Complex numbers: modulus and argument made easy

A complex = a point in the plane

$z = a + ib$ corresponds to the point $M(a, b)$ or vector $\vec{OM}$.

• Modulus $|z|$ = length of $\vec{OM}$
• Argument $\arg(z)$ = angle between the real axis and $\vec{OM}$ (in radians)

Formulas

$|z| = \sqrt{a^2 + b^2}, \quad \tan(\arg z) = \dfrac{b}{a} \; (\text{if } a > 0)$

Polar form

$z = |z| \cdot (\cos \theta + i \sin \theta) \quad \text{where } \theta = \arg z$

Example: $z = 1 + i$

• $|z| = \sqrt{2}$
• $\arg z$: $\tan \theta = 1$, and $a = 1 > 0$, so $\theta = \pi/4$.
• Polar: $z = \sqrt{2}\left(\cos \dfrac{\pi}{4} + i \sin \dfrac{\pi}{4}\right)$.

Useful properties

Formula	Use
$\|z_1 z_2\| = \|z_1\| \|z_2\|$	Modulus of a product
$\arg(z_1 z_2) = \arg z_1 + \arg z_2$	Argument of a product
$\|z^n\| = \|z\|^n$	Power
$\arg(z^n) = n \arg z$	Power
$\|\bar{z}\| = \|z\|$	Conjugate has same modulus
$\arg(\bar{z}) = -\arg z$	Opposite argument

Watch out for the quadrant when $a < 0$

If $a < 0$, add $\pi$ to the $\arctan(b/a)$ result to land in the right quadrant.

Example: $z = -1 + i$. Here $a = -1, b = 1$.

• $\arctan(1/(-1)) = -\pi/4$. But $(-1, 1)$ is in the 2nd quadrant.
• Corrected argument: $-\pi/4 + \pi = 3\pi/4$.

Common mistakes

1. Confusing modulus and argument.
2. Using $\arctan$ without thinking about the quadrant.
3. Forgetting modulus ≥ 0 always.

?À toi de jouer

What is the modulus of z = 3 + 4i?

A.5B.7C.√7D.25

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