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# Rational number

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A rational number is a number that can be expressed as the ratio between two integers, usually written as a / b, where the denominator? (here b) is not equal to zero. Rational numbers are commonly called "fractions".

Mathematically we may define them as an ordered pair of integers (a, b), with b not equal to zero. We can define addition and multiplication upon these pairs with the following rules:

  (a, b) + (c, d) = (a * d + b *c, b * d)
  (a, b) * (c, d) = (a * c, b * d)

To conform to our expectation that 2/4 = 1/2, we define an equivalence relation ~ upon these pairs with the following rule:

  (a, b) ~ (c, d) if, and only if, a * d = b * c.

So defined, i.e., by the quotient set defined by ~, the set of rational numbers, Q, forms a field. It may be shown that Q is the smallest field which contains the integers.

### See also:

-- integer -- irrational number -- real number --

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Last edited July 18, 2001 6:42 am (diff)
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