A real number that is not rational. A famous proof, sometimes attributed to Pythagoras, shows that $\sqrt{2}$ is irrational; the method can also be used to show that numbers such as $\sqrt{3}$ and $\sqrt{7}$ are also irrational. It follows that numbers like $1+\sqrt{2}$ and $1/(1+\sqrt{2})$ are irrational. The proof that e is irrational is reasonably easy, and in 1761 Lambert showed that π‎ is irrational. There are uncountably many irrationals, compared with denumerably many rationals, so in a sense ‘most’ real numbers are irrational. Any real number with a non-recurring decimal representation is irrational. See also transcendental.