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".
### See also:

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, denoted by **Q**, forms a field.
It may be shown that **Q** is the smallest field which contains the integers.

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