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 and b0.


  1. 1.

    2p is irrational for p=2,3,,

  2. 2.

    π,e, and 2p for p=2,3,, are irrational,

  3. 3.

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


  1. 1.

    It a is a real number and an is irrational for some n=2,3,, then a is irrational (proof (

  2. 2.

    The sum, difference, product, and quotient (when defined) of two numbers, one rational and another irrational, is irrational. (proof (

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