If z = x + iy is a complex number. Then, the modulus of z, denoted by | z |, is the distance of z from the origin in the Argand plane, and it is a non-negative real number equal to $\sqrt{\mathrm{x}^2+\mathrm{y}^2}$. 

i.e. |z| =$\sqrt{x^2+y^2}$. 

Every complex number can be represented as a point in the argand plane with the x-axis as the real axis and the y-axis as the imaginary axis.

$|z|=\sqrt{x^2+y^2}=r$ (length r from origin to point (x,y))

Properties of Modulus

i) $|z| \geq 0$
ii) $|z|=0$, iff $z=0$ and $|z|>0$, iff $z \neq 0$
iii) $-|z| \leq \operatorname{Re}(z) \leq|z|$ and $-|z| \leq \operatorname{Im}(z) \leq|z|$

iv) $|z|=|\bar{z}|=|-z|=|-\bar{z}|$
v) $z \bar{z}=|z|^2$
vi) $\left|z_1 z_2\right|=\left|z_1\right|\left|z_2\right| . \quad$ Thus, $\left|z^n\right|=|z|^n$
vii) $\left|\frac{z_1}{z_2}\right|=\frac{\left|z_1\right|}{\left|z_2\right|}$
viii) $\left|z_1 \pm z_2\right| \leq\left|z_1\right|+\left|z_2\right|_{\text {(Triangle inequality) this can be generalised for } \mathrm{n} \text { complex numbers. }}$
ix) $\left|z_1 \pm z_2\right| \geq\left|\left|z_1\right|-\left|z_2\right|\right|$