2.1 numbers and their practical applications

7/28/2019 2.1 Numbers and Their Practical Applications

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THINKING

QUANTITATIVELY


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Goals:

1.To integrate quantitative reasoning with

logical reasoning by investigating the

concept of numbers; and

2.To develop methods for interpreting large

and small numbers.


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This section includes a discussion of:

The history of numbers

How numbers are used

The modern system of numbers


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No single, formal definition of the concept of

numbers exists. The term numbers is used to

describe many different ideas.

The concept of numbers has evolved over

time, is ever-evolving and is not a fixed idea.

It developed in parallel with methods forwriting numerals which are symbols that

represent numbers.


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Numbers were originally used for simple

counting.

One of the earliest questions was "How many?"

Humans have been answering this question for

thousands of years.

However, thousands of years ago, there were no

numbers to represent 2 or 3.


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Numeral systems relied on tallies with fingers

or toes, piles of stones, or notches cut on a

bone or a piece of wood.

But these systems are inadequate for large

numbers.

To simplify the process of counting, counts are

groupedby 2s, 3s, then eventually, by 5s,

10s, and 20s.


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In 3000 B.C., the

Egyptians and

Babylonians

independently

introduced the first

numeral system to gobeyond tallying.


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456 + 265

=

=


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Modern numerals trace directly to the work of Hindu

mathematicians in India in the first few centuries A.D.

In A.D. 800, Hindu numerals became part of the Arabculture when major Hindu works on astronomy were

translated into Arabic.

The Arabs then led the development of mathematics

during the next several centuries. The written shapesof the Hindu numerals slowly changed over time. As a

result, modern numerals are called Hindu-Arabic

numerals.

These numerals took their current form when great

works of Arab mathematicians were translated into

Latin in about A.D. 1200.


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1. Counting. Cardinal numbers

answer the question how

many?.2. Ordering. Ordinal numbers

indicate the order of members

in a set.

3. Labeling. Nominal numbers

are used as labels or names.


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Many other systems of numerals came into use. The

Roman numerals, developed in 500 B.C., were used

in ancient Greece and Rome, and became dominantin Europe for more than 1,000 years.

It is an additive numeral system, in which values are

determined by adding the values of individualsymbols. The position of the symbol does not affect

its value.

It does not have a symbol forzero.


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Subtraction was introduced only in the 16th or

17th century, but by then Hindu-Arabic numerals

were far more common.

They were less useful because: (1) writing large

numbers is extremely difficult, and (2) they offer

no convenient way to represent fractions.

They are still used for decorative or artistic

purposes, or to denote book chapters.


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The Hindu-Arabic system is a decimal, or base-10,

place-value system. Decimal(from Lat. decimus

meaning tenth). The value of a numeral

depends on its place or position.

The symbols 0,1,2,,9 are called digits (from Lat.

digituswhich means finger)


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Place-value systems require a symbol forzero,

which is crucial in the development of the

modern number system.

Zero became a meaningful number only in A.D.

600 when Hindu mathematicians introduced it.

The Mayan civilization in America however

independently developed the idea of zero as

much as 500 years earlier.


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The Babylonians in 2000 B.C. invented the method

of writing fractions, which was nearly identical to

the modern method of writing decimal fractions.

However, the Babylonian system was based on

powers of 60 instead of powers of 10.

The method of writing fractions with a numerator

and denominator was probably developed by

Hindu mathematicians.


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The binary system is a place-value system

that uses only two symbols, 0 and 1, called

bits or binary digits. It is easy to convert base-2 numeral to base-

10 numeral and vice-versa.

A numeral in any base can be represented in

any other base.


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Convert 156 to base-2 numeral.

Recall: 156 = 1*100 + 5*10 + 6*1

= 1*102 + 5*101 + 6*100

We wish to find n and ai for i = 0,,n such that

156 = an*2n + an-1*2n-1+ + a1*21 + a0*20

where each ai is either 0 or 1.


