in mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients (or equivalently â by clearing denominators â with integer coefficients). numbers such as  that are not algebraic are said to be transcendental. almost all real and complex numbers are transcendental. (here "almost all" has the sense "all but a countable set"; see properties.) == examples == *the rational numbers, expressed as the quotient of two integers a and b, b not equal to zero, satisfy the above definition because x=a/b is the root of bx-a.some of the following examples come from hardy and wright 1972:159â160 and pp. 178â179 *the quadratic surds (irrational roots of a quadratic polynomial ax^2 + bx + c with integer coefficients a, b, and c) are algebraic numbers. if the quadratic polynomial is monic (a = 1) then the roots are quadratic integers. *the constructible numbers are those numbers that can be constructed from a given unit length using straightedge and compass and their opposites. these include all quadratic surds, all rational numbers, and all numbers that can be formed from these using the basic arithmetic operations and the extraction of square roots. (note that by designating cardinal directions for 1, â1, i, and -i, complex numbers such as 3+\sqrt{2}i are considered constructible.) *any expression formed using any combination of the basic arithmetic operations and extraction of nth roots gives an algebraic number. *polynomial roots that cannot be expressed in terms of the basic arithmetic operations and extraction of nth roots (such as the roots of x^5 - x + 1 ). this happens with many, but not all, polynomials of degree 5 or higher. *gaussian integers: those complex numbers a+bi where both a and b are integers are also quadratic integers. *trigonometric functions of rational multiples of \pi (except when undefined). for example, each of \cos(\pi/7), \cos(3\pi/7), \cos(5\pi/7) satisfies 8x^3 - 4x^2 - 4x + 1 = 0. this polynomial is irreducible over the rationals, and so these three cosines are conjugate algebraic numbers. likewise, \tan(3\pi/16), \tan(7\pi/16), \tan(11\pi/16), \tan(15\pi/16) all satisfy the irreducible polynomial x^4 - 4x^3 - 6x^2 + 4x + 1, and so are conjugate algebraic integers. *some irrational numbers are algebraic and some are not: **the numbers \sqrt{2} and \sqrt[3]{3}/2 are algebraic since they are roots of polynomials x^2 - 2 and 8x^3 - 3, respectively. **the golden ratio \phi is algebraic since it is a root of the polynomial x^2 - x - 1. **the numbers \pi and [[e (mathematical constant)|e]
Algebraic_numbers