# Complex Conjugate

In mathematics, the complex conjugate of a complex number is the number with equal real part and imaginary part equal in magnitude but opposite in sign.

For example, the complex conjugate of 3 + 4i is 3 − 4i.

In polar form, the conjugate of $\rho e^{i \phi}$ is $\rho e^{-i \phi}$. This can be shown using Euler's formula.

Complex conjugates are important for finding roots of polynomials. According to the complex conjugate root theorem, if a complex number is a root to a polynomial in one variable with real coefficients (such as the quadratic equation or the cubic equation), so is its conjugate.