Unit 10 · Complex Numbers　复数

By the end of this chapter you can:

1. Write a complex number in algebraic form $a + bi$ and identify its real and imaginary parts
2. Add, subtract, multiply, divide, and conjugate complex numbers
3. Compute the modulus $|z|$ and locate $z$ in the complex plane
4. Solve quadratics that have a negative discriminant

Exam weight on past CSCA papers: ~4% (1–2 of 48 MCQs). Often a quick win.

10.1　Algebraic Form　代数形式

The imaginary unit is defined by
$$i^{2} = -1.$$

Every complex number can be written uniquely as
$$z = a + bi$$
where $a, b \in \mathbb{R}$. We call $a$ the real part ($\operatorname{Re} z$) and $b$ the imaginary part ($\operatorname{Im} z$).

⚠️ Subtle but exam-friendly: $\operatorname{Im}(3 + 5i) = 5$, not $5i$. The imaginary part is the real number $b$, not the term $bi$.

When is a complex number...

Equality

Two complex numbers are equal iff their real and imaginary parts agree separately:
$$a + bi = c + di \iff a = c \ \text{and}\ b = d.$$

This is how you solve equations like $x + (y-1)i = 3 - 2i$: match parts to get $x = 3$, $y = -1$.

Conjugate　共轭

$$\overline{z} = \overline{a + bi} = a - bi.$$

Three useful facts (memorise):

10.2　Arithmetic with Complex Numbers　复数的四则运算

Addition and subtraction — combine like terms

$$(a + bi) + (c + di) = (a+c) + (b+d)i$$
$$(3 + 5i) - (1 - 2i) = 2 + 7i$$

Multiplication — FOIL, then use $i^{2} = -1$

$$(a + bi)(c + di) = ac + adi + bci + bd\,i^{2} = (ac - bd) + (ad + bc)i$$

🔑 Powers of $i$ cycle every 4:
$$i^{1} = i,\ i^{2} = -1,\ i^{3} = -i,\ i^{4} = 1,\ i^{5} = i, \ldots$$
To compute $i^{n}$, divide $n$ by $4$ and use the remainder. E.g. $i^{2026}$: $2026 = 4(506) + 2$, so $i^{2026} = i^{2} = -1$.

Division — multiply top and bottom by the conjugate

$$\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^{2} + d^{2}}$$

The denominator is now real, so we can split into real and imaginary parts.

Worked Example 10.2.A

Compute $\dfrac{2 + i}{1 - i}$.

Solution. Multiply top and bottom by the conjugate $1 + i$:
$$\frac{2 + i}{1 - i} \cdot \frac{1 + i}{1 + i} = \frac{(2 + i)(1 + i)}{(1)^{2} + (1)^{2}} = \frac{2 + 2i + i + i^{2}}{2} = \frac{2 + 3i - 1}{2} = \frac{1 + 3i}{2} = \tfrac{1}{2} + \tfrac{3}{2}i.$$

Worked Example 10.2.B — Q46 on Jan paper

Find $m$ if $z$ is a root of $x^{2} + mx + 4 = 0$ and $z$ lies on the line $x - y = 0$.

Step 1. The discriminant: real roots ↔ $m^{2} - 16 \ge 0$. The line $y = x$ in the complex plane corresponds to numbers of the form $z = t + ti$ (real and imaginary parts equal). For a non-real root, we need $m^{2} - 16 < 0$, i.e. $|m| < 4$.

Step 2. A real-coefficient quadratic with one non-real root has its conjugate as the other root. So the roots are $z = t + ti$ and $\overline{z} = t - ti$.

Step 3. Product of roots: $z \overline{z} = t^{2} + t^{2} = 2t^{2} = \dfrac{c}{a} = 4 \implies t^{2} = 2 \implies t = \pm \sqrt{2}$.

Step 4. Sum of roots: $z + \overline{z} = 2t = -\dfrac{b}{a} = -m \implies m = -2t = \mp 2\sqrt{2}$.

Answer: $m = \pm 2\sqrt{2}$ → choice (b).

10.3　Modulus and the Complex Plane　模与复平面

Complex plane (Argand diagram)

Plot $z = a + bi$ as the point $(a, b)$:

• horizontal axis = real part
• vertical axis = imaginary part

       Im
        ▲
        │
      b ●───── z = a + bi
        │   /
        │  /  |z|
        │ /
        │/
        ●───────────────▶ Re
        O      a

Modulus　模

$$|z| = |a + bi| = \sqrt{a^{2} + b^{2}}$$

Geometrically, $|z|$ is the distance from $z$ to the origin in the complex plane.

Useful facts:

• $|z| \ge 0$, equals $0$ only when $z = 0$
• $|z|^{2} = z \cdot \overline{z}$
• $|z_{1} z_{2}| = |z_{1}|\, |z_{2}|$ and $\left|\dfrac{z_{1}}{z_{2}}\right| = \dfrac{|z_{1}|}{|z_{2}|}$

Worked Example 10.3.A

Find $|3 - 4i|$.

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

Worked Example 10.3.B — Geometric locus

What does $|z - 1| = 2$ describe?

$|z - 1|$ is the distance from $z$ to the point $1 + 0i$ in the complex plane. Setting this equal to $2$ describes a circle of radius $2$ centred at $(1, 0)$.

Try it!　自测练习

Q1. Write $\dfrac{1}{2 + i}$ in algebraic form.

Q2. Compute $(1 + i)^{4}$.

Q3. Find $i^{77}$.

Q4. Find $|z|$ if $z = (1 + i)(1 - 2i)$.

Q5. Solve $x^{2} - 2x + 5 = 0$ over $\mathbb{C}$.

📌 Chapter summary

Topic	Key skill
Algebraic form	$z = a + bi$, separate parts; equality compares parts separately
Conjugate	$\overline{a+bi} = a - bi$; product $z \overline{z} = a^{2} + b^{2}$ is real
Multiplication	FOIL + replace $i^{2}$ with $-1$; powers of $i$ cycle every 4
Division	multiply by conjugate of denominator
Modulus	$|z| = \sqrt{a^{2}+b^{2}}$ = distance to origin
Locus	$|z - z_{0}| = r$ is a circle of radius $r$ centred at $z_{0}$

Connection to earlier units:

• The fact "real quadratics with negative discriminant have complex-conjugate roots" ties Unit 1 (inequalities → discriminant sign) directly to this unit.
• $|z|^{2} = z\overline{z}$ is the same identity as $a^{2} + b^{2} = (a + bi)(a - bi)$ from Unit 1's factoring exercises.