An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, , is an algebraic number, because it is a root of the polynomial . That is, it is a value for x for which the polynomial evaluates to zero. As another example, the complex number is algebraic because it is a root of . All integers and rational numbers are algebraic, as are all roots of integers. Real and complex numbers that are not algebraic, such as and , are called transcendental numbers. The set of algebraic numbers is countably infinite and has measure zero in the Lebesgue measure as a subset of the uncountable complex numbers. In that sense, almost all complex numbers are transcendental. Examples All rational numbers are algebraic. Any rational number, expressed as the quotient of an integer and a (non-zero) natural number , satisfies the above definition, because is the root of a non-zero polynomial, namely . Quadratic irrational numbers, irrational solutions of a quadratic polynomial with integer coefficients , , and , are algebraic numbers. If the quadratic polynomial is monic (), the roots are further qualified as quadratic integers. Gaussian integers, complex numbers for which both and are integers, are also quadratic integers. This is because and are the two roots of the quadratic . A constructible number can be constructed from a given unit length using a straightedge and compass. It includes all quadratic irrational roots, all rational numbers, and all numbers that can be formed from these using the basic arithmetic operations and the extraction of square roots. (By designating cardinal directions for 1, −1, , and −, complex numbers such as are considered constructible.) Any expression formed from algebraic numbers using any combination of the basic arithmetic operations and extraction of th roots gives another algebraic number. Polynomial roots that cannot be expressed in terms of the basic arithmetic operations and extraction of th roots (such as the roots of ). That happens with many but not all polynomials of degree 5 or higher. Values of trigonometric functions of rational multiples of (except when undefined): for example, , , and satisfy . This polynomial is irreducible over the rationals and so the three cosines are conjugate algebraic numbers. Likewise, , , , and satisfy the irreducible polynomial , and so are conjugate