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Rational Numbers (Advanced)

The rational numbers are those numbers which can be expressed as a ratio between two integers. For example, the fractions 1 3 and 1111 8 are both rational numbers. All the integers are included in the rational numbers, since any integer z can be written as the ratio z 1 .

Numbers which cannot be written as a ratio of integers are called irrational .

All decimals which terminate are rational numbers (since 8.27 can be written as 827 100 .) Decimals which have a repeating pattern after some point are also rationals: for example,

0.0 83333333 = 1 12 .

The set of rational numbers is closed under all four basic operations: that is, given any two rational numbers, their sum, difference, product, and quotient is also a rational number (as long as we don't divide by 0 .)

The Venn diagram below shows the relationships of the various sets of numbers.

Venn diagram showing subset relationships in real numbers, rational numbers, integers, natural numbers