Number Theory 1

© Laurel Clifford Creative Commons BY-SA

“The concept of number is the obvious distinction between the beast and man. Thanks to

number, the cry becomes a song, noise acquires rhythm, the spring is transformed into a dance, force becomes dynamic, and outlines figures.”—Joseph Marie de Maistre

Number Theory From the day we become aware of the world around us, we begin recognizing quantity and

number. Whether it be the number of toys in our room or cereal bites on our high chair tray,

we learn to count. Ancient peoples used pebbles, sticks, knots in string, tally marks in clay, then formal symbols and numeration systems to record the quantities around them. As the

quantities we deal with become more complicated, we develop new numbers to record them.

Our modern number system is a product of millennia of thought and theory. In this chapter,

we examine the numbers we work with and what they mean.

Natural Numbers A sheepherder looks out at their flocks, and notes how many sheep they have, but how to record this quantity? A set of numbers is required, and some sort of symbol to represent

these numbers. Our society uses the Hindu-Arabic numerals you have seen since you were a

child, with digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. But what do these numbers mean? How do they

behave? What operations can be performed with them, and what do the results look like? Once the sheepherder had leisure time to think, they pondered the meaning of the numbers

they used.

Around 300 BC, the Greek mathematician Euclid summarized the known mathematics in his work The Elements. Normally thought of as a work of geometry, The Elements also includes

sections on number and number theory. Euclid defined concepts about Natural Numbers, a

set with which you are very familiar with, N = {1, 2, 3, 4, …}. Euclid called the first

number, 1, the “unit:”

A unit is that by virtue of which each of the things that exist is called one.1

A number is a multitude composed of units.

We can think of the number 2 as composed of units, where the unit is 1, simply by recalling

that 1 + 1 = 2. Euclid also defined even and odd numbers using definitions that will seem

very familiar to you as well, where an even number can be divided by 2 and an odd number

cannot be divided by 2, and differs from an even number by… a unit! Surprised? Just in these first few definitions in The Elements, you can see the effect of Greek mathematics on

your own mathematical education.

Every natural number greater than 1 is either prime or composite. Euclid defined prime

numbers as being “measured” only by 1, meaning the only factors of the number are 1 and

itself. He defined natural numbers that were not prime as composite. Another way to define

prime numbers is to state that prime numbers have only two unique factors, and thus

composite numbers have more than two unique factors. With these definitions, we can

Definition 1. (n.d.). Euclid's Elements, Book VII, Definitions 1 and 2. Retrieved June 16, 2014, from

http://aleph0.clarku.edu/~djoyce/java/elements/bookVII/defVII1.html

2

answer the question: Is 1 a prime number? Ask yourself: how many factors does 1 have? It

has only one factor, so it is not a prime number, because prime numbers have two factors. It is not composite either, as it doesn’t have more than two factors. Mathematicians view the

number 1 as a special number, giving it the same title Euclid did: 1 is the unit. Another way

to view this unit concept is to think that the 1 item you have represents whatever units you

are using to count (ounces, milligrams, feet, pickles, cats, whatever noun you are counting).

The number 1 is neither prime nor composite, what about other numbers, like 57? Is 57

prime or composite? You may be pondering ideas such as: it is not an even number, so it is

not divisible by 2; could it be divisible by 3? How can you tell without digging out your calculator? Perhaps you know the divisibility test for determining if numbers are divisible

by 3: Add the digits of the number: 5 + 7 = 12. If the result is divisible by 3, then so is the

original number: since 12 is divisible by 3, so is 57. If you grab your calculator, you can see

that 57 = 3 × 19. Arabic mathematicians of the middle ages proved divisibility tests, as did Fibonacci. The table below summarizes several divisibility tests:

A number is divisible by: if…

2 the ones’ digit is even (divisible by 2).

3 the sum of the digits is divisible by 3.

4 the last two digits form a number divisible by 4.

5 the ones’ digit is 0 or 5 (divisible by 5).

8 the last three digits form a number divisible by 8.

9 the sum of the digits is divisible by 9.

10 the ones’ digit is 0.

