algebraic number
noun1
Algebra, Mathematics, Number theory, Sciences 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 (φ) is an algebraic number since it is a solution of the quadratic equation x²-x-1=0, whose coefficients are integers.
- The square root of a rational number, √, is an algebraic number since it is a solution of the quadratic equation nx²-m=0, whose coefficients are integers.
3 more examples
- Thus, the equation x-eʸ#61;0 is satisfied for x#61;1,#92;y#61;0 and for no other pair of algebraic numbers.1918, The American Mathematical Monthly, volume 25, Mathematical Association of America, page 435:
- As a matter of fact, it is not true that the algebraic numbers above can be factored uniquely, but the first case of failure occurs when p = 23.1921, L. J. Mordell, Three Lectures on Fermat's Last Theorem, page 16:
- (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).1991, P. M. Cohn, 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
- Derived terms
- algebraic number theory
- Related terms
- algebraic integer
- Hyponyms
- phi, golden ratio, algebraic integer
- Coordinate terms
- transcendental number