Overview
In mathematics, the conjugate of an expression of the form a + b \sqrt d is a - b \sqrt d, provided that \sqrt d does not appear in and . One says also that the two expressions are conjugate. In particular, the two solutions of a quadratic equation are conjugate, as per the \pm in the quadratic formula x = \frac -b \pm \sqrt b^2 - 4ac 2a . Complex conjugation is the special case where the square root is i = \sqrt -1 , the imaginary unit.
Properties
As (a + b \sqrt d)(a - b \sqrt d) = a^2 - b^2 d and (a + b \sqrt d) + (a - b \sqrt d) = 2a, the sum and the product of conjugate expressions do not involve the square root anymore. This property is used for removing a square root from a denominator, by multiplying the numerator and the denominator of a fraction by the conjugate of the denominator (see Rationalisation). An example of this usage is: \frac a + b \sqrt d x + y\sqrt d = \frac (a + b \sqrt d)(x - y \sqrt d) (x + y \sqrt d)(x - y \sqrt d) = \frac ax - dby + (xb - ay) \sqrt d x^2 - y^2 d . Hence: \frac 1 a + b \sqrt d = \frac a - b \sqrt d a^2 - db^2 . A corollary property is that the subtraction: : (a+b\sqrt d) - (a-b\sqrt d)= 2b\sqrt d, leaves only a term containing the root.
See also
* Conjugate element (field theory), the generalization to the roots of a polynomial of any degree