algebraic number
Noun
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Noun
algebraic number (plural algebraic numbers)
- (algebra, number theory) 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 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.
- 1918, The American Mathematical Monthly, Volume 25, Mathematical Association of America, page 435 ↗,
- Thus, the equation x - e^y = 0 is satisfied for x = 1,\ y = 0 and for no other pair of algebraic numbers.
- 1921, Louis J. Mordell, Three Lectures on Fermat's Last Theorem, page 16 ↗,
- 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.
- 1991, Paul 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:
- (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:
- German: algebraische Zahl
- Italian: numero algebrico
This text is extracted from the Wiktionary and it is available under the CC BY-SA 3.0 license | Terms and conditions | Privacy policy 0.004