The divisibility test for 7 is not given here. The work involved in determining divisibility by

7 is complicated, and arguably we’re better off dividing the number by 7 to test it! What

about a divisibility test for 6? Consider that 6 is product of 2 and 3, 6 = 2 × 3, so to be divisible by 6, a number would need to pass the divisibility tests for both 2 and 3.

Look back at the table, and notice that some of the tests focus just on the last digit, while

some use the sum of the digits. Our numeration system is based on sets of 10, with place values 1, 10, 100, 1000, and so forth. The number 10 is a product of 2 and 5 (10 = 2 × 5), so

any place value other than 1 is divisible by 2 and 5. Thus the divisibility tests for 2 and 5

only look at the digit in the ones’ place. The number 10 is 9 + 1. To test divisibility by 9,

each digit is added, as it represents that extra amount “off” from 9 in each place value. Understanding the reasoning behind the test will help you remember the test.

Example 1: Divisibility Tests

Use the divisibility tests to determine if 1,158,962,874,003 is a composite number.

This number is too big to put in a basic 4-function calculator to divide! We can reject

divisibility by 2, as the last digit (3) is odd. Similarly, it is not divisible by 5 or 10. To test

divisibility by 3, add the digits: 1+1+5+8+9+6+2+8+7+4+0+0+3 =54, 54 is divisible by 3, so 1,158,962,874,003 is divisible by 3 as well, and thus is a composite number (it has more than

2 factors).

Number Theory 3

Try it now 1: Use the divisibility tests to determine whether 2, 3, 4, 5, 6, 8, 9, or 10 divide the following:

a. 1,256,957,844,024

b. 3,984,670,912,570

The Greek mathematician Eratosthenes (275-194 BC) devised a

'sieve' to discover prime numbers. A sieve2 is like a strainer that

you drain spaghetti through when it is done cooking. The water

drains out, leaving your spaghetti behind. Eratosthenes's sieve drains out composite numbers and leaves prime numbers behind.

To use the sieve of Eratosthenes to find the prime numbers up to 100, make a chart of the

first one hundred whole numbers (1-100):

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

Cross out 1, because it is not prime, it is a unit. Circle 2, because it is the smallest positive

even prime. Now cross out every multiple of 2. Circle 3, the next prime. Then cross out all

of the multiples of 3; some multiplies of 3, like 6, may have already been crossed out because they are even. Circle the next open number, 5. Now cross out all of the multiples of 5. Circle

the next open number, 7. Now cross out all of the multiples of 7. Circle any number that is

left, and you have circled all the prime numbers from 1 to 100.

Why didn’t we have to look for multiples higher than 7? You may have noticed that the first

multiple of 7 you had left to cross out was 49, and 49 = 7 × 7. Every multiple of 7 that was

less than 49 was already crossed out because they had a smaller co-factor that was already

removed. For example, 35 was already removed when we removed multiples of 5. You may also notice that 49 = 72, and 7 is the square root of 49. The largest prime that we test when

looking for factors of a number will be less than or equal to the square root of the number.

Try it now 2: Sieve of Eratosthenes Go to the website

http://nlvm.usu.edu/en/nav/frames_asid_158_g_1_t_1.html?open=instructions&from=topic_t

_1.html and use the applet to find the primes less than 100 via the Sieve of Eratosthenes by

setting the rows to 10, and clicking on 2, then 3, then 5, then 7. Notice how the number of

multiples removed gets smaller as the factors get larger.

2 Sieve picture by Donovan Govan. CC-BY-SA-3.0, via Wikimedia Commons

4

After completing the exercise above, you should see a table listing the prime numbers less

than 100:

You may notice patterns and pairs of primes. Twin primes are consecutive prime numbers

such as 11 and 13, and 41 and 43. It is been conjectured that there are an infinite number of

twin primes, but this has never been proven. Mersenne primes are prime numbers with the form 2n – 1, 1 less than a power of 2. How many Mersenne primes are there in the table?

The largest prime number known to date, discovered in January 2013, is a Mersenne prime,

257,885,161 – 1, which has 17,425,170 digits, and is also known as a titanic prime.3

The Fundamental Theorem of Arithmetic states that every composite number can be

expressed as a unique product of prime numbers, which means that there is only one way to

factor the number as primes (reordering of factors does not count as a different way). For

example, the composite number 100 can be written as 5×5×2×2 or 5222, but there is no other list of prime factors for 100; they all will include two 5s and two 2s.

