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In constructive mathematics

In constructive mathematics, excluded middle is not valid, so it is not true that every real number is rational or irrational. Thus, the notion of an irrational number bifurcates into multiple distinct notions. One could take the traditional definition of an irrational number as a real number that is not rational. However, there is a second definition of an irrational number used in constructive mathematics, that a real number r {\displaystyle r} is an irrational number if it is apart from every rational number, or equivalently, if the distance | r − q | {\displaystyle \vert r-q\vert } between r {\displaystyle r} and every rational number q {\displaystyle q} is positive. This definition is stronger than the traditional definition of an irrational number. This second definition is used in Errett Bishop 's proof that the square root of 2 is irrational.