568 18564 31824 43758 48620 43758 31824 18564 8568 3060 816 153 18 1

1 19 171 969 3876 11628 27132 50388 75582 92378 92378 75582 50388 27132 11628 3876 969 171 19 1

Appendix D: Pascal's Triangle to Row 19

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Appendix E. Data Powers of Ten

SI-Prefixes

Number Prefix Symbol Number Prefix Symbol

d deci- da deka-

c centi- h hecto-

m milli- k kilo-

u( ) micro- M mega-

n nano- G giga-

p pico- T teta-

f femto- P peta-

a atto- E exa-

z zepto- Z zeta-

y yocto- Y yotta-

The following list is a collection of estimates of the quantities of data contained by the various media.Each is rounded to be a power of 10 times 1, 2 or 5.

The numbers quoted are approximate. In fact a kilobyte is 1024 bytes not 1000 bytes.

([email protected])Bytes (8 bits) Terabyte (1 000 000 000 000 bytes)

0.1 bytes: A binary decision 1 Terabyte: An automated tape robot ORAll the X-ray films in a largetechnological into paper and printed ORDaily rate of EOS data (1998)

1 byte: A single character

10 bytes: A single word 2 Terabytes: An academic researchlibrary OR A cabinet full of Exabytetapes


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http://[email protected]/

100 bytes: A telegram OR A punchedcard

10 Terabytes: The printed collection ofthe US Library of Congress

Kilobyte (1000 bytes) 50 Terabytes: The contents of a largeMass Storage System

1 Kilobyte: A very short story Petabyte (1 000 000 000 000 000 bytes)

2 Kilobytes: A Typewritten page 1 Petabyte: 3 years of EOS data (2001)

10 Kilobytes: An encyclopaedic pageOR A deck of punched cards

2 Petabytes: All US academic researchlibraries

50 Kilobytes: A compressed documentimage page

20 Petabytes: Production of hard-diskdrives in 1995

100 Kilobytes: A low-resolutionphotograph

200 Petabytes: All printed material OR

200 Kilobytes: A box of punched cards Production of digital magnetic tape in1995

500 Kilobytes: A very heavy box ofpunched cards

Exabyte (1 000 000 000 000 000 000bytes)

Megabyte (1 000 000 bytes) 5 Exabytes: All words ever spoken byhuman beings.

1 Megabyte: A small novel OR A 3.5inch floppy disk

Zettabyte (1 000 000 000 000 000 000000 bytes)

2 Megabytes: A high resolutionphotograph

Yottabyte (1 000 000 000 000 000 000000 000 bytes)

5 Megabytes: The complete works ofShakespeare OR 30 seconds ofTV-quality video

10 Megabytes: A minute of high-fidelitysound OR A digital chest X-ray

Etymology of Units

20 Megabytes: A box of floppy disks

50 Megabytes: A digital mammogram

100 Megabytes: 1 meter of shelvedbooks OR A two-volume encyclopaedicbook

200 Megabytes: A reel of 9-track tapeOR An IBM 3480 cartridge tape

1.Kilo Greek khilioi = 1000



500 Megabytes: A CD-ROM OR Thehard disk of a 1995 PC

2.Mega Greek megas = great, e.g.,

Alexandros Megos

Gigabyte (1 000 000 000 bytes) 3.Giga Latin gigas = giant

1 Gigabyte: A pickup truck filled withpaper OR A symphony in high-fidelitysound OR A

4.Tera Greek teras = monster

2 Gigabytes: 20 meters of shelved booksOR A stack of 9-track tapes

5.Peta Greek pente = five, fifth prefix,peNta -

N = peta

5 Gigabytes: An 8mm Exabyte tape 6.Exa Greek hex = six, sixth prefix,Hexa - H = exa

10 Gigabytes: Remember, in standard French, theinitial H is silent, so they wouldpronounce Hexa as Exa. It is far easierto call it Exa for

20 Gigabytes: A good collection of theworks of Beethoven OR 5 Exabyte tapesOR A

everyone's sake, right?

50 Gigabytes: A floor of books ORHundreds of 9-track tapes

7.Zetta almost homonymic with GreekZeta, but last letter of the Latin alphabet

100 Gigabytes: A floor of academicjournals OR A large ID-1 digital tape

8.Yotta almost homonymic with Greekiota, but penultimate letter of the Latinalphabet.

