Number Theory and the Real Number System

Theodore Vassiliadis

An introduction to number theory• Prime numbers• Integers, rational numbers, irrational

numbers, and real numbers• Properties of real numbers• Rules of exponents and scientific notation

WHAT YOU WILL LEARN

Number TheoryThe study of numbers and their properties.The numbers we use to count are called

natural numbers, , or counting numbers.

{1,2,3, 4,5,...}

FactorsThe natural numbers that are multiplied

together to equal another natural number are called factors of the product.

Example: The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and

24.

DivisorsIf a and b are natural numbers and the

quotient of b divided by a has a remainder of 0, then we say that a is a divisor of b or a divides b.

Prime and Composite NumbersA prime number is a natural number greater

than 1 that has exactly two factors (or divisors), itself and 1.

A composite number is a natural number that is divisible by a number other than itself and 1.

The number 1 is neither prime nor composite, it is called a unit.

Rules of Divisibility

285The number ends in 0 or 5.5

844 since 44 4

The number formed by the last two digits of the number is divisible by 4.

4

846 since 8 + 4 + 6 = 18

The sum of the digits of the number is divisible by 3.

3

846The number is even.2

ExampleTestDivisible by

Divisibility Rules, continued

730The number ends in 0.10

846 since 8 + 4 + 6 = 18

The sum of the digits of the number is divisible by 9.

9

3848since 848 8

The number formed by the last three digits of the number is divisible by 8.

8

846The number is divisible by both 2 and 3.

6

ExampleTestDivisible by

The Fundamental Theorem of Arithmetic

Every composite number can be expressed as a unique product of prime numbers.

This unique product is referred to as the prime factorization of the number.

Division Method1. Divide the given number by the smallest

prime number by which it is divisible.2. Place the quotient under the given number.3. Divide the quotient by the smallest prime

number by which it is divisible and again record the quotient.

4. Repeat this process until the quotient is a prime number.

Example of division methodWrite the prime factorization of 663.

The final quotient 17, is a prime number, so we stop. The prime factorization of 663 is 3 •13 •17

13

3

17

221

663

Finding the LCM of Two or More Numbers Determine the prime factorization of each

number. List each prime factor with the greatest

exponent that appears in any of the prime factorizations.

Determine the product of the factors found in step 2.

Example (LCM)Find the LCM of 63 and 105.

63 = 32 • 7105 = 3 • 5 • 7

Greatest exponent of each factor:32, 5 and 7

So, the LCM is 32 • 5 • 7 = 315.

Whole NumbersThe set of whole numbers contains the set of

natural numbers and the number 0.Whole numbers = {0,1,2,3,4,…}

IntegersThe set of integers consists of 0, the natural

numbers, and the negative natural numbers. Integers = {…–4, –3, –2, –1, 0, 1, 2, 3 4,…}On a number line, the positive numbers

extend to the right from zero; the negative numbers extend to the left from zero.

Writing an InequalityInsert either > or < in the box between the paired numbers to make the statement correct.

a) 3 1 b) 9 7 3 < 1 9 < 7c) 0 4 d) 6 8 0 > 4 6 < 8

The Rational Numbers• The set of rational numbers, denoted by

Q, is the set of all numbers of the form p/q, where p and q are integers and q 0.

• The following are examples of rational numbers:

1

3,

3

4,

7

8, 1

2

3, 2, 0,

15

7

FractionsFractions are numbers such as:

The numerator is the number above the fraction line.

The denominator is the number below the fraction line.

1

3,

2

9, and

9

53.

Reducing FractionsIn order to reduce a fraction to its lowest

terms, we divide both the numerator and denominator by the greatest common divisor.

Example: Reduce to its lowest terms.

Solution:

72

81

72 72 9 8

81 81 9 9

Terminating or Repeating Decimal NumbersEvery rational number when expressed as

a decimal number will be either a terminating or a repeating decimal number.

Examples of terminating decimal numbers are 0.7, 2.85, 0.000045

Examples of repeating decimal numbers 0.44444… which may be written

0.4,

and 0.2323232323... which may be written 0.23.

