Conjugate and Modulus

Two derived quantities attached to a complex number are indispensable for division, for measuring size, and for the polar form.

The conjugate

Conjugate

The conjugate of $z=a+bi$ is

$\bar z=a-bi,$

obtained by keeping the real part and reversing the sign of the imaginary part.

Its central property is that $z$ times its conjugate is a non-negative real number:

Other useful properties (all following directly from the definition):

• $\overline{z_1+z_2}=\bar z_1+\bar z_2$ and $\overline{z_1 z_2}=\bar z_1\,\bar z_2$,
• $\overline{\bar z}=z$,
• $z+\bar z=2a=2\operatorname{Re}(z)$ and $z-\bar z=2bi=2i\operatorname{Im}(z)$,
• $z$ is real $\iff z=\bar z$; $z$ is purely imaginary $\iff z=-\bar z$.

The modulus

Modulus

The modulus (or absolute value) of $z=a+bi$ is the non-negative real number

$|z|=\sqrt{a^{2}+b^{2}}.$

The modulus measures the "size" of $z$; it is the distance of the point $(a,b)$ from the origin in the Argand diagram (Section 5). Key facts:

• $|z|^{2}=z\,\bar z=a^{2}+b^{2}$,
• $|z|=|\bar z|$, …