Complex Numbers – Cambridge 9709 (Pure Mathematics 3, Topic 3.9)
1. Key Definitions & Fundamental Facts
Complex number : \(z = a + bi\) with \(a,b\in\mathbb R\) and \(i^{2}=-1\).Real part : \(\operatorname{Re}(z)=a\).Imaginary part : \(\operatorname{Im}(z)=b\).Modulus (absolute value) : \(|z| = \sqrt{a^{2}+b^{2}}\).Argument : \(\arg(z)=\theta\) is the oriented angle measured from the positive real axis to the line joining the origin to the point \((a,b)\).
Conjugate : \(\overline{z}=a-bi\).
Geometrically, \(\overline{z}\) is the reflection of \(z\) in the real axis; \(|\overline{z}|=|z|\) and \(\arg(\overline{z})=-\theta\).Equality of complex numbers (syllabus requirement):
\[
z_{1}=z_{2}\;\Longleftrightarrow\;
\operatorname{Re}(z_{1})=\operatorname{Re}(z_{2})\;\text{and}\;
\operatorname{Im}(z_{1})=\operatorname{Im}(z_{2}).
\]
Argand diagram showing a point \(z=a+bi\), its modulus \(|z|\), argument \(\theta\), and conjugate \(\overline{z}\) (reflection in the real axis).
2. Forms of a Complex Number
Form Expression When to use
Rectangular (Cartesian)
\(z = a + bi\)
Addition, subtraction, comparison, finding real/imag parts.
Polar (modulus‑argument)
\(z = r(\cos\theta + i\sin\theta)\)
Multiplication, division, powers, roots, trigonometric identities.
“cis” shorthand
\(\displaystyle \operatorname{cis}\theta \equiv \cos\theta+i\sin\theta,\qquad
z = r\,\operatorname{cis}\theta\)
Compact notation for polar form; essential for De Moivre’s theorem.
3. Converting Between Forms
3.1 Rectangular → Polar
Given \(z=a+bi\):
\[
r = \sqrt{a^{2}+b^{2}},\qquad
\theta = \operatorname{atan2}(b,a)
\]
Using \(\operatorname{atan2}\) (preferred on calculators/computers): it automatically places \(\theta\) in the correct quadrant.Manual method (if \(\operatorname{atan2}\) is unavailable):
Compute \(\theta_{0}= \tan^{-1}\!\bigl(\dfrac{b}{a}\bigr)\) (principal value \(-\frac{\pi}{2}<\theta_{0}<\frac{\pi}{2}\)).
Adjust for the quadrant:
If \(a>0\): \(\theta=\theta_{0}\).
If \(a<0\) and \(b\ge0\): \(\theta=\theta_{0}+\pi\).
If \(a<0\) and \(b<0\): \(\theta=\theta_{0}-\pi\).
If \(a=0\) and \(b>0\): \(\theta=\frac{\pi}{2}\).
If \(a=0\) and \(b<0\): \(\theta=-\frac{\pi}{2}\).
3.2 Polar → Rectangular
\[
a = r\cos\theta,\qquad b = r\sin\theta.
\]
Insert the known \(r\) and \(\theta\) into the definitions of cosine and sine.
Example – Conversion
Convert \(z = -3+4i\) to polar form.
\(r=\sqrt{(-3)^{2}+4^{2}}=5.\)
\(\theta=\operatorname{atan2}(4,-3)=\pi-\tan^{-1}\!\frac{4}{3}\approx 2.2143\text{ rad}.\)
Hence \(z=5\operatorname{cis}2.2143\) (or \(5(\cos2.2143+i\sin2.2143)\)).
4. Arithmetic Operations
Choose the form that makes the operation simplest.