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Now, 156 = 128 + 16 + 8 + 4

= 1*128 + 0*64 + 0*32 + 1*16 + 1*8

+ 1*4 + 0*2 + 0*1

= 1*27 + 0*26 + 0*25 + 1*24 + 1*23

+ 1*22 + 0*21 + + 0*20

Thus, 156 = 100111002


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Convert 10112 to base 10.

10112 = 1*23 + 0*22 + 1*21 + 1*20

= 8 + 0 + 2 + 1

= 13


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The Babylonian system is a base-60 system.

Vestiges of this remain in time keeping, and in

angle measurement.

22*(60)2 + 11*(60)1 + 23*(60)0

= 79,200 + 660 + 23 = 79883


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The Mayans used base-20 system.

1 times 160,000 = 160,000

10 times 8,000 = 80,0002 times 400 = 800

14 times 20 = 80

3 times 1 = 3


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The Natural Numbers

We build the modern number system beginning with

numbers used for counting.

Counting numbers, or natural numbers, comprise the

set {1,2,3,4,}.

Natural numbers are further categorized, according to

theirfactors or divisors, as eitherprime or composite.


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Fundamental Theorem of Arithmetic: Every composite

number can be uniquely expressed as a product of

prime numbers. (Prime Factorization)

Example: 1476 = (2)(738)

= (2)(2)(369)

= (2)(2)(3)(123)

= (2)(2)(3)(3)(41)


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Negative numbers came out of subtracting natural

numbers.

Examples of uses of negative numbers:1. In commerce, where debts and losses are

represented by negative numbers.

2. In temperature and elevation measurements

The set of all numbers that result from adding or

subtracting natural numbers is called the set of

integers.


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The integers include the natural numbers, also called

positive integers, zero, and the negatives of all

natural numbers, or negative integers.

The set ofwhole numbers is composed of zero and

the positive integers.


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Properties of Integers:

1. Every integer, except 0, has a sign which indicates

whether it is positive (+) or negative (-). 0 is neither

positive nor negative.

2. Every integer n has a magnitude (or absolute value),

denoted |n| which indicates how far it lies from 0 on the

number line.

|-5| = 5, |5| = 5

0 1-1-2-3-4-5 2 3 4 5


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The set of all possible outcomes of dividing integers

(except dividing by 0) is called the rational numbers.

Rational(from the word ratio which refers to the

division of two numbers)

The set of rational numbers is the set of all numbers

that can be expressed in the form x/ywhere bothx

and yare integers and y 0.

The set of integers is a subset of the set of rational

numbers.


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At one time in ancient Greece, all numbers

were believed to be rational numbers. A secret society of followers of Pythagoras

(500 B.C.) believed that numbers had special

and mystical meanings.


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1 was considered divine.

Even numbers were considered feminine.

Odd numbers besides 1 were considered masculine.

The number 5, sum of the first feminine and

masculine numbers, represented marriage.

7 represented the seven planets known to the

Greeks; the belief that 7 was a lucky number

probably came from them.


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The motto of the Pythagoreans: All is number.

Their sacred belief was that all numbers were either

whole, by which they meant the natural numbers(they did not recognize zero or negative numbers), or

fractions made by the division of whole numbers.

But using the Pythagorean Theorem, they realized thata right triangle with two sides of length 1 has a third

side of length equal to the square root of 2 , which

they could not express as a fraction.


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Eventually they proved that the square root of 2 cannot

be expressed by dividing two whole numbers, that is, it

is an irrational number.

They attempted to keep this as a secret because their

fundamental beliefs may be challenged, and even killed

one of their members, Hippasus, for telling others of

their discovery.

1

1c

12 + 12 = c2


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The combination of the rational and irrational

numbers is called the real numbers.

Another way to describe real numbers is as

the rational numbers and everything in

between.

Each point on the number line has a

corresponding real number, and vice versa.


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Finding a real number that is a square root of a

negative number is impossible. Thus another type of

non-real numbers, called imaginary numbers, was

invented to solve this problem.

