Abstract
Mathematical constants like π, e and γ arise frequently in number theory and other areas of mathematics and physics. Mathematicians have long wondered whether such numbers are irrational or perhaps even transcendental, that is, not algebraic. Because they are solutions of polynomial equations with rational coefficients, algebraic numbers form a countable subset of the complex numbers. Therefore, most complex numbers are transcendental, although, for any given number, it is usually difficult to figure out whether it is transcendental. This essay is about the arithmetic notion of periods, a countable subalgebra P of the complex numbers defined around 1999 by Maxim Kontsevich and Don Zagier [4]. Periods contain all algebraic numbers but also many other transcendental numbers important for number theory. This notion of periods generalizes in algebraic geometry and yields the theory of periods and period domains for algebraic varieties, see [2]. Kontsevich and Zagier define periods as those complex numbers whose real and imaginary parts are values of absolutely convergent integrals
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