Almost every real number is irrational
Irrationals are reals that are not ratios of integers—π, e, the golden ratio φ, √2 among them. Their decimals never terminate or repeat. Cantor’s contrast of uncountable reals with countable rationals implies almost all reals fall in the irrational camp.
Geometrically, two segments whose length ratio is irrational are called incommensurable: no unit, however tiny, fits a whole number of times into both. Irrationals can also be written as endless continued fractions, sometimes periodic ones. Conversely, any decimal that stops or repeats represents a rational number, a provable fact rather than a definition.
The first proof of their existence is credited to a Pythagorean, probably Hippasus of Metapontum in the 5th century BC, perhaps while studying the sides of a pentagram. His argument takes an isosceles right triangle in lowest whole-number terms. Since the hypotenuse squared equals twice a leg squared, the hypotenuse must be even; writing it as twice some integer then forces the leg to be even as well, contradicting lowest terms. Greeks called such ratios alogos, inexpressible, and legend says fellow Pythagoreans threw Hippasus overboard for undermining their creed that everything reduces to whole numbers, though another version has him merely exiled.
The crisis fed a wider debate. Zeno of Elea questioned whether quantities are really built from discrete units, and Eudoxus of Cnidus answered with a theory of proportion that separated continuously varying magnitudes from jumping numbers. Greek mathematics consequently leaned hard on geometry, which may explain why we still say x squared and x cubed. Theodorus of Cyrene proved the roots of whole numbers irrational up to 17 and apparently halted there.
In India, square-root problems appear in the Shulba Sutras of 800 BC or earlier; Brahmagupta and Bhāskara I worked on surd arithmetic in the 7th century AD, and Madhava's Kerala school later found infinite series for π. Medieval Islamic algebra treated irrationals as algebraic objects, and the Persian Al-Mahani classified quadratic and cubic irrationals, counting integers and fractions as rational and roots of non-squares as irrational.
Source: Irrational number
