Algebraic form, conjugation, modulus, etc.

A complex number is often written in the algebraic (or standard) form as $z = a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit with $i^2 = -1$. The conjugate of $z$ is $\overline{z} = a - bi$. The modulus (or absolute value) of $z$ is $|z| = \sqrt{a^2 + b^2}$.

Important properties

• The conjugate of $z = a + bi$ is $\overline{z} = a - bi$.

• The modulus $|z|$ represents the distance from the origin to the point $(a, b)$ in the complex plane.

• $z \cdot \overline{z} = |z|^2$.

• $|z| \geq 0$ and $|z| = 0$ only if $z = 0$.

• For any two complex numbers $z$ and $w$, $|z \cdot w| = |z| \cdot |w|$.