An irrational number is a real number that cannot be expressed as a fraction 
for integers 
and 
with 
. Irrational numbers have decimal
expansions that neither terminate nor become periodic. Every irrational number
is either an algebraic number or a real transcendental
number. The irrational numbers form a dense set
in the real line, since every nonempty open
interval contains an irrational number.
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 
,
e, and pi. The Erdős-Borwein
constant
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
(OEIS A065442; Erdős 1948, Guy 1994), where 
is the number of divisors of 
, and a set of generalizations (Borwein
1992) are also known to be irrational (Bailey and Crandall 2002).
A real mth root of an integer 
, where 
is a positive integer,
is irrational unless 
is the 
th
power of an integer. Numbers
of the form 
,
where 
is the logarithm, are irrational if 
and 
are integers greater than 1 and
one 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
integer 
.
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 (Apéry 1979, van der Poorten 1979). Rivoal (2000) proved that there
are infinitely many integers 
such that 
is irrational. Subsequently, he also showed that
at least one of 
,
,
..., 
is irrational (Rivoal 2001). Fauzan (2026) announced a proof
that zeta(5) is irrational.
Sun (2026) and OpenAI (2026) reported proofs of the irrationality of Catalan's constant. As of Oct. 7, 2026, independent specialist verification of the complete arguments had not been reported, and VibeMathed (2026) had not independently rebuilt the accompanying OpenAI Lean formalization.
By Gelfond's theorem, every value of 
is transcendental
if 
is algebraic with 
, 1 and 
is algebraic but not a
rational number. Each real value is therefore
irrational. This establishes the irrationality of Gelfond's
constant 
(using the principal value 
) and Gelfond-Schneider
constant 
.
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
 |
(4)
|
where 
is a positive integer and 
are integers, the real
roots 
are either integral or irrational. For real 
, 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 have eventually periodic regular continued fractions.
Hurwitz's irrational number theorem gives bounds of the form
 |
(5)
|
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.
The series
 |
(6)
|
where 
is the divisor function, is irrational for 
and 2.
et 
." Astérisque 61, 11-13, 1979.Bailey,
D. H. and Crandall, R. E. "Random Generators and Normal Numbers."
Exper. Math. 11, 527-546, 2002. 
Is Irrational." Preprint. 17 Sep 2026. 
,
,
,
and 
."
Amer. Math. Monthly 108, 222-231, 2001.Manning, H. P.
and 
."
§12-13 in 
,
,
..., 
."
25 Apr 2001. 
." Mitt. Math. Ges. Hamburg 18, 151-158,
1999.Sun, Z.-W. "Catalan's Constant Is Irrational." 3 Sep
2026. 
."
Math. Intel. 1, 196-203, 1979.VibeMathed. "Irrationality
of Catalan's Constant." Oct. 7, 2026.