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 [set_complement 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.