Definition Let $z$ be a complex number, and let $\ccj{z}$ be the complex conjugate of $z$ . Then the modulus, or absolute value, of $z$ is defined as $$ |z| := \sqrt{z\ccj{z}}. $$ There is also the notation $$\mod{z}$$ for the modulus of $z$ .

If we write $z$ in polar form as $z = re^{i\phi}$ with $r\ge 0,\; \phi\in[0,\,2\pi)$ , then $|z| = r$ . It follows that the modulus is a positive real number or zero. Alternatively, if $a$ is the real part of $z$ , and $b$ the imaginary part, then \begin{eqnarray} \label{eq100} |z| &=& \sqrt{a^2+b^2}, \end{eqnarray}which is simply the Euclidean norm of the point $(a,\,b)\in \sR^2$ . It follows that the modulus satisfies the triangle inequality $$ |z_1+z_2| \le |z_1|+|z_2|, $$ also $$|\Re{z}| \le |z|,\quad |\Im{z}| \le |z|,\quad |z| \le |\Re{z}|+|\Im{z}|.$$

Modulus is multiplicative: $$|z_1z_2| = |z_1|\cdot|z_2|, \quad \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$$

Since $\sR\subset\sC$ , the definition of modulus includes the real numbers. Explicitly, if we write $x\in\sR$ in polar form, $x = re^{i\phi}$ , $r > 0$ , $\phi\in[0,2\pi)$ , then $\phi = 0$ or $\phi=\pi$ , so $e^{i\phi}=\pm 1$ . Thus,