irrational (original) (raw)

An irrational number is a real number which cannot be represented as a ratio of two integers. That is, if x is irrational, then

with a,b∈ℤ and b≠0.

Examples

1. 1. 2p is irrational for p=2,3,…,
2. 1. π,e, and 2p for p=2,3,…, are irrational,
3. 1. It is not known whether Euler’s constant is rational or irrational.

Properties

1. 1. It a is a real number and an is irrational for some n=2,3,…, then a is irrational (proof (http://planetmath.org/IfAnIsIrrationalThenAIsIrrational)).
2. 1. The sum, difference, product, and quotient (when defined) of two numbers, one rational and another irrational, is irrational. (proof (http://planetmath.org/RationalAndIrrational)).