Imaginary numbers are numbers that represent the

square root of negative numbers.

A special number called i(for imaginary) is defined to

be the number whose square is negative 1, that is

i2 = 1


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Since i2 * i = 1 , then i3 = i2 * i = 1 * i = i

and i4 = i3 * i = i* i = i2 = (1 ) = 1

Imaginary numbers cannot be shown on a real

number line because they are not real numbers.

The complex numbers are numbers that include all

the real numbers and all the imaginary numbers.

A complex number is in the form a + bi, where aand b are real numbers and i2 = 1.

Examples: iii 2

1,3,7,23


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Complex Numbers

Real Numbers Imaginary Numbers

Irrational Rational

Integers Other fractions

Negative 0 Positive

Prime 1 Composite


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It is difficult to generate the sequence of prime

numbers.

Basic question: How many primes are there? Euclid(c. 300 B.C.) proved that there are infinitely many

primes.

Erathosthenes, Greek mathematician who lived in

the third century B.C., devised a systematic methodfor generating primes, called the Sieve of

Erathosthenes.


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Given a list of natural numbers from 1 to n.

Cross out 1 because it is neither prime nor composite.

The next number 2 is prime; cross out all subsequent multiples of 2

because they are composite. (We call 2 a sieve number because it helpsus sift through or remove other numbers in the list.)

The next number 3 is prime; then cross out all subsequent multiples of 3.

Move to the next number that has not been crossed out. Use 5 as a sieve

number and cross out all multiples of 5 that have not been crossed outyet.

Continue this process until we reach the end of the list.

The numbers that remain after all the crossings out are the primes on

the list.


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1 16 31 46 61 76 91

2 17 32 47 62 77 92

3 18 33 48 63 78 93

4 19 34 49 64 79 94

5 20 35 50 65 80 95

6 21 36 51 66 81 96

7 22 37 52 67 82 97

8 23 38 53 68 83 98

9 24 39 54 69 84 99

10 25 40 55 70 85 100

11 26 41 56 71 86 101

12 27 42 57 72 87 102

13 28 43 58 73 88 103

14 29 44 59 74 89 104

15 30 45 60 75 90 105

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In principle, the sieve of Erathosthenes could be used

on a list of numbers of any length.

The method, however, is extremely tedious. Kulik, a 19th century Austrian astronomer, spent 20

years using this method to find all primes between 1

and 100 million!

Moreover, the library to which he gave his

manuscripts lost the sections containing the primes

between 12,642,000 and 22,852,800.


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A formula to produce primes could generate

lists with much less effort.

Mathematicians have searched in vain for such

a formula for more than 2,000 years!

A few formulas work over a limited range ofnumbers before failing.


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Example: the expression n2 n + 41

successfully produces primes for small values

of n. This formula fails after n = 41, that is, itgenerates only 41 primes before it produces a

composite number. In addition, the formula

misses all primes less then 41.

Other formulas also fail. Mathematicians

believe that a suitable formula does not exist.


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In cryptography, wherein messages are written in code to

protect privacy and maintain security.

A security system can use large composite numbers as a

lock. The two primes multiplied to make the compositerepresent the keys.

Because there is no efficient way to find the prime

factorization, the lock can be opened only by people who

hold the keys. Research seeks efficient methods of factoring large

numbers, and computers are getting faster. But as larger

and larger primes are found, more inviolable locks can be

designed.


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Why the Search for Large Primes?

To continue a long standing tradition Euclid wanted to

characterizeperfectnumbers.

A numberp is perfect if it is the sum of all its divisors,

exceptp itself.

6 = 1 + 2 + 3

28 = 1 + 2 + 4 + 7 + 14

the next two are 496 and 8128

p = 2n-1(2n 1) where 2n 1 is prime (Mersenne)


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French monk Marin Mersenne (1588-1648) stated in the

preface to his Cogitata Physica-Mathematica (1644) that

the numbers 2

n

1 were prime for n = 2, 3, 5, 7, 13, 17, 19,31, 67, 127 and 257 and not prime for all numbers less than

257.

