The conjugate of a complex number *a + bi* is *a - bi*. There are various ways of denoting this, but the modern standard is, for the number **A**, A-bar.

The conjugate of a real number is, therefore, itself, hence this isn't often seen. The conjugate of a pure imaginary number is it's own negative - again, fairly useless.

To generalize the first definition to quaternions, the conjugate of *a + bi + cj + dk* is *a - bi - cj - dk*. This is frequently encountered in the multiplicate inverse:

a^{-1} = ~a / (a · ~a)

Where ~a is the conjugate of a.