REAL NUMBERS

(as opposed to fake numbers?)

Objective• TSW identify the parts of the Real

Number System• TSW define rational and irrational

numbers• TSW classify numbers as rational

or irrational

Real Numbers• Real Numbers are every number.

• Therefore, any number that you can find on the number line.

• Real Numbers have two categories.

What does it Mean?• The number line goes on forever.• Every point on the line is a REAL

number.• There are no gaps on the number

line.• Between the whole numbers and the

fractions there are numbers that are decimals but they don’t terminate and are not recurring decimals. They go on forever.

Real NumbersREAL NUMBERS

-8

 

-5,632.1010101256849765… 

61

49%

π

 

549.23789

154,769,852,3541.333

Two Kinds of Real Numbers

• Rational Numbers

• Irrational Numbers

Rational Numbers• A rational number is a real

number that can be written as a fraction.

• 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

Integers

One of the subsets of rational numbers

What are integers?• Integers are the whole numbers and

their opposites.• Examples of integers are 6-120186-934

• Integers are rational numbers because they can be written as fraction with 1 as the denominator.

Types of Integers• Natural Numbers(N):

Natural Numbers are counting numbers from 1,2,3,4,5,................N = {1,2,3,4,5,................}

• Whole Numbers (W): Whole numbers are natural numbers including zero. They are 0,1,2,3,4,5,...............W = {0,1,2,3,4,5,..............} W = 0 + N

WHOLENumber

s

REAL NUMBERS

IRRATIONALNumbers

NATURALNumbers

RATIONALNumbers

INTEGERS

Irrational Numbers• An irrational number is a

number that cannot be written as a fraction of two integers.

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

A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.

Caution!

Irrational numbers can be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an irrational number.

Examples of Irrational Numbers

•  • Pi

 

Try this!• a) Irrational

• b) Irrational

• c) Rational

• d) Rational

• e) Irrational

66 e)

d)25 c)

12 b)

2 a)

115

Additional Example 1: Classifying Real Numbers

Write all classifications that apply to each number.

5 is a whole number that is not a perfect square.

5

irrational, real

–12.75 is a terminating decimal.–12.75rational, real

16 2

whole, integer, rational, real= = 24

2 16

2

A.

B.

C.

A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.

State if each number is rational, irrational, or not a real number.

21

irrational

0 3rational

0 3 = 0

Additional Example 2: Determining the Classification of All Numbers

A.

B.

not a real number

Additional Example 2: Determining the Classification of All Numbers

4 0C.

State if each number is rational, irrational, or not a real number.

Objective• TSW compare rational and

irrational numbers• TSW order rational and irrational

numbers on a number line

Comparing Rational and Irrational Numbers

• When comparing different forms of rational and irrational numbers, convert the numbers to the same form.

Compare -3 and -3.571 (convert -3 to -3.428571…

-3.428571… > -3.571

37

37

Practice•  

Ordering Rational and Irrational Numbers

• To order rational and irrational numbers, convert all of the numbers to the same form.

• You can also find the approximate locations of rational and irrational numbers on a number line.

Example• Order these numbers from least to

greatest. ¹/₄, 75%, .04, 10%, ⁹/₇

¹/₄ becomes 0.2575% becomes 0.750.04 stays 0.0410% becomes 0.10⁹/₇ becomes 1.2857142…

Answer: 0.04, 10%, ¹/₄, 75%, ⁹/₇

PracticeOrder these from least to greatest: