Meaning of algebraic number | Babel Free

Definitions

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.

Equivalents

Examples

“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.”

“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.”

“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.”

“(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).”

CEFR level