Operation Rectangular Form Polar Form
Addition / Subtraction
\((a+bi)\pm(c+di)= (a\pm c)+(b\pm d)i\)
– (normally performed in rectangular form)
Multiplication
\((a+bi)(c+di)= (ac-bd)+(ad+bc)i\)
\(r_{1}\operatorname{cis}\theta_{1}\times r_{2}\operatorname{cis}\theta_{2}= (r_{1}r_{2})\operatorname{cis}(\theta_{1}+\theta_{2})\)
Division
\(\dfrac{a+bi}{c+di}= \dfrac{(a+bi)(c-di)}{c^{2}+d^{2}}\)
\(\dfrac{r_{1}\operatorname{cis}\theta_{1}}{r_{2}\operatorname{cis}\theta_{2}}=
\left(\dfrac{r_{1}}{r_{2}}\right)\operatorname{cis}(\theta_{1}-\theta_{2})\)
Conjugate
\(\overline{z}=a-bi\)
\(\overline{z}=r\operatorname{cis}(-\theta)\)
Modulus
\(|z|=\sqrt{a^{2}+b^{2}}\)
\(|z|=r\)
Worked Example – Division in Polar Form
Find \(\displaystyle\frac{3+4i}{1-2i}\).
Convert each number to polar form:
\[
3+4i = 5\operatorname{cis}\!\bigl(\tan^{-1}\tfrac{4}{3}\bigr)=5\operatorname{cis}0.9273,
\qquad
1-2i = \sqrt5\operatorname{cis}\!\bigl(\tan^{-1}\tfrac{-2}{1}\bigr)=\sqrt5\operatorname{cis}(-1.1071).
\]
Apply the division rule:
\[
\frac{3+4i}{1-2i}= \frac{5}{\sqrt5}\operatorname{cis}\!\bigl(0.9273-(-1.1071)\bigr)
=\sqrt5\operatorname{cis}(2.0344).
\]
Return to rectangular form (optional):
\[
\sqrt5\bigl(\cos2.0344+i\sin2.0344\bigr)\approx 0.6+1.8i .
\]
5. Modulus‑Argument (Polar) Form – Summary
Every non‑zero complex number can be written uniquely as
\[
z = r\operatorname{cis}\theta = r(\cos\theta+i\sin\theta),
\]
where
\(r=|z|>0\),
\(\theta=\arg(z)\) is taken in the principal interval \(-\pi<\theta\le\pi\).
6. Loci in the Complex Plane
Typical A‑Level questions ask for the set of points \(z\) satisfying a given condition. The table below lists the most common forms together with their geometric interpretation.
Condition
Geometric description
Typical algebraic form
\(|z-c| = k\) (\(k>0\))
Circle centred at \(c\) with radius \(k\).
—
\(\operatorname{Re}\!\big((z-z_{1})\overline{(z-z_{2})}\big)=0\)
Perpendicular bisector of the segment joining \(z_{1}\) and \(z_{2}\).
—
\(\arg(z-a)=\alpha\)
Ray starting at \(a\) making angle \(\alpha\) with the positive real axis.
—
\(|z-z_{1}|+|z-z_{2}| = 2a\) (\(2a> |z_{1}-z_{2}|\))
Ellipse with foci \(z_{1},z_{2}\) and major‑axis length \(2a\).
—
\(|z-z_{1}|\;|z-z_{2}| = k\)
Rectangular hyperbola (focus‑directrix form).
—
Example – Locus of a Fixed Argument
Find the locus of points satisfying \(\arg(z-2i)=\frac{\pi}{4}\).
Interpretation: a ray starting from the point \(2i\) (i.e. \((0,2)\)) making an angle \(\pi/4\) with the positive real axis.
Equation in Cartesian form: \(\displaystyle \frac{y-2}{x}= \tan\frac{\pi}{4}=1\;\Longrightarrow\; y = x+2.\)
Hence the locus is the straight line \(y=x+2\) for \(x\ge0\) (the ray part).
