Editing Number (section)

==Subclasses of the complex numbers==

===Algebraic, irrational and transcendental numbers===
[[Algebraic number]]s are those that are a solution to a polynomial equation with integer coefficients. Real numbers that are not rational numbers are called [[irrational number]]s. Complex numbers which are not algebraic are called [[transcendental number]]s. The algebraic numbers that are solutions of a [[monic polynomial]] equation with integer coefficients are called [[algebraic integer]]s.

===Periods and exponential periods===
{{Main|Period (algebraic geometry)}}
A period is a complex number that can be expressed as an [[integral]] of an [[algebraic function]] over an algebraic [[Domain of a function|domain]]. The periods are a class of numbers which includes, alongside the algebraic numbers, many well known [[Mathematical constant|mathematical constants]] such as the [[Pi|number ''π'']]. The set of periods form a countable [[Ring (mathematics)|ring]] and bridge the gap between algebraic and transcendental numbers.<ref name=":1">{{Citation |last1=Kontsevich |first1=Maxim |title=Periods |date=2001 |work=Mathematics Unlimited — 2001 and Beyond |pages=771–808 |editor-last=Engquist |editor-first=Björn |url=https://link.springer.com/chapter/10.1007/978-3-642-56478-9_39 |access-date=2024-09-22 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-642-56478-9_39 |isbn=978-3-642-56478-9 |last2=Zagier |first2=Don |editor2-last=Schmid |editor2-first=Wilfried}}</ref><ref>{{Cite web |last=Weisstein |first=Eric W. |title=Algebraic Period |url=https://mathworld.wolfram.com/AlgebraicPeriod.html |access-date=2024-09-22 |website=mathworld.wolfram.com |language=en}}</ref> 

The periods can be extended by permitting the integrand to be the product of an algebraic function and the [[Exponential function|exponential]] of an algebraic function. This gives another countable ring: the exponential periods. The [[E (mathematical constant)|number ''e'']] as well as [[Euler's constant]] are exponential periods.<ref name=":1" /><ref>{{Cite journal |last=Lagarias |first=Jeffrey C. |date=2013-07-19 |title=Euler's constant: Euler's work and modern developments |journal=Bulletin of the American Mathematical Society |volume=50 |issue=4 |pages=527–628 |doi=10.1090/S0273-0979-2013-01423-X |arxiv=1303.1856 |issn=0273-0979}}</ref> 

===Constructible numbers===
Motivated by the classical problems of [[Straightedge and compass construction|constructions with straightedge and compass]], the [[constructible number]]s are those complex numbers whose real and imaginary parts can be constructed using straightedge and compass, starting from a given segment of unit length, in a finite number of steps.

===Computable numbers===
{{Main|Computable number}}
A '''computable number''', also known as ''recursive number'', is a [[real number]] such that there exists an [[algorithm]] which, given a positive number ''n'' as input, produces the first ''n'' digits of the computable number's decimal representation. Equivalent definitions can be given using [[μ-recursive function]]s, [[Turing machine]]s or [[λ-calculus]]. The computable numbers are stable for all usual arithmetic operations, including the computation of the roots of a [[polynomial]], and thus form a [[real closed field]] that contains the real [[algebraic number]]s.

The computable numbers may be viewed as the real numbers that may be exactly represented in a computer: a computable number is exactly represented by its first digits and a program for computing further digits. However, the computable numbers are rarely used in practice. One reason is that there is no algorithm for testing the equality of two computable numbers. More precisely, there cannot exist any algorithm which takes any computable number as an input, and decides in every case if this number is equal to zero or not.

The set of computable numbers has the same cardinality as the natural numbers. Therefore, [[almost all]] real numbers are non-computable. However, it is very difficult to produce explicitly a real number that is not computable.