Rational Vs. Irrational

Rational Vs. Irrational Rational Vs. Irrational Rational Vs. Irrational Rational Vs. Irrational

Making sense of rational and Irrational numbers

Making sense of rational and Irrational numbers

AnimalReptile

Biologists classify animals based on shared characteristics. The horned lizard is an animal, a reptile, a lizard, and a gecko.

You already know that some numbers can be classified as whole numbers, integers, or rational numbers. The number 2 is a whole number, an integer, and a rational number. It is also a real number.

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The set of real numbers is all numbers that can be written on a number line. It consists of the set of rational numbers and the set of irrational numbers.

Irrational numbersRational numbers

Real Numbers

Integers

Wholenumbers

Recall that rational numbers can be written as the quotient of two integers (a fraction) or as either terminating or repeating decimals.

3 = 3.84 5

= 0.623

1.44 = 1.2

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. For example, 2 is not a perfect square, so 2 is irrational.

Additional Example 1: Classifying Real NumbersWrite 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.

Check It Out! Example 1

Write all classifications that apply to each number.

9

whole, integer, rational, real

–35.9 is a terminating decimal.–35.9rational, real

81 3

whole, integer, rational, real

= = 39 3

81 3

A.

B.

C.

9 = 3

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 3

rational

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 0

C.

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

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

23

irrational

9 0

undefined, so not a real number

Check It Out! Example 2

A.

B.

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

64 81

rational

8 9

=8 9

64 81

C.

Check It Out! Example 2

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

Lesson Quiz

Write all classifications that apply to each number.

1. 2. –

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

3. 4.

2

4 • 9

16 2

25 0

not a real number rational

real, irrational real, integer, rational