Finally, by 1947 Mersenne's range, n < 258, had been

completely checked and it was determined that the correct

list is: n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107 and 127.


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To collect rare and beautiful items

Euclid initiated the study of Mersennes in 300 BC andsince then there are less than 50 Mersenne primes

discovered

In math, there is a definite notion of beauty, for

example elegant proofs are regarded to be short but

concise and clear. And Mersenne primes have one of

the simplest forms for primes: 2n 1


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For the money

The Electronic Frontier Foundation announced that itwas giving $100,000 for the first to come up with a ten-

million digit prime, $150,000 for the first hundred-

million digit prime and $250,000 for the first billion

digit prime.
http://www.eff.org/http://www.eff.org/


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GIMPS is a collaborative project of volunteers who usefreely

available computer software to search for Mersenne prime

numbers. In 23 August 2008, the 45th Mersenne prime 243,112,609 1

with 12,978,189 digits, was discovered at the University of

California Los Angeles (UCLA)s Department of Mathematics,where Edson Smith was responsible for installing and

maintaining the GIMPS software on their computers,

allowing GIMPS to win the $100,000.


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GIMPS gave $50,000 to the UCLA Department of

Mathematics, $25,000 was donated to a math-related

charity selected by GIMPS founder George Woltman & theremaining $25,000 was paid to Odd Magnar Strindmo for his

discovery of M47, Hans-Michael Elvenich for M46, the

University of Central Missouri (M44 and M43), Dr. Martin

Nowak (M42), Josh Findley (M41), Michael Shafer and his

selected charity (M40) and Michael Cameron (M39).


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After the 23rd Mersenne prime was found at the

University of Illinois, the mathematics department was

so proud that the chair of their department, Dr. Bateman,had their postage meter changed to stamp "211213-1 is

prime" on each envelope.


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The 25th and 26th Mersenne primes were found by high-

school students Laura Nickel and Landon Curt Noll, who,

though they had little understanding of the mathematics

involved, used Lucas' simple test on the local university'smainframe (CSUH's CDC 174) to find the next two

primes. Their discovery of the first prime made the national

television news and the front page of the New York

times. They went their separate ways after finding the firstprime, but Noll kept the program running to find the

second--so Noll claims complete ownership. Noll searched

later, and though he never found another Mersenne prime,

he is one of a team that holds the record for the largestnon-Mersenne prime. He currently works for Silicon

Graphics. (http://primes.utm.edu/mersenne/index.html)


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Infinitymay be the most astonishing aspect of

the concept of numbers.

Georg Cantor (1845-1918) began a serious

study of infinity over a century ago. His results

shocked the mathematical world at that time.

Cardinalityis another term for the number of

elements of a set.


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One way is by counting the elements of each set and

see whether the count is the same for both sets.

Another way is to determine if there is a one-to-onecorrespondence between the members of the two

sets.

{r, e, a, d, y}

{I, m, n, o, p}


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The set of natural numbers and the set of even integers

have the same cardinality.1 2 3 4 5 6

2 4 6 8 10 12 ...

Similarly, there are as many natural numbers as odd

numbers, and as many natural numbers as multiples of 3,

etc. Galileo in 1638 considered these as unexplainable

paradoxes and chose not to work with infinity further.


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After 250 years, Cantor took these paradoxes as starting

points for further work and invented a new arithmetic,

called transfinite arithmetic, that applies to infinity.

The cardinality of the natural numbers is symbolized byo(pronounced aleph naught or aleph null).

We have the following results:

o+ 1=o and o+ o=o


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The infinite array above contains all positive rational numbers. We have the

one-to-one correspondence:

1 2 3 4 5 6 7 8 ...

1/1 1/2 2/1 3/1 2/2 1/3 1/4 2/3 ...

1 2 3 4 5

1 1/1 1/2 1/3 1/4 1/5

2 2/1 2/2 2/3 2/4 2/5

3 3/1 3/2 3/3 3/4 3/5

4 4/1 4/2 4/3 4/4 4/55 5/1 5/2 5/3 5/4 5/5

.


