# Rational and Irrational Numbers | Mathematics

December 22, 2015

Rational and Irrational Numbers | Mathematics

As you know, the word rational means “logical” and the word irrational means “illogical.” Similarly, numbers hold the same definitions. Understanding and recognizing rational and irrational numbers is a high emphasis topic on the TASC Test Assessing Secondary Completion™ Mathematics subtest.

Let’s dive in and analyze each of these terms individually.

## Rational Numbers

According to MathWarehouse.com, a rational number is any number that can be written as a fraction with a denominator greater than zero.

Study these examples of rational numbers:

• 19 can be written as:  191
• -32 can be written as:  -321
• .75 can be written as:  34
• √9 (square root) = 3 which can be written as:  31

Rational numbers have either a limited number of decimals (such as  = 0.75) or a repeating number of sequenced decimals (such as  = 0.333333333…). It is important to note that repeating decimals – decimals that repeat the same sequence of digits indefinitely – are rational numbers.

For example:

• .125125125125… repeats in a sequence of 125
• .333333333333… repeats in a sequence of 3

## Irrational Numbers

Irrational numbers are numbers that cannot be written as a fraction with a denominator greater than zero.

Study these examples of irrational numbers:

• √11 cannot be simplified into a fraction like √9
• π (pi = 3.14159265359…) cannot be simplified into a fraction
• .0303003000030330… cannot be simplified into a fraction
•  has a denominator of zero

Numbers that repeat indefinitely with non-repeating digits are irrational numbers, as seen in the example of .0303003000030330…

Test your knowledge of rational and irrational numbers by quizzing yourself at KhanAcademy.org.

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