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.

• Other irrational square root numbers are:

    √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.

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