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The complex conjugate for a given complex number is the number with the same real part but a negative imaginary part. In polars, this is the equivalent of having the radius remain the same but the argument becoming negative. For an example, the conjugate of

$z=-4+3i=5e^{i\arctan\left(-\tfrac34\right)}=\left(5,\arctan\left(-\tfrac34\right)\right)$

is

$\bar{z}=-4-3i=5e^{i\arctan\left(\tfrac34\right)}=\left(5,\arctan\left(\tfrac34\right)\right)$

Complex conjugates are useful for defining complex division and for finding roots of polynomials, as well as defining the inner product of complex vectors.

Properties

• $\overline{z+w}=\bar{z}+\overline{w}$
• $\overline{z\cdot w}=\bar{z}\cdot\overline{w}$
• $\bar{z}=z$ iff $z\in\R$
• $\|\bar{z}\|^2=\bar{z}\cdot z=z\cdot\overline{z}$
• $\overline{\overline{z}} = z$
• $a^{\bar{z}}=\overline{a^z}$
• $\log_a(\bar{z})=\overline{\log_a(z)}$
• $f(\bar{z})=\overline{f(z)}$ if $f$ is holomorphic
• $\frac{a+bi}{c+di}=\frac{a+bi}{c+di}\cdot\frac{\overline{c+di}}{\overline{c+di}}=\left( \frac{ac+bd}{c^2+d^2} \right) + \left( \frac{bc-ad}{c^2+d^2} \right)i$