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irrational number

A real number that can't be written as one whole number divided by another; in other words, a real number that isn't a rational number. The decimal expansion of an irrational numbers doesn't come to an end or repeat itself (in equal length blocks), though it may have a pattern such as 0.101001000100001... The vast majority of real numbers are irrational, so that if you were to pick a single point on the real number line at random the chances are overwhelmingly high that it would be irrational. Put another way, whereas the set of all rationals is countable, the irrationals form an uncountable set and therefore represent a larger kind of infinity. Indeed, as the Harvard logician Willard Van Orman Quine pointed out: "The irrationals exist in such variety ... that no notation whatever is capable of providing a separate name for each of them."

There are two types of irrational number: algebraic numbers, such as the square root of 2, which are the roots of algebraic equations, and the transcendental numbers, such as π and e, which aren't. In some cases it isn't known if a number is irrational or not; undecided cases include 2e, πe, and the Euler-Mascheroni constant, γ (gamma).

An irrational number raised to a rational power can be rational; for instance, √2 to the power 2 is 2. Also, an irrational number to an irrational power can be rational. What kind of number is √2√2? The answer is irrational. This follows from the so-called Gelfond-Schneider theorem, which says that if A and B are roots of polynomials, and A is not 0 or 1 and B is irrational, then AB must be irrational (in fact, transcendental).

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