Complex conjugate ($\overline{x}$)

The complex conjugate of $z=x+iy$ is $\overline{z}=x-iy$. It keeps the real part and negates the imaginary part. In the complex plane, this is a reflection across the horizontal axis. Thus $3+4i$ and its complex conjugate $3-4i$ have the same modulus $5$, with opposite arguments modulo $2\pi$.

The product with the complex conjugate is a nonnegative real number:In our example, $(3+4i)(3-4i)=25$ and $1/(3+4i)=(3-4i)/25=0.12-0.16i$. Its modulus is $1/5$. Geometrically, taking the multiplicative inverse reverses the argument and reciprocates the modulus; the reflected direction alone is only part of the operation.

Applying the complex conjugate twice returns the starting point. Moreover, a complex number equals its own complex conjugate exactly when its imaginary part is zero. The horizontal axis is therefore characterized algebraically by $z=\overline{z}$.

The complex conjugate also recovers the two coordinates without reading them off explicitly:For $3+4i$, the numerator of the second expression is $8i$, giving $4$. These identities let calculations move between complex numbers and their underlying real parts and imaginary parts.

The complex conjugate of a complex number is obtained by changing the sign of its imaginary part. A complex number is typically expressed in the form: \[ z = a + bi \] where: - \( a \) is the real part, - \( b \) is the imaginary part, and - \( i \) is the imaginary unit with the property \( i^2 = -1 \).