Complex numbers: complete lesson (i, algebraic, modulus, argument, exponential)

The number i

There is a number $i$ such that $i^2 = -1$. It is neither positive nor negative.

The set $\mathbb{C}$ of complex numbers contains all $z = a + i b$ with $a, b \in \mathbb{R}$.

• $a = \text{Re}(z)$: real part
• $b = \text{Im}(z)$: imaginary part (a real number, not $ib$!)

Algebraic form and operations

Sum: $(a + ib) + (c + id) = (a+c) + i(b+d)$.

Product: $(a + ib)(c + id) = (ac - bd) + i(ad + bc)$.

Conjugate: $\overline{z} = a - ib$. Then $z \overline{z} = a^2 + b^2 = |z|^2$.

Inverse: $\dfrac{1}{z} = \dfrac{\overline{z}}{|z|^2}$.

Modulus and argument

$|z| = \sqrt{a^2 + b^2}$: distance from $z$ to the origin in the complex plane.

$\arg(z) = \theta$ such that $z = |z|(\cos\theta + i \sin\theta)$. The angle between the real axis and $[Oz]$.

Properties:

• $|z_1 z_2| = |z_1| \cdot |z_2|$
• $\arg(z_1 z_2) = \arg(z_1) + \arg(z_2) \pmod{2\pi}$
• $|z^n| = |z|^n$, $\arg(z^n) = n \arg(z)$

Trigonometric and exponential forms

Trigonometric: $z = |z|(\cos\theta + i\sin\theta)$

Exponential: $z = |z| e^{i\theta}$ (with $e^{i\theta} = \cos\theta + i\sin\theta$, Euler’s formula)

De Moivre’s formula

$(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$

In exponential form: $(e^{i\theta})^n = e^{in\theta}$.

Solving $z^2 = w$ in $\mathbb{C}$

If $w = r e^{i\alpha}$: the 2 square roots are $\pm \sqrt{r} \, e^{i\alpha/2}$.

Real-coefficient quadratic

$az^2 + bz + c = 0$ with $\Delta = b^2 - 4ac$.

• $\Delta \geq 0$: two real solutions (usual formulas).
• $\Delta < 0$: two complex conjugate solutions $z = \dfrac{-b \pm i\sqrt{-\Delta}}{2a}$.

Geometric representation

$z = a + ib$ ↔ point $M(a, b)$ in the complex plane.

• $|z|$ = $OM$
• $\arg(z)$ = angle $(\vec{u}, \overrightarrow{OM})$
• $|z_2 - z_1|$ = distance between $M_1$ and $M_2$
• $\arg(z_2 - z_1)$ = angle of vector $\overrightarrow{M_1 M_2}$

n-th roots of unity

The $n$ solutions of $z^n = 1$ are $z_k = e^{2i k \pi / n}$ for $k = 0, 1, \ldots, n-1$.

Common mistakes

1. Writing $\sqrt{-1}$ instead of $i$ (bad notation).
2. Confusing imaginary part with $ib$ ($\text{Im}(2 + 3i) = 3$, not $3i$).
3. Forgetting the $\pmod{2\pi}$ in arguments.
4. Adding moduli of a sum ($|z_1 + z_2| \neq |z_1| + |z_2|$ in general).