What Is an Irrational Number? Quantities No Fraction Can Express

A rational number is a ratio of two whole numbers, and the rationals are dense, meaning that between any two of them lies another, so it is not obvious that anything is left out. An irrational number is a quantity that is not such a ratio, and the standard example is the length of the diagonal of a square with sides of one unit, which the theorem about right-angled triangles makes the square root of two. Writing such a number as a decimal produces an expansion that never terminates and never settles into a repeating cycle, which is equivalent to not being a ratio, since any terminating or repeating decimal can be converted into a fraction by a standard procedure. That equivalence is the practical test and the proof of irrationality is the mathematical one.

The argument that the square root of two is irrational is short and is a standard example of proof by contradiction:

• •Suppose it equals a fraction in lowest terms, so the numerator and denominator share no common factor
• •Squaring gives that the numerator squared equals twice the denominator squared, so the numerator squared is even
• •A number whose square is even must itself be even, so the numerator is even and can be written as twice something
• •Substituting shows the denominator squared is also even, so the denominator is even too
• •Both are therefore even, which contradicts the assumption that the fraction was in lowest terms
• •The supposition is impossible, so no such fraction exists

The result is attributed to the Pythagorean school in ancient Greece and is reported to have caused genuine difficulty, since their philosophy held that everything could be expressed in ratios of whole numbers, and a length that could not was a direct contradiction of a foundational commitment rather than a technical curiosity. A much-repeated story has the discoverer drowned at sea, either as punishment or by divine displeasure, and the story is late, poorly attested and probably legend, though the intellectual crisis it dramatises appears to have been real. The consequence within Greek mathematics was substantial, contributing to a shift towards geometric rather than numerical treatment of quantity, since lengths could be handled geometrically without needing to be expressed as numbers at all.

These numbers are unavoidable in ordinary calculation and are handled by approximation. Any measurement is a rational number, since it is recorded to finite precision, so an irrational quantity is never used directly and a decimal approximation is used instead, with the precision chosen to suit the purpose. Engineering and physics use a handful of digits, and the number of digits genuinely needed for practical work is far smaller than most people assume, with a couple of dozen sufficing for calculations at the scale of the solar system. Continued fractions give the best rational approximations for a given size of denominator and produce the familiar simple fractions used for pi. Computation of enormous numbers of digits continues and serves as a test of hardware and algorithms rather than answering any mathematical question, since nothing depends on knowing them.

The irrationals turn out to be overwhelmingly more numerous than the rationals, which was established in the nineteenth century and is counterintuitive. The rationals can be arranged in a list that includes every one of them, which makes them countable despite being infinite. The real numbers cannot be listed, which Cantor demonstrated by showing that any proposed list omits a number constructible from it, and since the rationals are countable the irrationals must account for the difference. That means almost every real number is irrational in a precise sense. A further distinction separates algebraic irrationals, which are solutions of polynomial equations with whole number coefficients, from transcendental ones which are not, and the transcendental numbers include pi and the base of natural logarithms and are themselves the overwhelming majority.

Such a number is not a ratio of whole numbers, which is equivalent to its decimal expansion never terminating and never repeating. The proof for the square root of two supposes a fraction in lowest terms and shows both parts must be even, which is a contradiction. The story of the discoverer drowned at sea is late and probably legend. Almost every real number is irrational, in a sense that can be made precise.