You may recall from previous math courses using factor trees to determine prime factors of a number. Consider the number 240. We can recognize that it is divisible by 10, and factor

it as 24×10, but we haven’t completed the prime factorization until we have factored the 24

and 10 into their respective prime factors:

Factor trees created using the applet at:

http://nlvm.usu.edu/en/nav/frames_asid_202_g_2_t_1.html?from=topic_t_1.html

3 Weisstein, Eric W. "Titanic Prime." From MathWorld--A Wolfram Web Resource.

http://mathworld.wolfram.com/TitanicPrime.html

Number Theory 5

You can also recognize that 240 is even, thus divisible by 2, and divide by 2 until you reach a factor that is no longer divisible by 2, then divide by 3, and so forth, similar to the process

used in the Sieve of Eratosthenes. This process is also known as casting out 2s:

Both factoring methods express to us that the prime factorization is the same, 24 = (24)(3)(5).

Try it now 3: Find the prime factorization of the following numbers:

a. 38

b. 21 c. 360

Prime factorization helps us determine the greatest common divisor (GCD), sometimes

known as the greatest common factor (GCF) of two or more numbers. The greatest

common divisor (GCD) is the largest natural number that divides (“goes into” evenly, with

no remainder) the given numbers. Since the GCD looks for common divisors, it is useful in

problem solving when breaking larger amounts into smaller subsets: the GCD will be the size

of the largest common subset.

If we consider the numbers 240 and 360, we can find their prime factorizations express their

factorizations using a Venn diagram:

240 = (24)(3)(5) 360 = (23)(32)(5)

Notice that 240 = (2)(2)(2)(2)(3)(5)

and 360 = (2)(2)(2)(3)(3)(5) so they have (2)(2)(2), a 3, and a 5 in common:

The intersection of the prime factorizations is

the GCD: (2)(2)(2)(3)(5) = 120 and is noted as GCD(240, 360) = 120.

The same Venn diagram can be used to find the Least Common Multiple (LCM) of the two

numbers. The Least Common Multiple (LCM) is the smallest natural number that is a

240: 360:

2 3

2

2

3

2

5

6

multiple of the given numbers (the result from multiplying each of the given numbers by a

number). The LCM is useful in problem solving for predicting common repetitions of both values. The LCM can be located from the Venn diagram by listing all the factors shown in

the sets, the union of the prime factorizations. The LCM of 240 and 360 would be

(2)(2)(2)(2)(3)(3)(5) = (24)(32)(5) = 720, and is noted as LCM(240, 360) = 720. The LCM

represents every factor from each factorization with the highest exponent for repeated common factors.

Compare the product of the LCM and GCD of 240 and 360 with the product of 240 and 360:

LCM(240, 360) × GCD(240, 360) = (2)(2)(2)(2)(3)(3)(5)× (2)(2)(2)(3)(5) = (27)(33)(52) While 240×360 = (2)(2)(2)(2)(3)(5)×(2)(2)(2)(3)(3)(5) = (27)(33)(52)

They both equal the same result, (27)(33)(52) = 86400. The product of the two values is the

same as the product of their LCM and GCD. This relationship is helpful for checking

accuracy of results as well as finding either the LCM or GCD if you know one of them. For example, if you know the GCD(240, 360) = 120, then take the product of 240 and 360, which

is 86400, and divide it by 120: 86400 ÷ 120 = 720, which is the LCM(240, 360).

Try it now 4:

a. Find the GCD(144, 15) and the LCM(144, 15) b. Verify the results by finding the product of 144×15 and the product of the

LCM×GCD.

Consider the values 38 and 21, and their prime factorizations: 38 = (2)(19) and 21 = (3)(7).

Organizing their prime factors into a Venn diagram gives us:

In the intersection of the two sets, where we would normally locate the GCD, there are no values. This

empty intersection tells us the GCD(38, 21) = 1, as

every number has a factor of 1. When the GCD of

two values is 1, we say the values are relatively

prime. Notice that the LCM, the union of the sets,

is the product of the factors, LCM(38, 21) =

(2)(19)(3)(7) = 798, which is the same as the

product of 38 and 19: 38×19 = 798.

