In mathematics, an irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore, irrational numbers, when written as decimal numbers, do not terminate, nor do they repeat. For example, the number π starts with 3.14159265359, but no finite number of digits can represent it exactly and it does not end in a segment that repeats itself infinitely often. The same can be said for any irrational number - wikipedia

As a consequence of Cantor's proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational.

When the ratio of lengths of two line segments is irrational, the line segments are also described as being incommensurable, meaning they share no measure in common.

Numbers which are irrational include the ratio π of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two; in fact all square roots of natural numbers, other than of perfect squares, are irrational.