Modulus and Argument

For $z = a + bi$ with $a, b \in \mathbb{R}$, the modulus is

Geometrically, $|z|$ is the distance from the point $(a, b)$ to the origin on the Argand plane, a direct application of Pythagoras. Three consequences:

1. $|z| \geq 0$ with equality iff $z = 0$.
2. $|z|^2 = a^2 + b^2 = z\bar{z}$ (linking modulus back to the conjugate identity from Lesson 03).
3. $|z\,w| = |z|\,|w|$, the modulus of a product is the product of the moduli (proof in later lessons).

Distance interpretation. For two complex numbers $z_1, z_2$, the quantity $|z_1 - z_2|$ is the Euclidean distance between them on the Argand plane. This is why $|z - z_0| = r$ describes a circle of radius $r$ centred at $z_0$.

$|z| = \sqrt{a^2 + b^2}$ (Pythagoras on the Argand plane) · $|z|^2 = z\bar{z}$ (from Lesson 03) · $|z| \geq 0$; zero iff $z = 0$ · $|z_1 - z_2|$ = distance between $z_1$ and $z_2$

Pause, copy $|z| = \sqrt{a^2+b^2}$, the identity $|z|^2 = z\bar{z}$, the non-negativity of modulus, and $|z_1-z_2|$ as a distance into your book.

We just saw that $|z| = \sqrt{a^2+b^2}$ gives the distance from the origin, with $|z|^2 = z\bar{z}$ and $|z_1-z_2|$ being the distance between two points. That raises a question: how do we find the direction (argument) of $z$, and how does the quadrant affect the answer? This card answers it → use the reference angle $\alpha = \arctan(|b/a|)$, then adjust sign and magnitude by quadrant to get $\theta \in (-\pi, \pi]$.

The argument $\arg(z)$ is the angle $\theta$, measured anticlockwise from the positive real axis, to the line segment from $0$ to $z$. Because rotating by $2\pi$ returns to the same point, $\arg z$ is defined only modulo $2\pi$. The principal argument is the unique value in $(-\pi, \pi]$.

Why $\arctan(b/a)$ alone is not enough. The calculator's $\arctan$ function returns values in $(-\pi/2, \pi/2)$, only half the plane. For $z = -1 + i$ in quadrant 2, $b/a = -1$ so $\arctan(-1) = -\pi/4$, but the true angle to $(-1, 1)$ is in the upper-left, i.e., $3\pi/4$. The fix: identify the quadrant first, then build the argument from a reference angle.

Boundary cases. Positive real: $\arg = 0$. Negative real: $\arg = \pi$. Positive imaginary: $\arg = \pi/2$. Negative imaginary: $\arg = -\pi/2$. The number $z = 0$ has no argument.

Principal argument $\theta \in (-\pi, \pi]$ · Reference angle $\alpha = \arctan(|b|/|a|) \in [0, \pi/2]$ · Q1: $\theta = \alpha$; Q2: $\theta = \pi - \alpha$; Q3: $\theta = -(\pi - \alpha)$; Q4: $\theta = -\alpha$ · Always sketch $z$ on the Argand plane before computing $\arg z$

Pause, copy the principal argument range $\theta \in (-\pi,\pi]$, the reference-angle formula $\alpha = \arctan(|b/a|)$, and the four-quadrant adjustment rules (Q1: $\theta=\alpha$; Q2: $\theta=\pi-\alpha$; Q3: $\theta=-(\pi-\alpha)$; Q4: $\theta=-\alpha$) into your book.