Irrational number

An ir­ra­tional num­ber is a real num­ber that is not a ra­tio­nal num­ber. This set is gen­er­ally de­noted us­ing ei­ther \(\mathbb{I}\) or \(\overline{\mathbb{Q}}\), the lat­ter of which rep­re­sents it as the com­ple­ment of the ra­tio­nals within the re­als.

In the Cauchy se­quence defi­ni­tion of real num­bers, the ir­ra­tional num­bers are the equiv­alence classes of Cauchy se­quences of ra­tio­nal num­bers that do not con­verge in the ra­tio­nals. In the Dedekind cut defi­ni­tion, the ir­ra­tional num­bers are the one-sided Dedekind cuts where the set \(\mathbb{Q}^\ge\) does not have a least el­e­ment.

Prop­er­ties of ir­ra­tional numbers

Ir­ra­tional num­bers have dec­i­mal ex­pan­sions (and in­deed, rep­re­sen­ta­tions in any base \(b\)) that do not re­peat or ter­mi­nate.

The set of ir­ra­tional num­bers is un­countable.


  • Pi is irrational

    The num­ber pi is fa­mously not ra­tio­nal, in spite of jok­ing at­tempts at leg­is­la­tion to fix its value at 3 or 227.

  • The square root of 2 is irrational

    The num­ber whose square is 2 can’t be writ­ten is a quo­tient of nat­u­ral numbers


  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.