The complex conjugate of a complex number z = a + bi is the number z̄ = a − bi, obtained by negating the imaginary part while keeping the real part unchanged. Geometrically, the conjugate reflects the complex number across the real axis in the complex plane (Argand diagram). This operation is fundamental in complex analysis and has many applications in mathematics, physics, and engineering.

Complex conjugates are used extensively in signal processing, quantum mechanics, and electrical engineering. One of the most important properties is that multiplying a complex number by its conjugate always produces a non-negative real number equal to the square of the magnitude: z · z̄ = |z|². This property is crucial for dividing complex numbers and simplifying complex expressions.

Complex conjugates play a vital role in many areas of science and engineering. In electrical engineering, they are used to calculate power in AC circuits, where the complex conjugate of current is multiplied by voltage to find apparent power. In signal processing, the Fourier transform uses conjugates to analyze frequency components of signals.

In quantum mechanics, the probability density of a wave function is found by multiplying the wave function by its complex conjugate. This ensures the result is always a real, non-negative value representing probability. Conjugates are also essential for finding roots of polynomials, as complex roots of real polynomials always occur in conjugate pairs.