Complex Conjugate

Complex Conjugate

In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs. For example, 3 + 4i and 3 − 4i are complex conjugates.

The conjugate of the complex number

where and are real numbers, is

For example,

An alternative notation for the complex conjugate is . However, the notation avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of complex conjugation. The star-notation is preferred in physics while the bar-notation is more common in pure mathematics. If a complex number is represented as a 2×2 matrix, the notations are identical.

Complex numbers are considered points in the complex plane, a variation of the Cartesian coordinate system where both axes are real number lines that cross at the origin, however, the -axis is a product of real numbers multiplied by +/- . On the illustration, the -axis is called the real axis, labeled Re, while the -axis is called the imaginary axis, labeled Im. The plane defined by the Re and Im axes represents the space of all possible complex numbers. In this view, complex conjugation corresponds to reflection of a complex number at the x-axis, equivalent to a degree rotation of the complex plane about the Re axis.

In polar form, the conjugate of is . This can be shown using Euler's formula.

Pairs of complex conjugates are significant because the imaginary unit is qualitatively indistinct from its additive and multiplicative inverse, as they both satisfy the definition for the imaginary unit: . Thus in most "natural" settings, if a complex number provides a solution to a problem, so does its conjugate, such as is the case for complex solutions of the quadratic formula with real coefficients.