When solving problems involving the LCM or GCD, we determine if we are looking for

subsets (smaller sets) of the values, which would suggest a divisor or the GCD, or larger

multiples (larger sets) of the values, which would suggest a multiple or the LCM.

Example 2: Hot Dogs vs. Buns

Suppose that you like a particular specialty kind of hot dog that comes in packages of 10, and

you buy buns in packages of 8. How many whole packages of each should you purchase so that each hot dog has a bun?

38: 21:

7

3 19

2

Number Theory 7

Notice that we are buying whole packages, and more than one package, so we will end up

with multiples of hot dogs and buns (we are not breaking packages up; grocery stores take issue with that sort of thing!). Since we don’t necessarily need to feed an army, we are

looking for the least common multiple:

LCM(10, 8) = (5)(2)(2)(2) = 40 We need 40 hot dogs and 40 buns, so 4 packages of

hot dogs and 5 packages of buns.

The Venn diagram may seem overkill here, but did you catch how the number of packages relates to the

non-common factor(s)?

Example 3: Garden Plots

A large field measures 70 feet by 525 feet. If you divide it up into equal square garden plots,

what size is the largest possible plot if the side lengths are natural numbers?

The clue word in this problem is “divide.” We’re dividing the larger dimensions into smaller sizes, so we are looking for the GCD.

GCD(70, 525) = (7)(5) = 35

So the plots should be 35 feet by 35 feet in size.

From the Venn diagram, we can see the non-

common factors tell us how many plots will fit in

the field: there will be 2 plots by 15 plots, or 30 total plots in the field.

Try it now 5: a. Kris and Mickey are running laps around the same track. Kris can run one lap in 8

minutes but Mickey takes 12 minutes. If they both start at the same place, the same

time, and run in the same direction, at what time will they first pass each other?

b. What is the largest size of equal square tiles that could be used to make a checkerboard pattern on a floor measuring 128 inches by 96 inches?

Beyond Natural Numbers When counting sheep, natural numbers work quite well for the sheepherder as there is no meaning to part of a sheep; sheep are whole numbers. If you’re keeping track of your sheep

by making a tally mark on a clay tablet, and you have no sheep, then you make no tally mark.

A picture counting system needs no symbol for 0. However, when working with larger and

larger numbers, making tally marks becomes cumbersome, you end up creating a positional system, as the Babylonians did, and 0 becomes important, not just to mean 0 sheep, but as a

10: 8:

2

2 5 2

70: 525:

5

2 7

5

3

8

placeholder in place value. The addition of the symbol of 0 to the natural numbers creates

the set of Whole Numbers, W = {0, 1, 2, 3, 4, …}. But will whole numbers be sufficient for all the counting and mathematical operations we need to do?

An important concept for operations with number sets is the idea of closure. A set of

numbers is closed under an operation if you take two numbers from the set, perform the operation, and the result is also part of the set. Consider the set of Whole Numbers and the

operation of addition. If you add two whole numbers, will you always get a whole number?

That is, is whole number + whole number = whole number? For example, 3 + 5 = 8, a

whole number. Hopefully you intuitively say the whole numbers are closed under the

operation of addition, although we have not proven it.

But are the whole numbers closed under the operation of subtraction? Is whole number –

whole number = whole number? For example, 3 – 5 = -2, but wait! The result here is not a whole number, but is instead a negative number. We have a counterexample. There is no

whole number that results from 3 – 5. If we consider a number line, showing whole

numbers, view 3 – 5 as starting at 3 and moving left 5, there is nowhere to move to as the

number line ends at 0:

Lack of closure for whole numbers suggests there’s another number system out there that includes both the whole numbers and their opposites: Integers:

Extending the number line to the left past 0, and using a (-) sign to show direction leftward, we can use integers to illustrate 3 – 5 = -2, 2 units to the left of 0. Integers are closed under

the operation of subtraction.

