In elementary mathematics each group of three digits in a number is called a period (for example, ones period, thousands period, millions period).
In this definition, "rational" can be exchanged for "algebraic" without changing the meaning, since irrational algebraic numbers and functions are themselves expressible as integrals of rational functions over rational domains.
The set of periods is denoted by and places in the number hierarchy as
where denotes the algebraic numbers.
Sums and products of periods remain periods, so the periods form an algebra.
Besides the algebraic numbers, the following numbers are known to be periods:
The periods can be extended to the exponential periods by permitting the product of an algebraic function and the exponential function of an algebraic function as an integrand. This extension includes all algebraic powers of e, the gamma function of rational arguments, and values of Bessel functions. If Euler's constant is added as a new period, then according to Kontsevich and Zagier "all classical constants are periods in the appropriate sense".
It is conjectured that, if a period is given by two different integrals, then either integral can be transformed into the other using only the linearity of integrals, changes of variables, and the Newton-Leibniz formula.
A useful property of algebraic numbers is that equality between two algebraic expressions can be determined algorithmically. It is conjectured that this is also possible for periods.