Irrational (mathematics)

An irrational number is a number that cannot be expressed as a fraction of two integers. Unlike Rational numbers, which have decimal representations that either terminate or repeat endlessly in a pattern, irrationals have non-repeating, non-terminating decimals stretching infinitely without ever settling into a cycle.

The most famous examples are Pi, the ratio of a circle's circumference to its diameter, and Euler's number, the base of natural logarithms. The square root of 2 was the first number proven irrational by ancient Greek mathematicians—a discovery that shook their belief that all quantities could be expressed as ratios.

Irrationals are surprisingly common: the square roots of most non-perfect squares are irrational, as are transcendental numbers like π and e. Together, rationals and irrationals form the Real numbers, the continuous number line familiar from everyday geometry.

The existence of irrationals reveals something profound: Numbers are richer and stranger than simple fractions suggest. They underpin calculus, trigonometry, and countless applications in physics and engineering where precision demands we work with these elusive, infinite decimal expansions.

Related

Rational numbers Real numbers Transcendental numbers Prime numbers Golden ratio