Multiplication of Fractions

Division of Fractions

a

b

c

d

a c

b d

ac

bd, b 0, d 0

a

b

c

d

a

b

d

c

ad

bc, b 0, d 0, c 0

Example: Multiplying FractionsEvaluate the

following.

a)

b)

2

3

7

16

2

3

7

16

27316

14

48

7

24

1

3

4

2

1

2

13

4

2

1

2

7

45

2

35

84

3

8

Example: Dividing FractionsEvaluate the

following.a)

b)

2

3

6

7

2

3

6

7

2

37

6

2736

14

18

7

9

5

8

4

5

5

8

4

5

5

85

4

5584

25

32

Addition and Subtraction of Fractions

a

c

b

c

a b

c, c 0;

a

c

b

c

a b

c, c 0

Example: Add or Subtract Fractions

Add:

Subtract:

4

9

3

9

4

9

3

9

4 3

9

7

9

11

16

3

16

11

16

3

16

11 3

16

8

16

1

2

Fundamental Law of Rational NumbersIf a, b, and c are integers, with b 0, c 0,

then

a

b

a

bc

c

acbc

ac

bc.

Example:Evaluate:

Solution:

7

12

1

10.

7

12

1

10

7

125

5

1

106

6

35

60

6

60

29

60

Irrational NumbersAn irrational number is a real number

whose decimal representation is a nonterminating, nonrepeating decimal number.

Examples of irrational numbers: 5.12639573...

6.1011011101111...

0.525225222...

Radicals are all irrational

numbers. The symbol is called the radical sign. The number or expression inside the radical sign is called the radicand.

2, 17, 53

Perfect SquareAny number that is the square of a natural

number is said to be a perfect square.The numbers 1, 4, 9, 16, 25, 36, and 49 are

the first few perfect squares.

Real NumbersThe set of real numbers is formed by the

union of the rational and irrational numbers.The symbol for the set of real numbers is .

Relationships Among Sets

Irrational numbers

Rational numbers

Integers

Whole numbersNatural numbers

Real numbers

Properties of the Real Number System Closure

If an operation is performed on any two elements of a set and the result is an element of the set, we say that the set is closed under that given operation.

Commutative PropertyAddition

a + b = b + a for any real numbers a and b.

Multiplication a • b = b • a for any real numbers a and b.

Example8 + 12 = 12 + 8 is a true statement.5 9 = 9 5 is a true statement.

Note: The commutative property does not hold true for subtraction or division.

Associative PropertyAddition (a + b) + c = a + (b +

c),

for any real numbers a, b, and c.

Multiplication (a • b) • c = a • (b • c),

for any real numbers a, b, and c.

Example(3 + 5) + 6 = 3 + (5 + 6) is true.

(4 6) 2 = 4 (6 2) is true.

Note: The associative property does not hold true for subtraction or division.

Distributive PropertyDistributive property of multiplication over

addition

a • (b + c) = a • b + a • c

for any real numbers a, b, and c.

Example: 6 • (r + 12) = 6 • r + 6 • 12 = 6r + 72

ExponentsWhen a number is written with an exponent,

there are two parts to the expression: baseexponent

The exponent tells how many times the base should be multiplied together.

45 44444

Scientific NotationMany scientific problems deal with very large

or very small numbers.93,000,000,000,000 is a very large number.0.000000000482 is a very small number.

Scientific Notation continuedScientific notation is a shorthand method

used to write these numbers.

9.3 1013 and 4.82 10–10 are two examples of numbers in scientific notation.

To Write a Number in Scientific Notation1. Move the decimal point in the original number to

the right or left until you obtain a number greater than or equal to 1 and less than 10.

2. Count the number of places you have moved the decimal point to obtain the number in step 1.If the decimal point was moved to the left, the count is to be considered positive. If the decimal point was moved to the right, the count is to be considered negative.

3. Multiply the number obtained in step 1 by 10 raised to the count found in step 2. (The count found in step 2 is the exponent on the base 10.)

ExampleWrite each number in scientific notation.

a) 1,265,000,000.1.265 109

b) 0.0000000004324.32 1010

To Change a Number in Scientific Notation to Decimal Notation1. Observe the exponent on the 10.2. a) If the exponent is positive, move the

decimal point in the number to the right the same number of places as the exponent. Adding zeros to the number might be necessary.

b) If the exponent is negative, move the decimal point in the number to the left the same number of places as the exponent. Adding zeros might be necessary.

ExampleWrite each number in decimal notation.

a) 4.67 105

467,000

b) 1.45 10–7

0.000000145