complex conjugate

The complex conjugate of a number is its mirror image across the real axis: you keep the real part and flip the sign of the imaginary part. The conjugate of a + bi is a - bi. Geometrically it is the reflection of the point in the horizontal axis, like looking at the number in a mirror laid along the real number line.

The conjugate of z is written z with a bar over it, often typed as conj(z) or z-bar. Its great usefulness comes from one fact: a number times its conjugate is always a non-negative real number. Concretely, (a + bi)(a - bi) = a^2 + b^2, which is exactly the square of the distance from the origin. This is the trick used to divide complex numbers and to rationalize denominators.

Conjugates also explain why non-real roots of real polynomials come in matched pairs: if a polynomial has real coefficients and a + bi is a root, then a - bi must be a root too. This is called the complex conjugate root theorem.