Complex multiplication as rotation & scaling
Multiplying by re^(iα) scales by r and rotates by α about the origin — that is the entire geometric content of complex multiplication. It makes plane transformations algebraic: rotation about a point a is z ↦ a + e^(iα)(z − a), which is translate, rotate, translate back, written in one line.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Read the multiplier in polar formIts modulus is the scale factor and its argument is the rotation angle.
- Recognise the special casesMultiplying by i rotates 90° anticlockwise; by −1 rotates 180°; by a positive real just scales.
- Rotate about a general pointSubtract the centre, multiply, add it back: w = a + e^(iα)(z − a).
- Compose transformations by multiplyingTwo successive rotations multiply their factors, so the angles add automatically.
Worked example
Rotate the point z = 3 + i by 90° anticlockwise about the point a = 1 + i.
- Translate so a is at the origin: z − a = (3 + i) − (1 + i) = 2.
- Rotate by 90°: multiply by i, giving 2i.
- Translate back: 2i + (1 + i) = 1 + 3i.
- Check: the distance from a was 2 before and |1 + 3i − (1+i)| = |2i| = 2 after ✓.
Answer. The image is 1 + 3i — the same distance from the centre, turned a quarter turn.
Where marks get dropped
These are the specific errors that cost credit on complex multiplication as rotation & scaling questions — QED's rubric penalises each of them separately.
- Rotating about the origin when the centre is elsewhere. You must translate first, or the point lands in the wrong place.
- Using a clockwise angle without negating it. Positive arguments rotate anticlockwise by convention.
- Forgetting that the modulus scales too. Multiplying by 2i both doubles the distance and rotates 90°.
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Complex multiplication as rotation & scaling — frequently asked questions
How does this relate to rotation matrices?
Multiplication by cos α + i sin α corresponds exactly to the 2×2 rotation matrix [[cos α, −sin α],[sin α, cos α]]. Complex numbers are a compact encoding of that algebra.
What transformation is conjugation?
Reflection in the real axis: z̄ flips the sign of the imaginary part. Combining it with multiplication gives every reflection in a line through the origin.
Why is this useful in graphics?
Because composing rotations reduces to multiplying complex numbers — cheaper and more numerically stable than repeated matrix products. In 3D the same idea extends to quaternions.
The rest of Complex Numbers
Arithmetic, the Argand plane, polar form and De Moivre. Each subtopic below has its own method, worked example and mark-losing traps.
- 1The imaginary unit i & complex arithmetic
- 2Conjugates & division
- 3Modulus, argument & the Argand diagram
- 4Polar & exponential form re^{iθ}
- 5Multiplying & dividing in polar form
- 6De Moivre’s theorem & powers
- 7nth roots & roots of unity
- 8Solving polynomial equations over ℂ
- 9The fundamental theorem of algebra & conjugate roots
- 10Euler’s formula & trigonometric identities
- 11Loci & regions in the complex plane
- 12Complex multiplication as rotation & scaling
- 13Applications: phasors & AC circuits
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