A reduced quadratic irrational is a root of a quadratic equation with integer coefficients whose ‘algebraic conjugate’ (i.e. the other root of the quadratic) lies in the interval (−1, 0).

For example, 1 + \sqrt{2} is a reduced quadratic irrational because it is a root of x^{2} − 2x − 1, and the other root 1 − \sqrt{2} = −.414 … is in (−1, 0). On the other hand, \sqrt{17} is a quadratic irrational because it satisfies x^{2} − 17 = 0, but is not a reduced quadratic irrational as the other root (−\sqrt{17}) is not in (−1, 0).