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LearnAQA7367First teaching 2017Paper 1 · Compulsory contentComplex numbers

Paper 1 · Module

Complex numbers

5 subtopics≈ 195 min in total

Start the module: Arithmetic and conjugates Open graphing calculator

Arithmetic and conjugates
Meet $i$ , where $i^2=-1$ . Add, subtract and multiply complex numbers like brackets, divide using the conjugate, and solve equations by equating real and imaginary parts.
≈ 35 min
The Argand diagram, modulus and argument
Plot $z=x+iy$ as the point $(x,y)$ . Its distance from the origin is the modulus $|z|$ , and its angle from the positive real axis is the argument $\arg z$ , with $-\pi<\arg z\le\pi$ .
≈ 35 min
Polynomial equations and square roots
Every quadratic now has two roots. For polynomials with real coefficients, complex roots come in conjugate pairs, which lets you find all the roots from one. Square roots of $a+bi$ come from equating real and imaginary parts.
≈ 40 min
Polar and exponential forms
Write $z$ as $r(\cos\theta+i\sin\theta)$ or $re^{i\theta}$ . In these forms multiplying is easy: multiply the moduli and add the arguments. Dividing: divide the moduli and subtract the arguments.
≈ 40 min
Loci and regions in the Argand diagram
Read $|z-a|$ as "the distance from $z$ to $a$ " and $\arg(z-a)$ as "the direction from $a$ to $z$ ". Then $|z-a|=k$ is a circle, $|z-a|=|z-b|$ a perpendicular bisector, and $\arg(z-a)=\alpha$ a half-line.
≈ 45 min

AQA 7367 Complex numbers — A Level Further Mathematics · Salam Academy