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This essential trains for: SAT-I, GMAT, AMC-8, AMC-10, Math Kangaroo 5-6, Math Kangaroo 7-8.

Definition: Rational numbers are numbers that result from the division of two integer numbers as long as the divisor is non-zero.

Example:

equation

Fact: A rational number can be written as a fraction or as a decimal number (by performing the long division). This decimal number may be terminating or non-terminating.

Fact: For any rational number, the corresponding non-terminating decimal part must have a repeating part.

Fact: Any non-terminating, non-repeating decimal is not a rational number.

Example: 0.313113111311113...

Of course, in the example above the digits 1 and 3 do repeat but they do so without forming a pattern of fixed finite length.

Fact: An integer number is, of course, also a rational number (with denominator 1).

Mathematical definition using symbols pertaining to set theory:

equation

The set of rational numbers is denoted by Q.

The set of integer numbers is denoted by Z.