Introduction to Real Numbers Introduction to Real Numbers and and

Their PropertiesTheir Properties

Two Kinds of Real Numbers

• Rational Numbers

• Irrational Numbers

Rational Numbers

• A rational number is a real number that can be written as a ratio of two integers.

• A rational number written in decimal form is terminating or repeating.

Examples of Rational Numbers

•16•1/2•3.56

•-8•1.3333…•- 3/4

Irrational Numbers

• An irrational number is a number that cannot be written as a ratio of two integers.

• Irrational numbers written as decimals are non-terminating and non-repeating.

Examples of Irrational Numbers

• Square roots of non-perfect “squares”

• Pi

17

What are integers?• Integers are the

whole numbers and their opposites.

• Examples of integers are

6-120186-934

Using Exponents

If “a” is a real number and “n” is a natural number, then an = a•a•a•••a•a (n factors of a).

where n is the exponent, a is the base, and an is an exponential expression. Exponents are also called powers.

To find the value of a whole number exponent:

100 = 1, 20 = 1, 80 = 1, #0 = 1101 = 10, 21 = 2, 81 = 8, #1 = #102 = 10 x 10 = 100, 22 = 2 x 2 = 4, 82 = 8 x 8 = 64103 = 10 x 10 x 10 = 1000, 23 = 2 x 2 x 2 = 8104 = 10 x 10 x 10 x 10 = 10,000 24 = 2 x 2 x 2 x 2 = 16(-10)3 = (-10)(-10)(-10) (12).5 =

Using the Identity Properties “additive identity”

Zero is the only number that can be added to any number to get that number.

0 is called the “identity element for addition” a + 0 = a Example 1: 4 + 0 = 4

“multiplicative identity”

One is the only number that can be multiplied by any number to get that number.

1 is called the “identity element for multiplication”

a • 1 = a Example 2: 4 • 1 = 4

The Real Number SystemThe Real Number SystemReal Numbers

Rational Numbers Irrational Numbers

3

1/2-2

15%2/3

1.456

-0.7

0

√3 2π

−√5 2

3π4

The Real Number The Real Number SystemSystem Real Numbers

Rational Numbers Irrational Numbers

31/2 -2

15%

2/3

1.456

- 0.7

0

√3 2π

−√5 2

3π4

Integers

The Real Number SystemReal Numbers

Rational Numbers Irrational Numbers

31/2

-2

15%

2/3

1.456

- 0.7

0

√3 2π

−√5 2

3π4

Integers Whole

The Real Number SystemReal Numbers

Rational Numbers Irrational Numbers

31/2

-2

15%

2/3

1.456

- 0.7

0√3 2

π−√5 2

3π4

Integers Whole Natural

Finding Additive inversesFinding Additive inverses

For any real number x, the number –x is the For any real number x, the number –x is the

additive inverse of x.additive inverse of x.

Example 1:Example 1:Number

Inverse Additive

6 - 6- 4 4

- 8.7 8.70 0

2

32

3−

Symbol Meaning Example

= is equal to 4 = 4

≠ is not equal to 4 ≠ 5

< is less than 4 < 5

≤ is less than or equal -4 ≤ -3

> is greater than -4 > -5

≥ is greater than or equal -8 ≥ - 10

Number Reciprocal or Inverse Additive Inverse

−6 6

0.05 20 -0.05

0 none 0

2

5− 5

2− 2

51

6−

11

77

11

7

11−