An algebraic period is a complex number whose real part and imaginary part can each be expressed as an absolutely convergent integral
 |
where 
is a rational function with rational
coefficients and 
is a finite union of domains defined by finitely many inequalities
between polynomials with rational
coefficients (Kontsevich and Zagier 2001).
Periods (which are countable) were defined to fill in a gap between algebraic numbers (which do not contain many mathematical constants) and the transcendental numbers (which are not countable). In particular, any algebraic number is a period and any number that is not a period is a transcendental number (Waldschmidt 2006), so there is a "gap" between those two statements in the sense that algebraic periods may be algebraic or transcendental. As a result, Kontsevich and Zagier (2001) propose their Principle 1: "Whenever you meet a new number, and have decided (or convinced yourself) that it is transcendental, try to figure out whether it is a period."
Periods form a ring since sums and products of periods are also periods. However, this class of numbers is larger and less well understood than the ring of algebraic numbers. Periods have explicit integral descriptions, but need not be constructible numbers in the straightedge-and-compass sense. It is conjectured that equality of any two numbers which have been expressed as periods can be verified. Most of the important constants of mathematics belong to the class of periods (Kontsevich and Zagier 2001).
Examples of periods include
 |
for integers 
, where 
is the Riemann zeta
function, and ![[Gamma(p/q)]^q](/images/equations/AlgebraicPeriod/Inline6.svg)
for positive integers 
and 
, where 
is the gamma function
(Kontsevich and Zagier 2001).
Chaitin's constant 
is not a period because it is a noncomputable
number, whereas every real period is a computable
number (Yoshinaga 2008). In particular, a period can be approximated by an algorithm to any prescribed positive rational number error tolerance. The noncomputability
of 
rules out such an algorithm.
It is not known if 
,
, or the Euler-Mascheroni
constant 
are periods, though it is conjectured that 
and 
are not.
