Editing Algebraic number (section)

{{Short description|Complex number that is a root of a non-zero polynomial in one variable with rational coefficients}}
{{Distinguish|Algebraic solution}}
{{Use shortened footnotes|date=September 2024}}
[[File:Isosceles right triangle with legs length 1.svg|thumb|200px|The square root of 2 is an algebraic number equal to the length of the [[hypotenuse]] of a [[right triangle]] with legs of length 1.]]

An '''algebraic number''' is a number that is a [[root of a function|root]] of a non-zero [[polynomial]] in one variable with [[integer]] (or, equivalently, [[Rational number|rational]]) coefficients.  For example, the [[golden ratio]], <math>(1 + \sqrt{5})/2</math>, is an algebraic number, because it is a root of the polynomial {{math|''x''{{sup|2}} − ''x'' − 1}}. That is, it is a value for x for which the polynomial evaluates to zero.  As another example, the [[complex number]] <math>1 + i</math> is algebraic because it is a root of {{math|''x''{{sup|4}} + 4}}.

All integers and rational numbers are algebraic, as are all [[nth root|roots of integers]]. Real and complex numbers that are not algebraic, such as [[pi|{{pi}}]] and {{mvar|[[e (mathematical constant)|e]]}}, are called [[transcendental number]]s.

The [[set (mathematics)|set]]  of algebraic (complex) numbers is [[countable set|countably infinite]] and has [[measure zero]] in the [[Lebesgue measure]] as a [[subset]] of the [[uncountable set|uncountable]] complex numbers. In that sense, [[almost all]] complex numbers are [[transcendental number|transcendental]]. Similarly, the set of algebraic (real) numbers is countably infinite and has Lebesgue measure zero as a subset of the real numbers, and in that sense almost all real numbers are transcendental.