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Complex Numbers and Polynomials

Quick questions on Roots of unity explained: H2 Further Mathematics

7short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the roots as powers of one root?
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Writing ω=ei 2π/n\omega = \mathrm{e}^{i\,2\pi/n} (the first primitive root), the full set is 1,ω,ω2,…,ωn−11, \omega, \omega^2, \dots, \omega^{n-1}. Each root is a power of ω\omega, which makes algebra with them compact.
What is the sum of the roots of unity?
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The nn roots of unity sum to zero for n≥2n \geq 2:
What is the nth roots of a general complex number?
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To solve zn=wz^n = w where w=r eiϕw = r\,\mathrm{e}^{i\phi}, write w=r ei(ϕ+2kπ)w = r\,\mathrm{e}^{i(\phi + 2k\pi)} and take nnth roots:
What is uneven spacing?
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The arguments must step by exactly 2πn\dfrac{2\pi}{n}; an arithmetic slip breaks the regular polygon.
What is q1?
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Write the general form of the nnth roots of unity. [1 mark]
What is q2?
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State the sum of the five fifth roots of unity. [1 mark]
What is q3?
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What modulus does each fourth root of 81eiθ81\mathrm{e}^{i\theta} have? [1 mark]

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