# algebraic number

﻿
algebraic number
1. a root of an algebraic equation with integral coefficients.
2. root1 (def. 10b).
[1930-35]

* * *

real number for which there exists a polynomial equation with integer coefficients such that the given real number is a solution. Algebraic numbers include all of the natural numbers, all rational numbers, some irrational numbers, and complex numbers of the form pi + q, where p and q are rational, and i is the square root of −1. For example, i is a root of the polynomial x2 + 1 = 0. Numbers, such as that symbolized by the Greek letter π, that are not algebraic are called transcendental numbers. The mathematician Georg Cantor (Cantor, Georg) proved that, in a sense that can be made precise, there are many more transcendental numbers than there are algebraic numbers, even though there are infinitely many of these latter.

* * *

Universalium. 2010.

### Look at other dictionaries:

• algebraic number — n. a root of a polynomial equation with coefficients that are rational numbers …   English World dictionary

• Algebraic number — In mathematics, an algebraic number is a complex number that is a root of a non zero polynomial in one variable with rational (or equivalently, integer) coefficients. Complex numbers such as pi that are not algebraic are said to be transcendental …   Wikipedia

• algebraic number — noun root of an algebraic equation with rational coefficients • Hypernyms: ↑irrational number, ↑irrational * * * noun : a root of an algebraic equation whose coefficients are rational * * * Math. 1. a root of an algebraic equation with integral… …   Useful english dictionary

• algebraic number — A real number that is the root of a polynomial equation. All rationals are algebraic (they are all roots of equations b x = a where a, b are integers). Reals that are not algebraic are called transcendental numbers …   Philosophy dictionary

• algebraic number — noun Date: 1904 a root of an algebraic equation with rational coefficients …   New Collegiate Dictionary

• algebraic number — al′gebra′ic num′ber n. 1) math. a root of an algebraic equation with integral coefficients 2) math. root I, 9), b) • Etymology: 1930–35 …   From formal English to slang

• algebraic number — noun A complex number that is a root of a polynomial equation with rational coefficients. Ant: transcendental number …   Wiktionary

• Algebraic number field — In mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector… …   Wikipedia

• Algebraic number theory — In mathematics, algebraic number theory is a major branch of number theory which studies the algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number… …   Wikipedia

• algebraic number theory — noun The subfield of number theory where algebraic numbers are studied using algebra …   Wiktionary