In algebra, a period is a number that can be written an integral of an algebraic function over an algebraic
domain. More specifically, a period is a real number

where
is a polynomial and
is a rational function on with rational coefficients.

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.
However, its elements are constructible and 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 a positive integer where is the Riemann zeta
function,
for
positive integers, and .

Belkale, P. and Brosnan, P. "Periods and Igusa Local Zeta Functions." Int. Math. Res. Not.49, 2655-2670, 2003.Kontsevich,
M. and Zagier, D. "Periods." In athematics
UnlimitedÑ-2001 and Beyond (Ed. B. Engquist and W. Schmid).
Berlin: Springer, pp. 771-808, 2001.Marcolli, M. "Feynman
Integrals and Motives." Europ. Congress Math. Eur. Math. Soc. Zürich,
pp. 293-332, 2010.Waldschmidt, M. "Transcendence of Periods:
The State of the Art." Pure Appl. Math. Quart.2, 435-463, 2006.Yoshinaga,
M. "Periods and Elementary Real Numbers." 03 May 2008. https://arxiv.org/abs/0805.0349.