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
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
The irrationality of pi itself was proven by Lambert
in 1760; for the general case, see Hardy and Wright (1979, p. 47). Apéry's
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).
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