# Rational Numbers

Created | Updated Dec 23, 2008

Numbers
| A History of Numbers
| Propositional Logic
| Logical Completeness
| The Liar's Paradox

Logical Consistency
| Basic Methods of Mathematical Proof
| Integers and Natural Numbers

Rational Numbers
| Irrational Numbers
| Imaginary Numbers
| The Euler Equation

Rational numbers sound very sensible and logical, but in fact the name means merely that the numbers are ratios. That is, they are one whole number divided by another whole number. More technically, they can all be expressed in the form ^{a}/_{b}, where a and b are both members of the set of integers, and b is non-zero. Examples of rational numbers are ^{1}/_{2}, 400, -^{4}/_{7}, -6. Integers are a special case of rational numbers where b, the denominator, is equal to 1.

If you express a rational number as a decimal, it will either be terminating, as in the case of ^{1}/_{5} = 0.2, or recurring, as in the case of ^{14}/_{33} = 0.42424242...

There is always more than one way of writing the same rational number. Two rational numbers are equivalent if ^{a}/_{b} = ^{c}/_{d}, where a, b, c and d are all integers, and b and d are not equal to zero. For example, ^{3}/_{6} = ^{1}/_{2}. One can obtain the most basic form of a rational number if a and b have no common factors. In this example, 3 and 6 have a common factor 3. Dividing the top and bottom of the fraction by 3 gives ^{1}/_{2}, which cannot be simplified.

The Pythagoreans determinedly believed that the set of rational numbers was the set of all numbers, and tried to hide any evidence to the contrary. See the entry on Irrational Numbers for more information.