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
$$x\ne \frac{a}{b}$$ 
with $a,b\in \mathbb{Z}$ and $b\ne 0$.
Examples

1.
$\sqrt[p]{2}$ is irrational for $p=2,3,\mathrm{\dots}$,

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

3.
It is not known whether Euler’s constant is rational or irrational.
Properties

1.
It $a$ is a real number and ${a}^{n}$ is irrational for some $n=2,3,\mathrm{\dots}$, then $a$ is irrational (proof (http://planetmath.org/IfAnIsIrrationalThenAIsIrrational)).

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  20130322 11:55:59 
Last modified on  20130322 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 