Negative numbers allow us to show distance in a direction opposite from what we call positive numbers, as we show on the left side of the number line. We use negative numbers

to model debt, money is going in the opposite direction from us! The Chinese (in 200 BC)

and Indians (in 620 AD) used negative numbers to model debt, although modern western

society avoided their formal use until the 19th century4. We use negative numbers to indicate direction in temperature (below 0) as well as in altitude (below sea level).

Are the integers closed under the operation of multiplication? Does an integer times an

integer always produce another integer? We can try an example: (-8) × 3 = -24, which is an integer. One example is not proof, but intuitively, we can argue that as multiplication can be

thought of as repeated addition (add 3 sets of -8, or -8 + -8 + -8 = -24), then as the integers

are closed under addition, they should be closed under multiplication.

4 Rogers, L. (n.d.). The History of Negative Numbers. : nrich.maths.org. Retrieved June 16, 2014, from

http://nrich.maths.org/5961

?

Number Theory 9

But what about division? Does an integer divided by an integer always produce an integer?

Consider our previous example and change the operation: (-8) ÷ 3. There is no integer result here as there will be a remainder: The integers are not closed under division.

So how do we split a debt of $8 among 3 people? We can have each person pay $2, but that

leaves debt remaining, while each person paying $3 pays too much. We need a value between $2 and $3. There are no integer numbers in between consecutive integers, so we

need a new number set. We need the answer to -8/3, a ratio between two integers, the

Rational Numbers.

Each Whole Number and Integer can be considered Rational Numbers as well, as they can be

expressed as the ratio between two integers. The following are rational numbers:

-2 as it can be written as -2/1

0 as it can be written as 0/5 1.4 as it can be written as 14/10

1/2 as it can be written as 2/4

You may notice that 1/2 is already a ratio between two integers, but it could also be expressed as 2/4, as well as 3/6, 100/200, -5/-10, 11/22, and so forth. Rational numbers do

not have unique representations as a particular rational number, such as 1/2, has numerous

equivalent rational forms.

Another property that rational numbers have: they are dense. A number set is considered to

be dense if between any two numbers you can find another number that is also a member of

that set. Based on this concept, are integers dense? Consider the number line: there are no

integers in between any two integers. For example, there is no integer between 4 and 5.

Consider the rational numbers: is there a rational number between 4/7 and 5/7? If we consider equivalent forms of 4/7 and 5/7:

4 8

7 14 and

5 10

7 14 , and in between them is

9

14

These forms came from multiplying the numerator and denominator by a common factor, 2.

A student suggested that midway between 4/7 and 5/7 should be 4.5/7, using a decimal form.

However, this number, 4.5/7 is not a ratio of two integers, to which the student said,

“Multiply the numerator and denominator by 10 to convert the decimal to an integer:”

4.5 10 45

7 10 70 , a rational number, which can be reduced to

9

14

We could apply this student’s technique with 4.3/7 or similar to locate another value, 43/70,

in between 4/7 and 5/7.

10

Try it now 6: Locate a rational number between 7/15 and 8/15.

Rational numbers can be expressed in decimal form. Recall that our place value system uses base 10 place values. A rational number can be viewed as a quotient: the numerator (top)

divided by the denominator (bottom). Use a calculator to find the decimal forms of the

following fractions and see if you can find a pattern and connection to the place value

system:

1 4 1 1 3 2 5

2 5 3 11 8 15 6

Every rational number in decimal form will either be a terminating (finite) or non-

terminating, repeating decimal. The rational numbers that terminate have denominators (the

divisor in the ratio) with only 2 and 5 as their prime factors. If you consider that our place

value system is based on multiples of 10, and prime factors of 10 are 2 and 5, it makes sense that to terminate, the denominator needs to be a factor of a multiple of 10.

A simple division process takes a rational number from fraction to decimal form. How do

we go backwards from decimal to fraction form?

If the decimal terminates, it is straight-forward: use the place value of the terminating

digit. For example, 0.875 terminates in the thousandths place, so it is 875/1000, which

reduces:

875 875 125 7

0.8751000 1000 125 8

If the decimal does not terminate, a little bit of algebra can help. Consider the repeating

decimal 0.88888888…. which can be written as 0. 8̅ (the bar over the value indicates that value repeats). We recognize that it is a rational number because it repeats. We know that

number exists, so we call it “n,”

n = 0.8888888….

