Ancient Greece
The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum), who probably discovered them while … See more

Square roots
The square root of 2 was likely the first number proved irrational. The golden ratio is another famous … See more

The decimal expansion of an irrational number never repeats (meaning the decimal expansion does not repeat the same number or sequence of numbers) or terminates (this means there is not a finite number of nonzero digits), unlike any rational number. … See more

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 … See more

An irrational number may be algebraic, that is a real root of a polynomial with integer coefficients. Those that are not algebraic are transcendental.
Algebraic See more

Dov Jarden gave a simple non-constructive proof that there exist two irrational numbers a and b, such that a is rational:
Consider √2 ; if this … See more

Since the reals form an uncountable set, of which the rationals are a countable subset, the complementary set of irrationals is uncountable.
Under the usual (Euclidean) distance function $${\displaystyle d(x,y)=\vert x-y\vert }$$ See more