7. De Moivre’s Theorem
For any integer \(n\) and any real \(\theta\):
\[
\bigl(\cos\theta+i\sin\theta\bigr)^{n}= \cos(n\theta)+i\sin(n\theta),
\qquad\text{or}\qquad
\bigl(r\operatorname{cis}\theta\bigr)^{n}=r^{n}\operatorname{cis}(n\theta).
\]
7.1 n‑th Roots of a Complex Number
If \(z=r\operatorname{cis}\theta\) with \(r>0\), the \(n\) distinct roots are
\[
z^{1/n}=r^{1/n}\operatorname{cis}\!\left(\frac{\theta+2k\pi}{n}\right),
\qquad k=0,1,\dots ,n-1.
\]
7.2 Expanding Trigonometric Powers
Equating real or imaginary parts after expanding \((\cos\theta+i\sin\theta)^{n}\) yields identities such as
\[
\cos^{3}\theta = \frac{1}{4}\bigl(3\cos\theta+\cos3\theta\bigr),\qquad
\sin^{4}\theta = \frac{3}{8}-\frac{1}{2}\cos2\theta+\frac{1}{8}\cos4\theta.
\]
8. Worked Example – Solving a Cubic Equation Using De Moivre
Problem: Find all complex solutions of \(z^{3}=8i\).
Write the right‑hand side in polar form:
\[
8i = 8\operatorname{cis}\frac{\pi}{2}.
\]
Apply De Moivre’s theorem for the cube root:
\[
z = 8^{1/3}\operatorname{cis}\!\left(\frac{\frac{\pi}{2}+2k\pi}{3}\right),
\qquad k=0,1,2.
\]
Compute \(8^{1/3}=2\) and the three arguments:
\(k=0:\;\theta_{0}= \dfrac{\pi/2}{3}= \dfrac{\pi}{6}\)
\(\displaystyle z_{0}=2\operatorname{cis}\frac{\pi}{6}=2\Bigl(\frac{\sqrt3}{2}+\frac12 i\Bigr)=\sqrt3+i.\)
\(k=1:\;\theta_{1}= \dfrac{\pi/2+2\pi}{3}= \dfrac{5\pi}{6}\)
\(\displaystyle z_{1}=2\operatorname{cis}\frac{5\pi}{6}=2\Bigl(-\frac{\sqrt3}{2}+\frac12 i\Bigr)=-\sqrt3+i.\)
\(k=2:\;\theta_{2}= \dfrac{\pi/2+4\pi}{3}= \dfrac{3\pi}{2}\)
\(\displaystyle z_{2}=2\operatorname{cis}\frac{3\pi}{2}=2(0- i)=-2i.\)
Hence the three solutions are
\[
z=\sqrt3+i,\quad z=-\sqrt3+i,\quad z=-2i.
\]
9. Quick‑Reference Checklist (Cambridge 9709 – 3.9)
Know the definitions of \(\operatorname{Re}(z),\operatorname{Im}(z),|z|,\arg(z),\overline{z}\) and the equality condition.
Convert between rectangular and polar forms; always check the quadrant (use atan2 when possible).
Use rectangular form for addition/subtraction; polar form for multiplication, division, powers, roots, and trigonometric expansions.
Remember the principal argument interval \(-\pi<\arg z\le\pi\) (or \(0\le\arg z<2\pi\) if you prefer).
Identify loci quickly:
\(|z-c|=k\) → circle.
\(\arg(z-a)=\alpha\) → ray from \(a\).
\(|z-z_{1}|+|z-z_{2}|=2a\) → ellipse.
\(|z-z_{1}|\,|z-z_{2}|=k\) → rectangular hyperbola.
Apply De Moivre’s theorem for:
powers: \((r\operatorname{cis}\theta)^{n}=r^{n}\operatorname{cis}(n\theta)\),
nth‑roots: \(r^{1/n}\operatorname{cis}\bigl(\frac{\theta+2k\pi}{n}\bigr)\),
deriving trigonometric identities.