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	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