Modulus and Argument of a Complex Number

Modulus and Argument of a Complex Number

Modulus. For $z=a+ib$, the modulus of $z$, written $|z|$ (also called $r$), is defined as

Geometrically, $|z|$ is the length $OP$ of the segment from the origin to the point $P(a,b)$ representing $z$ in the Argand plane: dropping perpendiculars from $P$ to each axis forms a right triangle with legs $|a|$ and $|b|$, so $OP=\sqrt{a^2+b^2}$ by Pythagoras — exactly the modulus. Because it is a square root of a sum of squares, the modulus is always a non-negative real number, and $|z|=0$ exactly when $z=0$ (both $a=0$ and $b=0$).

Standard properties (each provable from the definition and the algebra of §2):

• $|z|=|\bar z|$, since $a^2+(-b)^2=a^2+b^2$.
• $z\bar z=|z|^2$ (already met in §2.3).
• $|z_1z_2|=|z_1|\,|z_2|$ — the modulus of a product is the product of the moduli.
• $\left|\dfrac{z_1}{z_2}\right|=\dfrac{|z_1|}{|z_2|}$ for $z_2\neq0$.

$|3-4i|=\sqrt{3^2+(-4)^2}=\sqrt{9+16}=\sqrt{25}=5$.

Argument. The argument (also called the amplitude) of a non-zero $z=a+ib$, written $\theta=\arg(z)$, is the angle that the segment $OP$ makes with the positive real axis, measured anticlockwise as positive. It satisfies

A given $z\neq0$ has infinitely many possible arguments, differing by whole multiples of $2\pi$ (adding a full turn does not change the direction of $OP$); the one lying in the standard range $(-\pi,\pi]$ is called the principal argument.

Why the raw inverse tangent is not enough. The function $\tan^{-1}(b/a)$ always returns a value strictly between $-\pi/2$ and $\pi/2$, so it alone cannot distinguish a point in Quadrant I from the diametrically opposite point in Quadrant III (both give the same ratio $b/a$), nor Quadrant II from Quadrant IV. The correct principal argument therefore needs a quadrant-by-quadrant correction to the raw $\tan^{-1}(b/a)$ value, tabulated in the accompanying reference table for every quadrant and every axis. …