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Thus, there are as many natural numbers as

there are rational numbers!

And we have the result:

o x o= o and (o)2 = o

Similarly, (o)3 = o and so on.


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Cantor showed that some infinite sets have cardinality

greater than o.

He showed that the real numbers cannot be put into

one-to-one correspondence with the natural numbers,and that there are more irrationals than rationals.

In fact, between any two points in the number line, the

number of irrationals is greater than the number of all

rational numbers.

The cardinality of this new, higher infinity is designated

1.


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Thus the symbol is not enough to denote all

infinite numbers because infinity has more than one

level. Does a level of infinity exist between o and1?

The answer is unknown, but a set with such

cardinality has never been found.

The continuum hypothesis says that no set with

such cardinality exists.


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Yes, as Cantor proved.

In fact, he showed that an infinite number of higher

levels of infinity exist, and their cardinality might bedesignated

o,1, 2, 3, 4,...

But no one has ever been able to describe a set withan infinity higher than2.


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In ancient times, there was no way to express

extremely large or small numbers; in fact it was

unnecessary. Today, these seemingly incomprehensible numbers

are dealt with in the real world.

Goal: learn to think quantitatively by developing

methods for interpreting such numbers.


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tens or hundreds of billions of dollars of spending

and taxation

the collective impact of six billion people on the

environment

a nuclear weapon with one megaton of explosive

power

a computer with gigabytes of memory andprocessing times measured in nanoseconds,

microseconds or milliseconds

Can you assess the values of these numbers?


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Survival and prosperity in the modern world depend

on decisions that involve numbers that may, at first,

seem incomprehensibly large or small.

To make wise decisions, you must find ways of putting

such numbers into perspective.

Our task: to learn how to make extremely large orsmall numbers comprehensible by relating them to

numbers with which we are already familiar.


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Consider the following numbers:

The diameter of the Galaxy is about

1,000,000,000,000,000,000 kilometers The nucleus of a hydrogen atom has a diameter of

about 0.000000000000001 meters

These numbers are difficult to read and mostpeople will just skip right over them. There is a

better way of expressing such numbers.


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Laws of Exponents

103 = 10 x 10 x 10 = 1000;

103

x 105

= 1,000 x 100,000 = 100,000,000 = 108

; 105 x 103 = 100,000 x 0.001 = 100 = 102;

001.01000

1

10

110

3

3

4

3

7

10000,10

000,1

000,000,10

10

10

100,000,1100000,000,1101026

mnmn 101010

mn

m

n

10

10

10

000,999,9000,1000,000,10101037


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4 hundred billion = 400 x 109= 0.4 x 1012 = 4 x 1011

Numbers written with a number between 1 and 10

multiplied by a power of 10 are said to be inscientific notation.

A number written in scientific notation can be

quickly converted to ordinary notation. 3045 = 3.045 x 103; 0.000012 = 1.2 x 105

5.7 x 106 = 5,700,000; 2.26 x 104 = 0.000226


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(4 x 109) (3 x 1012) = (4 x 3)(109 x 1012)

= 12 x 1021= 1.2 x 1022

There is no shortcut for adding or subtracting

numbers in scientific notation.

34373

7

3

7

100.5105.01021

1010

4.82.4

104.8102.4

626100002.4200,000,4200000,000,4102104

6461098.3000,980,3000,20000,000,4102104


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The scientific notation simplifies writing

extremely large or small numbers.

Rounding and expressing numbers in scientificnotation allow quick approximationsof the

exact answers.

1312

66

1092.1102.19

104108.4321,997,3378,785,4


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The scientific notation makes extremely large or

small numbers deceptively easy to write.

Example: 10^26 does not lookmuch different from 10^20,

but is a million times larger

10^80 deceptively small but is larger than the totalnumber of atoms in the entire universe

2.1 numbers and their practical applications

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