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H2 Maths Notes 4 1 Complex Numbers Argand Diagrams

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H2 Maths Notes (JC 1-2): 4.1) Complex Numbers and Argand Diagrams

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07 Oct 2025, 00:00 Z

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Q: What does H2 Maths Notes (JC 1-2): 4.1) Complex Numbers and Argand Diagrams cover?
A: Cartesian and polar forms, loci, and standard manipulations for H2 Maths Topic 4.1.

Before you revise
Keep polar and exponential forms side by side-most Argand proofs demand switching between them quickly. Draw every locus and label arguments clearly.

Core Definitions

Complex number $z = x + iy$ with real part $x$ and imaginary part $y$ .
Modulus $\lvert z \rvert = \sqrt{x^2 + y^2}$

Part 16 of 32

Previous topicH2 Maths Notes (JC 1-2): 3) Vectors Next topicH2 Maths Notes (JC 1-2): 4) Complex Numbers

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; argument

\arg z

is the angle from positive real axis.

Polar form:

z = r(\cos \theta + i \sin \theta)

with

r = \lvert z \rvert \space \theta = \arg z

.

Exponential form:

z = r e^{i\theta}

.

Algebra of Complex Numbers

Addition/subtraction: combine real and imaginary parts.
Multiplication: $(a + ib)(c + id) = (ac - bd) + i(ad + bc)$ .
Division: multiply numerator and denominator by complex conjugate.
Conjugate $\bar{z} = x - iy$ ; satisfies $z \bar{z} = \lvert z \rvert^2$

Example -- Division

Compute $\dfrac{3 + 4i}{1 - 2i}$ . $\frac{3 + 4i}{1 - 2i} \times \frac{1 + 2i}{1 + 2i} = \frac{(3 + 4i)(1 + 2i)}{1 + 4} = \frac{3 + 6i + 4i + 8i^2}{5} = \frac{-5 + 10i}{5} = -1 + 2i.$

Polar Form and De Moivre

Convert using $r = \sqrt{x^2 + y^2}$ , $\theta = \operatorname{atan2}(y, x)$ .
De Moivre: $\bigl[r(\cos \theta + i \sin \theta)\bigr]^n = r^n (\cos n\theta + i \sin n\theta)$
Roots: $z^n = w$ has roots $r^{1/n} \left( \cos\frac{\theta + 2k\pi}{n} + i \sin\frac{\theta + 2k\pi}{n} \right)$

Example -- Cube roots

Solve $z^3 = 8(\cos \pi + i \sin \pi)$ .

$r = 8$ , $\theta = \pi$ .
Roots: $2 \left( \cos\frac{\pi + 2k\pi}{3} + i \sin\frac{\pi + 2k\pi}{3} \right)$ for $k = 0, 1, 2$ .
Gives $2\bigl( \cos \tfrac{\pi}{3} + i \sin \tfrac{\pi}{3} \bigr), 2\bigl( \cos \pi + i \sin \pi \bigr), 2\bigl( \cos \tfrac{5\pi}{3} + i \sin \tfrac{5\pi}{3} \bigr)$

Loci on the Argand Plane

$\lvert z - a \rvert = r$ represents a circle centre $a$ radius $r$ .
$\arg(z - a) = \theta$ is a ray from $a$ making angle $\theta$ .
Linear loci: express $z = x + iy$ , substitute into condition, separate into real/imaginary parts to get Cartesian equation of line/curve.

Example -- Perpendicular bisector

Locus of points equidistant from $2 + i$ and $-1 + 3i$ :

$\lvert z - (2 + i) \rvert = \lvert z - (-1 + 3i) \rvert.$

Let $z = x + iy$ . Squaring both sides yields $(x - 2)^2 + (y - 1)^2 = (x + 1)^2 + (y - 3)^2$ which simplifies to $6x - 4y + 2 = 0$ .

Inequalities and Regions

$\lvert z - a \rvert \leq r$ is a closed disk; $<$ gives interior.
Combined inequalities create lens or annulus regions; sketch boundaries first, then shade interior.
Argument inequalities produce sector regions between rays.

Calculator Workflow

Use GC complex mode to check arithmetic; ensure calculator is in radians for argument work.
Store complex numbers in memory to avoid transcription errors during De Moivre calculations.
For loci, use graphing mode with implicit equations to cross-check manual sketches.

Exam Watch Points

Always state principal value of argument in the interval requested (usually $-\pi < \arg z \leq \pi$ ).
When sketching loci, mark intercepts, centre, radius, and boundary style (solid/dashed).
Provide exact trigonometric forms-avoid rounding angles unless final answer requires degrees.
For De Moivre problems, list all distinct roots; avoid duplicates caused by missing the $2k\pi$ term.

Practice Quiz

Drill Argand plotting, modulus/argument regions, and transformation effects before tackling calculus.

Quick Revision Checklist

Convert between Cartesian, polar, and exponential forms fluently.
Perform conjugation, modulus, and division confidently.
Sketch loci derived from modulus and argument conditions with accurate annotations.
Apply De Moivre to powers and roots, checking for all distinct solutions.

Next steps: Keep the H2 Maths notes hub bookmarked and move on to Topic 4 - Complex numbers deep dive before tackling calculus in Topic 5.

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H2 Maths Notes (JC 1-2): 4.1) Complex Numbers and Argand Diagrams

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