algebraic number

Noun

 * 1)  A complex number (more generally, an element of a number field) that is a root of a polynomial whose coefficients are integers; equivalently, a complex number (or element of a number field) that is a root of a monic polynomial whose coefficients are rational numbers.
 * The golden ratio (&phi;) is an algebraic number since it is a solution of the quadratic equation $$ x^2 + x - 1 = 0 $$, whose coefficients are integers.
 * The square root of a rational number, $$\textstyle\sqrt{\frac m n}$$, is an algebraic number since it is a solution of the quadratic equation $$n x^2 - m = 0$$, whose coefficients are integers.
 * 1) * 1991,, Algebraic Numbers and Algebraic Functions, Chapman & Hall, page 83,
 * The existence of such 'transcendental' numbers is well known and it can be proved at three levels:
 * (i) It is easily checked that the set of all algebraic numbers is countable, whereas the set of all complex numbers is uncountable (this non-constructive proof goes back to Cantor).
 * The existence of such 'transcendental' numbers is well known and it can be proved at three levels:
 * (i) It is easily checked that the set of all algebraic numbers is countable, whereas the set of all complex numbers is uncountable (this non-constructive proof goes back to Cantor).

Translations

 * Chinese:
 * Mandarin:
 * Czech: algebraické číslo
 * Finnish: algebrallinen luku
 * Galician:
 * German: algebraische Zahl
 * Hungarian:
 * Icelandic: algebruleg tala, algebrutala
 * Italian:
 * Kazakh: алгебралық сан
 * Korean: 대수적 수(代數的數)
 * Latin: numerus algebraicus
 * Romanian: număr algebric
 * Russian: алгебраическое число
 * Serbo-Croatian: algebarski broj
 * Swedish:
 * Thai: จำนวนเชิงพีชคณิต
 * Turkish: cebirsel sayılar