Pure D: Complex Numbers (STEP 3)

Topic overview

De Moivre's theorem, roots of complex numbers, geometric applications

Key concepts

Verify STEP 2/3 separately. Complex numbers and 2×2 matrices may belong to STEP 2; De Moivre, general complex roots, planes and vector products, etc. belong to STEP 3.

Key points

◆Follow the prerequisite knowledge requirements given by the examination body; new concepts may only be used when the question provides sufficient guidance.
◆The examples on this site are training with a limited scope and do not represent full syllabus coverage or official difficulty.

Worked examples

Try each example yourself before checking the explanation.

1Written examples within the scope

For a positive integer $n$ , the $n$ -th roots of unity are the complex numbers $\omega_k = e^{2\pi i k/n}$ for $k = 0, 1, \ldots, n-1$ .

Show that $\displaystyle\sum_{k=0}^{n-1} \omega_k^j = \begin{cases} n & \text{if } n \mid j \\ 0 & \text{otherwise} \end{cases}$ for any integer $j$ .

Let $n = 5$ and $\omega = e^{2\pi i/5}$ . Show that $1 + \omega + \omega^2 + \omega^3 + \omega^4 = 0$ and hence find $\cos\dfrac{2\pi}{5} + \cos\dfrac{4\pi}{5}$ .

Show that $\omega$ satisfies $z^2 + z - 1 = 0$ in some quadratic equation, and hence find the exact value of $\cos\dfrac{2\pi}{5}$ .

Using $\cos\dfrac{2\pi}{5}$ , find the exact value of $\sin\dfrac{\pi}{5}$ and verify that a regular pentagon inscribed in a unit circle has side length $\sqrt{\dfrac{5-\sqrt{5}}{2}}$ .

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