Complex numbers (Further Pure 2)
| English | Chinese | Pinyin |
|---|---|---|
| de Moivre's theorem | 棣莫弗定理 | dì mò fú dìng lǐ |
| polar form | 极坐标形式 | jí zuò biāo xíng shì |
| argument | 辐角 | fú jiǎo |
| modulus | 模 | mó |
| imaginary part | 虚部 | xū bù |
| roots of unity | 单位根 | dān wèi gēn |
| unit circle | 单位圆 | dān wèi yuán |
Raising to a power, the easy way
- To work out $(\cos\theta + i\sin\theta)^{10}$ the long way, you'd multiply ten times.
- De Moivre's theorem 棣莫弗定理 does it in one line — and as a bonus, it splits a circle into equal pieces.
De Moivre's theorem
- A complex number in polar form 极坐标形式: $z = r(\cos\theta + i\sin\theta)$.
- De Moivre's theorem — for any integer $n$:
- Raising to a power multiplies the argument 辐角 by $n$ (and the modulus 模 by $r^n$).
Complex numbers
z = x + yi
Plot z as a point: its distance from O is the modulus, its angle the argument.
De Moivre's theorem states (cos θ + i sin θ)ⁿ equals:
Raising to the power n multiplies the argument by n: cos nθ + i sin nθ.
Expanding multiple angles
- Comparing real and imaginary parts 虚部 of $(\cos\theta + i\sin\theta)^n$ gives identities.
- Worked example: the real part of the cube gives $\cos 3\theta = 4\cos^3\theta - 3\cos\theta$.
Using de Moivre, cos 3θ expands to:
Take the real part of (cosθ + i sinθ)³: cos 3θ = 4cos³θ − 3cosθ.
The roots of unity 单位根
- The equation $z^n = 1$ has exactly $n$ solutions — the $n$th roots of unity.
- They lie on the unit circle 单位圆, equally spaced $\dfrac{360^\circ}{n}$ apart, starting at $1$.
The five solutions of $z^5 = 1$ sit on the unit circle, $72^\circ$ apart — a perfect pentagon.
Why equally spaced? Each root has modulus $1$ and arguments $\dfrac{360^\circ k}{n}$ for $k = 0, 1, \dots, n-1$ — so consecutive roots differ by the same angle.
How many distinct roots of unity does the equation z⁵ = 1 have?
zⁿ = 1 has exactly n roots, equally spaced on the unit circle; here n = 5.
The fifth-roots of unity are equally spaced on the unit circle. How many degrees apart are consecutive roots?
360°/5 = 72° between consecutive roots.
The n roots of unity are equally spaced around the unit circle.
They sit at equal angular gaps of 360°/n around the circle.
Common slip
De Moivre needs polar form first. $(1 + i)^8$ is not $1^8 + i^8$ — convert $1+i$ to $r(\cos\theta+i\sin\theta)$ with $r = \sqrt2$, $\theta = 45^\circ$, then apply the theorem.
- Summing a series of $\cos$/$\sin$ terms uses the complex sum $C + \mathrm{i}S$ of complex numbers.
Before applying de Moivre to (1 + i)⁸, you must first write 1 + i in ______ form.
Convert to r(cosθ + i sinθ) first, then raise to the power.
You've got it
- de Moivre: $(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$
- it expands multiple angles, e.g. $\cos 3\theta = 4\cos^3\theta - 3\cos\theta$
- the $n$ roots of unity are equally spaced $\dfrac{360^\circ}{n}$ apart on the unit circle