In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. Informally, this means numbers that cannot be represented as simple fractions. It can be deduced that they also cannot be represented as terminating or repeating decimals, but the idea is more profound than that. 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. Perhaps the most well known irrational numbers are π, e and 2.

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. A measure of a line segment I in this sense is a line segment J that "measures" I in the sense that some whole number of copies of J laid end-to-end occupy the same length as I.

History

The concept of irrationality was implicitly accepted by Indian mathematicians since the 7th century BC, when Manava (c. 750690 BC) was aware that the square roots of certain numbers such as 2 and 61 could not be exactly determined.

The first proof of the existence of irrational numbers is usually attributed to Hippasus of Metapontum, a Pythagorean who probably discovered them while identifying sides of the pentagram. The then-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. However, Hippasus, in the 5th century BC, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction. He did this by demonstrating that if the hypotenuse of an isosceles right triangle was indeed commensurable with an arm, then that unit of measure must be both odd and even, which is impossible. His reasoning is as follows:

• The ratio of the hypotenuse to an arm of an isosceles right triangle is a:b expressed in the smallest units possible.
• By the Pythagorean theorem: a² = 2b².
• Since a² is even, a must be even as the square of an odd number is odd.
• Since a:b is in its lowest terms, b must be odd.
• Since a is even, let a = 2y.
• Then a² = 4y² = 2b²
• b² = 2y² so b² must be even, therefore b is even.
• However we asserted b must be odd. Here is the contradiction.

Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible, but according to legend did not give Hippasus the respect he deserved. It is said that he made this discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans “…for having produced an element in the universe which denied the…doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.” Hippasus' discovery posed a very serious problem to Pythagorean mathematics, since it shattered the assumption that number and geometry were inseparable, a foundation of their theory.