04 real numbers

The Real Number System

Math 11 – College Algebra

“Education is not the filling of a bucket, but the lighting of a fire.”

- W.B. Yeats

Real Numbers

• Real numbers consist of all the rational and irrational numbers.

• The real number system has many subsets:– Natural Numbers – Whole Numbers – Integers

Natural Numbers

• Natural numbers are the set of counting numbers. {1, 2, 3,…}

Denoted by N N={1,2,3,...}

Set of Positive integers

Whole Numbers

• Whole numbers are the set of numbers that include 0 plus the set of natural numbers.

{0, 1, 2, 3, 4, 5,…}

Denoted by W W={0,1,2,3,...}

Integers

• Integers are the set of whole numbers and their opposites.

{…,-3, -2, -1, 0, 1, 2, 3,…}

Denoted by J J={-3,-2,-1,0,1,2,3,...}

Set of Positive Integers, Negative Integers and Zero

Rational Numbers

• Rational numbers are any numbers that can be expressed in the form of , where a and b are integers, and b ≠ 0. b

a

Denoted by Q

Q ={...,-52, -8/5,0,7/8,18,72,...}

Terminating Decimals

• Terminating decimals are decimals that contain a finite number of digits.

• Examples:36.80.1254.5

Repeating Decimals

• Repeating decimals are decimals that contain a infinite number of digits.

• Examples: 0.333… 7.689689…

FYI…The line above the decimals indicate that numberrepeats.

9.1

J and Q

J Q

Terminating Decimals

35

0.60

94 2.25

Consider the following rational numbers:

Nonterminating and Repeating Decimals

Consider the following rational numbers:

13

0.333...4711 4.272727...

Question

The Answer is

Irrational Numbers

2 1.41421356...

3.14159...Denoted

by H3{..., 11, 2, , 17,...}H

REAL NUMBERS

H Q

Irrational Numbers

• Irrational numbers are any numbers that cannot be expressed as .

• They are expressed as non-terminating, non-repeating decimals; decimals that go on forever without repeating a pattern.

• Examples of irrational numbers:– 0.34334333433334…– 45.86745893…– (pi)–

ba

2

ba

Venn Diagram of the Real Number System

Irrational NumbersRational Numbers

Example

• Classify all the following numbers as natural, whole, integer, rational, or irrational. (N, J, Q, H) List all that apply.

a. 117b. 0c. -12.64039…d. -½e. 6.36f. g. -3

To show how these number are classified, use the Venn diagram. Place the number where it belongs on the Venn

diagram.

94

21

Rational NumbersIntegers

Whole Numbers

NaturalNumbers

Irrational Numbers

-12.64039…

117

0

6.3694

-3

The Bigger PictureNatural Numbers

(N)Positive Integers

Zero (0)

Whole Numbers (W)

Negative Integers

Integers (J) Noninteger Rational Numbers

Rational Numbers (Q)

Irrational Numbers (H)

REAL NUMBERS (R)

FYI…For Your Information

• When taking the square root of any number that is not a perfect square, the resulting decimal will be non-terminating and non-repeating. Therefore, those numbers are always irrational.

Try This!

• Insert either or to make the statements correct

a.N__J d. {0}__Qb.Q__J e. { , , 3.5}__Hc.N__R

/

2

Try This!

• In each of the following, determine which one of the sets N,J,Q,H,R and is equal to the given set

a.J Q c. N H b.J Q d. H R

/

Try This!

• Insert either or to make the statements correct

a.N__J d. {0}__Qb.Q__J e. { , , 3.5}__Hc.N__R

/

2

Try This!

• In each of the following, determine which one of the sets N,J,Q,H,R and is equal to the given set

a.J Q c. N H b.J Q d. H R

/