Quadratic Irrational

In mathematics, a quadratic irrational (also known as a quadratic irrationality or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients. Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their common denominator, a quadratic irrational is an irrational root of some quadratic equation whose coefficients are integers. The quadratic irrationals form the real algebraic numbers of degree 2 and can, therefore, be expressed in this form:

for integers a, b, c, d; with b and d non-zero, and with c > 1 and square-free. This implies that the quadratic irrationals have the same cardinality as ordered quadruples of integers, and are therefore countable.

The rational numbers together with all quadratic irrationals with a given c form a field, called a real quadratic field. In particular, their inverses are of the same form, since

This field is often called the field obtained by adjoining √c to the rational numbers, and denoted Q(√c).

Quadratic irrationals have useful properties, especially in relation to continued fractions, where we have the result that all quadratic irrationals, and only quadratic irrationals, have periodic continued fraction forms. For example