An irrational number is a number that cannot be expressed as a fraction for any integers and . Irrational numbers have decimal expansions that neither terminate nor become periodic. Every transcendental number is irrational.
There is no standard notation for the set of irrational numbers, but the notations , , or , where the bar, minus sign, or backslash indicates the set complement of the rational numbers over the reals , could all be used.
The most famous irrational number is , sometimes called Pythagoras's constant. Legend has it that the Pythagorean philosopher Hippasus used geometric methods to demonstrate the irrationality of while at sea and, upon notifying his comrades of his great discovery, was immediately thrown overboard by the fanatic Pythagoreans. Other examples include , , , etc. The Erdős-Borwein constant
(OEIS A065442; Erdős 1948, Guy 1994), where is the numbers of divisors of , and a set of generalizations (Borwein 1992) are also known to be irrational (Bailey and Crandall 2002).
Numbers of the form are irrational unless is the th power of an integer. Numbers of the form , where is the logarithm, are irrational if and are integers, one of which has a prime factor which the other lacks. is irrational for rational . is irrational for every rational number (Niven 1956, Stevens 1999), and (for measured in degrees) is irrational for every rational with the exception of (Niven 1956). is irrational for every rational (Stevens 1999).
The irrationality of e was proven by Euler in 1737; for the general case, see Hardy and Wright (1979, p. 46). is irrational for positive integral . The irrationality of pi itself was proven by Lambert in 1760; for the general case, see Hardy and Wright (1979, p. 47). Apéry's constant (where is the Riemann zeta function) was proved irrational by Apéry (1979; van der Poorten 1979). In addition, T. Rivoal (2000) recently proved that there are infinitely many integers such that is irrational. Subsequently, he also showed that at least one of , , ..., is irrational (Rivoal 2001).
From Gelfond's theorem, a number of the form is transcendental (and therefore irrational) if is algebraic , 1 and is irrational and algebraic. This establishes the irrationality of Gelfond's constant (since ), and . Nesterenko (1996) proved that is irrational. In fact, he proved that , and are algebraically independent, but it was not previously known that was irrational.
Given a polynomial equation
where are integers, the roots are either integral or irrational. If is irrational, then so are , , and .
Irrationality has not yet been established for , , , or (where is the Euler-Mascheroni constant).
Quadratic surds are irrational numbers which haveperiodic continued fractions.
Hurwitz's irrational number theoremgives bounds of the form
for the best rational approximation possible for an arbitrary irrational number , where the are called Lagrange numbers and get steadily larger for each "bad" set of irrational numbers which is excluded.
where is the divisor function, is irrational for and 2.