= Complex conjugate
{title2=$\overline z$}
{wiki}
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 $2\pi$.
\Image[/complex-arithmetic-geometry.png]
{title=Addition translates a complex number; conjugation reflects it and preserves its modulus}
The product with the <complex conjugate> is a nonnegative <real number>:
$$
z\overline z=x^2+y^2=|z|^2,\qquad \frac1z=\frac{\overline z}{|z|^2}\quad(z\ne0).
$$
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:
$$
\operatorname{Re}z=\frac{z+\overline z}{2},\qquad \operatorname{Im}z=\frac{z-\overline z}{2i}.
$$
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>.
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