In mathematics, a quadratic irrational, also known as a quadratic surd or quadratic irrationality, is an irrational number that is the solution to some quadratic equation with rational coefficients. Since fractions can be cleared from a quadratic equation by multiplying both sides by their common denominator, this is the same as saying it is a root of a quadratic equation whose coefficients are integers. Quadratic irrationals therefore have the form
- <math>{a+\sqrt{b} \over c}</math>
for integers a, b, c. This implies that the quadratic irrationals have the same cardinality as ordered triples of integers, and are therefore countable. The quadratic rationals with a given b form a field, called a quadratic field. Quadratic irrationals are special numbers, 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
- <math>\sqrt{3}=1.732\ldots=[1;1,2,1,2,1,2,\ldots]</math>
A special case includes Pell's equation.
