# Rational and Irrational Numbers Defined and Explained  ## Rational Numbers Definition

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

All integers, positive or negative are rational numbers because they divide by 1.

Zero is a rational number that can be written as a ratio of two integer numbers, as 0/1.

Rational numbers can be written or expressed as either an ordinary (terminating) decimal or as an infinite repeating decimal.

Rational Numbers Examples:

• 5 can be written as a ratio of two numbers, 5/1.

It can be written as 5.0, an ordinary decimal.

• 1/11 is a ratio of two integers. Its decimal equivalent is

.090909… The decimal portion 09 repeats.

The decimal value never terminates.

• 2/3 is a ratio of two integer numbers. Its decimal

equivalent is .6666666… The decimal is infinite

repeating, but never terminates. It is a rational number.

• 0.5 is the decimal equivalent of two integer numbers, 1/2.

It is a decimal that terminates, but does not repeat.

It is a rational decimal.

• √4 can be written as a ratio of two numbers. The √4

is 2. 2 can be written as 2/1. There is no decimal place

value for 2 or 2/1. It is a rational square root number.
Some other rational square root numbers are:

√9, √16, √36, √49, √64.

• log10 x  is 2 when x = 100. 2 is an integer, it is rational. ## Irrational Numbers Definition

Real numbers that cannot be written as a ratio of two integers are irrational. They are decimal numbers that both do not terminate and do not repeat. Irrational numbers are infinite non-repeating decimal numbers.

Irrational Numbers Examples:

• √2 is not a ratio of two integers. Its decimal equivalent

is 1.4142135… The decimal never repeats and never     terminates. It cannot be written as a ratio of two integers.

It is an irrational square root number.

√3, √5, √7, √10, 2 + √2, 4 − √7.

• π is a math symbol with decimal value 3.1415926…

It cannot be written as a ratio of two integers. It is an

infinite non-repeating decimal number.

• e is a math symbol with decimal value 2.71828…

It cannot be written as a ratio of two integers. It is an

infinite non-repeating decimal number.

• log10 x  when x = 2 is 0.30102999566… Is not a ratio

of two integers. It is irrational. Top of Page