$\mathbb{C} = {a + bi : a,b \in \mathbb{R},\ i^2 = -1}$ is a field: closed under $+$ and $\times$, with commutativity, associativity, distributivity, and inverses.
Powers of $i$ cycle with period 4: $i^1=i,\ i^2=-1,\ i^3=-i,\ i^4=1$.
| Operation | Formula |
|---|---|
| Addition | $(a+c) + (b+d)i$ |
| Subtraction | $(a-c) + (b-d)i$ |
| Multiplication | $(ac-bd) + (ad+bc)i$ |
| Division | $\dfrac{z_1}{z_2} = \dfrac{z_1 \bar{z}_2}{ |
Division example: $\dfrac{2+3i}{1-2i} = \dfrac{(2+3i)(1+2i)}{5} = \dfrac{-4+7i}{5} = -\dfrac{4}{5} + \dfrac{7}{5}i$.
For $z = a+bi$: $\bar{z} = a - bi$.
$$z + \bar{z} = 2a = 2\,\text{Re}(z), \qquad z\bar{z} = a^2 + b^2 = |z|^2$$
$$\overline{z_1 z_2} = \bar{z}_1 \bar{z}_2, \qquad \overline{z_1+z_2} = \bar{z}_1 + \bar{z}_2$$
$$|z| = \sqrt{a^2 + b^2}, \qquad |z_1 z_2| = |z_1||z_2|, \qquad \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$$
The principal argument $\arg(z) \in (-\pi, \pi]$ satisfies $\cos\theta = a/|z|$, $\sin\theta = b/|z|$.
| Quadrant | Sign of $a,b$ | Correction |
|---|---|---|
| 1st | $+,+$ | $\arctan(b/a)$ |
| 2nd | $-,+$ | $\pi + \arctan(b/a)$ |
| 3rd | $-,-$ | $-\pi + \arctan(b/a)$ |
| 4th | $+,-$ | $\arctan(b/a)$ |
Example: $z = -1+i$: $|z| = \sqrt{2}$, in 2nd quadrant so $\arg(z) = \pi - \arctan(1) = \dfrac{3\pi}{4}$.
Argument properties:
$$\arg(z_1 z_2) = \arg(z_1) + \arg(z_2) \pmod{2\pi}$$
$$\arg!\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2) \pmod{2\pi}$$
Represent $z = a+bi$ as the point $(a,b)$ or the vector from the origin.
- Horizontal axis: $\text{Re}(z)$
- Vertical axis: $\text{Im}(z)$
- $|z|$ = distance from origin
- $\arg(z)$ = angle with positive real axis
Geometric interpretations:
- $z_1 + z_2$: parallelogram/vector addition
- $\bar{z}$: reflection in the real axis
- $-z$: reflection through the origin
- Multiplication by $i$: rotation by $\pi/2$ anticlockwise
| Condition | Locus |
|---|---|
| $ | z - z_0 |
| $ | z - z_1 |
| $\arg(z - z_0) = \theta$ | Ray from $z_0$ at angle $\theta$ (excluding $z_0$) |
| $\text{Re}(z) = c$ | Vertical line $x = c$ |
| $\text{Im}(z) = c$ | Horizontal line $y = c$ |
Locus example: Describe $|z - 2| = |z + 2i|$.
$(x-2)^2 + y^2 = x^2 + (y+2)^2 \Rightarrow -4x + 4 = 4y + 4 \Rightarrow y = -x$. This is the line $y = -x$ (perpendicular bisector of $2$ and $-2i$).
KEY TAKEAWAY: The Argand diagram is the geometric heart of complex number work. Always check which quadrant a complex number lies in before computing its argument.
COMMON MISTAKE: Using $\arg(z) = \arctan(b/a)$ without quadrant correction. This gives the wrong sign in the 2nd and 3rd quadrants.
VCAA FOCUS: Locus questions requiring you to identify and sketch circles, lines, and rays from algebraic conditions appear regularly in both Paper 1 and Paper 2.