If we multiply this number by 10 (and multiply both sides of the equation by 10), we have:

10n = 8.888888….

Notice how this moves the decimal place one place, and there are still an infinite amount of

repeating digits following. Writing the two equations together, we have:

10n = 8.8888888…..

n = 0.8888888….. subtract the two (left side – left side, right side – right side)

9n =8 as all the repeating digits will subtract out infinitely.

We now have an equation we can solve for our unknown number, n:

9n = 8 divide both sides by 9,

n = 8/9 which is a ratio of two integers.

Number Theory 11

We can check it by using our calculator and dividing 8 by 9. Your calculator may round the

last decimal place it gives you, but it should still be 0.88888…. repeating infinitely.

Example 4: Converting Repeating Decimals

Convert 0.8787878787… to rational number form.

We recognize that it is a rational number as it is a repeating, nonterminating decimal. We call this number “n”: n = 0.8787878787…

We also notice that two digits are repeating, so multiply this equation by 100:

100n = 87.87878787…, which moves the decimal place two places.

100n = 87.87878787…

n = 0.8787878787… Subtract and all the repeating decimal values will cancel out

99n = 87 Solve for n (divide by 99).

n = 87/99 Check with your calculator: do 87÷99.

Note that 87 29

99 33 since both the numerator and denominator are divisible by 3.

Example 5: Converting Repeating Decimals

Convert 0.62525252525… to rational number form.

We recognize that it is a rational number as it is a repeating, nonterminating decimal.

We call this number “n”: n = 0.62525252525…

We also notice that two digits are repeating (be careful as the 6 is not part of the repeating portions), so multiply this equation by 100:

100n = 62.525252525… which moves the decimal place two places.

100n = 62.525252525… n = 0.62525252525… Subtract and the repeating decimal values will cancel out

99n = 61.9 Solve for n (divide by 99).

n = 61.9/99 WAIT! That’s not done, as there’s a decimal on top.

Note that 61.9 10 619

99 10 990 and we now have a rational number (check it with your

calculator).

Converting Repeating Decimals to Rational Form: 1. Use “n” to represent the unknown rational form of the number.

2. Create a second equation by multiplying by a power of 10 based on the number of

repeating digits.

3. Subtract the two equations to cancel out the repeating digits (make sure the digits align in order to do so)

4. Solve for n, reducing as necessary.

12

Try it now 7: Convert each decimal to rational form:

a. 0.742 b. 0.7777777…. c. 0.7474747474… d. 0.742742742742...

Beyond Rational Numbers Ancient Greek mathematicians were very fond of rational numbers. When they discovered

that there were other numbers which were not rational, they swore that "terrible" discovery to secrecy. One story (most likely just a story, but dramatically exciting anyway) suggests they

murdered the man who let the secret out! Rather irrational of them.

We created rational numbers to attempt to find closure under the operation of division. Are rational numbers closed under division? Is a rational number divided by a rational number

always another rational number? Almost… there is one number that creates havoc for

division: division by 0. Rational numbers will never truly be closed under division because

of division by 0.

Rational numbers allowed for ratios to be expressed easily, they can't express every number.

The most obvious examples can be found in geometry.

Consider a square whose sides are all one unit long:

Then the distance along the diagonal can be determined by the

Pythagorean Theorem, a2 + b2 = c2: 12 + 12 = c2

1 + 1 = c2

2 = c2

We can use the square root operation to undo the squaring and solve for c:

2 c

So what does “c” equal? Is “c” a rational number? What is this number 2 ?

We know that 2 is bigger than 1, as 12 = 1. We also know it is smaller than 2, as 22 = 4. So

we need a rational number between 1 and 2.

If 2 is a rational number, then we can write it as a ratio between two integers, x and y:

2x

y

But it still has the square root in it, so let’s square both sides to get rid of the square root:

2

2

2x

y

2

22

x

y

Which is still kind of yucky, so let’s cross-multiply to get rid of the fraction: 2 22y x

1

1

?