200 Gigabytes: 50 Exabyte tapes

The first prefix is number-derived; second, third, and fourth are based on mythology.Fifth and sixth are supposed to be just that: fifth and sixth. But, with the seventh, another fork hasbeen taken.The General Conference of Weights and Measures (CGMP, from the French;they have been headquartered, since 1874, in Sevres on the outskirts of Paris) has now decided toname the

prefixes, starting with the seventh, with the letters of the Latin alphabet, but starting from the end.Now,that makes it all clear! Remember, both according to CGMP and SI, the prefixes refer to powers of 10.Mega is 106 , exactly 1,000,000, kilo is exactly 1000, not 1024.



Appendix F. Hierarchy of Numbers

0(zero) 1(one) 2(two) 3(three) 4(four)

5(five) 6(six) 7(seven) 8(eight) 9(nine)

101(ten) 102(hundred) 103(thousand)

Name American-French English -German

Million 106 106

Billion 109 109

Trillion 1012 1018

Quadrillion 1015 1024

Quintillion 1018 1030

Sextillion 1021 1036

Septillion 1024 1042

Octillion 1027 1048

Nonillion 1030 1054

Decillion 1033 1060

Undecillion 1036 1066

Duodecillion 1039 1072

Tredecillion 1042 1078

Quatuordecillion 1045 1084

Quindecillion 1048 1090

Sexdecillion 1051 1096


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Septendecillion 1054 10102

Octodecillion 1057 10108

Novemdecillion 1060 10114

Vigintillion 1063 10120

Googol 10100

Googolplex


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Appendix - z-score percentile for normal distribution

Percentile z-Score Percentile z-Score Percentile z-Score

1 -2.326 34 -0.412 67 0.44

2 -2.054 35 -0.385 68 0.468

3 -1.881 36 -0.358 69 0.496

4 -1.751 37 -0.332 70 0.524

5 -1.645 38 -0.305 71 0.553

6 -1.555 39 -0.279 72 0.583

7 -1.476 40 -0.253 73 0.613

8 -1.405 41 -0.228 74 0.643

9 -1.341 42 -0.202 75 0.674

10 -1.282 43 -0.176 76 0.706

11 -1.227 44 -0.151 77 0.739

12 -1.175 45 -0.126 78 0.772

13 -1.126 46 -0.1 79 0.806

14 -1.08 47 -0.075 80 0.842

15 -1.036 48 -0.05 81 0.878

16 -0.994 49 -0.025 82 0.915

17 -0.954 50 0 83 0.954

18 -0.915 51 0.025 84 0.994

19 -0.878 52 0.05 85 1.036

20 -0.842 53 0.075 86 1.08

21 -0.806 54 0.1 87 1.126

22 -0.772 55 0.126 88 1.175

23 -0.739 56 0.151 89 1.227

24 -0.706 57 0.176 90 1.282

25 -0.674 58 0.202 91 1.341


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26 -0.643 59 0.228 92 1.405

27 -0.613 60 0.253 93 1.476

28 -0.583 61 0.279 94 1.555

29 -0.553 62 0.305 95 1.645

30 -0.524 63 0.332 96 1.751

31 -0.496 64 0.358 97 1.881

32 -0.468 65 0.385 98 2.054

33 -0.44 66 0.412 99 2.326


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Appendix H - primes: Sieve of Eratosthenes (circle primes)

1

2 3 4 5 6 7 8 9 10

11

12 13 14 15 16 17 18 19 20

21

22 23 24 25 26 27 28 29 30

31

32 33 34 35 36 37 38 39 40

41

42 43 44 45 46 47 48 49 50

51

52 53 54 55 56 57 58 59 60

61

62 63 64 65 66 67 68 69 70

71

72 73 74 75 76 77 78 79 80

81

82 83 84 85 86 87 88 89 90

91

92 93 94 95 96 97 98 99 100

Directions: Skip the number 1, circle 2 and cross out evry second number after 2,Circle 3 and cross out every 3rd number after 3 (even if it had been crossed out before).Continue this procedure with 5, 7, and each succeeding number not crossed out.

Appendix H - primes: Sieve of Eratosthenes

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