# irrational

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

 $x\neq\frac{a}{b}$

with $a,b\in\mathbb{Z}$ and $b\neq 0$.

## Examples

1. 1.

$\sqrt[p]{2}$ is irrational for $p=2,3,\ldots$,

2. 2.

$\pi,e$, and $\sqrt[p]{2}$ for $p=2,3,\ldots$, are irrational,

3. 3.

It is not known whether Euler’s constant is rational or irrational.

## Properties

1. 1.

It $a$ is a real number and $a^{n}$ is irrational for some $n=2,3,\ldots$, then $a$ is irrational (proof (http://planetmath.org/IfAnIsIrrationalThenAIsIrrational)).

2. 2.

The sum, difference, product, and quotient (when defined) of two numbers, one rational and another irrational, is irrational. (proof (http://planetmath.org/RationalAndIrrational)).

 Title irrational Canonical name Irrational Date of creation 2013-03-22 11:55:59 Last modified on 2013-03-22 11:55:59 Owner yark (2760) Last modified by yark (2760) Numerical id 12 Author yark (2760) Entry type Definition Classification msc 11J82 Classification msc 11J72 Synonym irrational number Related topic TranscedentalNumber Related topic AlgebraicNumber Related topic Integer Related topic LindemannWeierstrassTheorem Related topic GelfondsTheorem Related topic ProofThatTheRationalsAreCountable