Number Theory 13

At this point, it still looks strange, so let’s remind ourselves about that idea earlier that every

number has a unique prime factorization. So whatever x is, it has a unique prime factorization. Squaring x doubles the number of prime factors. We don’t know what they

are, but we know there are an even number of them.

2 22y x

Same can be said for y and y2: it has an even number of prime factors. But if you look at the

left side of the equation, there is a 2 as well, an extra prime factor, so the left side has an odd number of prime factors, while the right side has an even number of prime factors. Since an

even number can’t equal an odd number, this situation is impossible. There is no rational

number for 2 and so 2 has to be irrational.

Challenge! 8

Apply the same argument to show that √5 is irrational and √4 is rational.

Another famous irrational number (also called a transcendental number)

is the ratio of the circumference of a circle to its diameter, 3.1415926….

or . It is a ratio, but cannot be expressed as the ratio of two integers.

If we consider or √2 = 1.41421356 …, we notice they are non-

terminating, non-repeating decimals, so they are not rational numbers.

The only way to express these numbers is to expand our number set beyond the rational

number set to include numbers these new numbers, known as Irrational Numbers. The

Real Number set includes both rational and irrational numbers. We can recognize irrational numbers because they will be decimals that DO NOT repeat or terminate.

As we’ve seen, roots are one common place that irrational numbers show up. Consider the

value of the following roots:

0 1 36 38 80

We can estimate the value of the last two roots, but cannot express that exactly using

decimals. We can find roots that are perfect squares.

12 22 32 42 52 62 72 82 92 102 112 122 132 142 152

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225

We know that 80 is close to 9, since 80 is close to 81. We know that 36 is close to 6 as

38 is close to 36.

Odd number of factors with 2

Even number of factors

14

Sometimes we can use perfect squares to simplify roots, if we recognize that the values have

perfect square factors.

For example, 80 16 5 and since 16 is the perfect square of 4, we can simply it:

80 16 5 16 5 4 5 . Similarly, 120 4 30 and since 4 is the perfect square of

2, we can simplify it: 120 4 30 2 30 . Is there more we could do? Does 30 have any

perfect square factors? No (we’re done).

Real Numbers and Number Properties At this point, we’ve looked at several number sets. Each number set enlarges the previous.

If we drew Venn diagram5 of the number sets, it would look like: (there’s a small error on

this diagram: what is labeled Natural includes 0, so it’s actually the Whole number set)

We’ve also looked a CLOSURE, whether the result of an operation is still a member of the set. Let’s revisit the closure of our number sets: Think about how the issue of closure drives

the creation of new number sets.

Number Set Addition Subtraction Multiplication Division

Natural Numbers

Closed Not Closed Closed Not Closed

Whole

Numbers

Closed Not Closed Closed Not Closed

Integers

Closed Closed Closed Not Closed

Rational

Numbers

Closed Closed Closed Closed except

for division by 0

Real Numbers Closed Closed Closed Closed except

for division by 0

5 Image copyright by Keith Enevoldson, http://thinkzone.wlonk.com/Numbers/NumberSets.htm

Number Theory 15

In addition to the closure properties, all real numbers illustrate the commutative properties of

addition and multiplication, reverse the order of addition or multiplication, same result:

Commutative Property: a + b = b + a ab = ba

All real numbers also have the associative property of addition and multiplication, regroup the terms added or multiplied, same result:

Associative Property: a + (b + c) = (a + b) + c a(bc) = (ab)c

The property of real numbers that ties multiplication and addition together is the distributive

property:

Distributive Property: a(b + c) = ab + ac

Try it now 9: We have commutative, associative and distributive properties for addition and multiplication.

Do they extend to other operations? Choose values for a, b, c, etc. and test the following

properties to see if they are true:

a. Distributive Property for Subtraction: a(b – c) = ab – ac

b. Distributive Property for Roots: √𝑎 + 𝑏 = √𝑎 + √𝑏

c. Distributive Property for Roots (II): √𝑎2 + 𝑏2 = √𝑎2 + √𝑏2

d. Commutative Property for Subtraction: a – b = b – a

e. Associative Property for Subtraction: a – (b – c) = (a – b) – c