nth roots & roots of unity
Every non-zero complex number has exactly n distinct nth roots, evenly spaced around a circle of radius ⁿ√r. The formula is z^(1/n) = ⁿ√r · e^(i(θ + 2πk)/n) for k = 0, 1, …, n−1 — and the 2πk is the whole point, since omitting it produces only one of the n roots.
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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.
- Write the number in polar formIncluding the +2πk in the argument before dividing.
- Take the real nth root of the modulusⁿ√r, a positive real number.
- Divide the argument by n for each k(θ + 2πk)/n for k = 0 to n−1. Beyond that the roots repeat.
- Check the geometryThe n roots lie on a circle of radius ⁿ√r, separated by 2π/n — they form a regular n-gon.
Worked example
Find all cube roots of 8.
- 8 = 8e^(i·0), so r = 8 and θ = 0. Cube root of the modulus: 2.
- Arguments: (0 + 2πk)/3 for k = 0, 1, 2 — that is 0, 2π/3 and 4π/3.
- k=0: 2e^(i0) = 2.
- k=1: 2e^(i2π/3) = 2(−1/2 + i√3/2) = −1 + i√3. k=2 gives the conjugate −1 − i√3.
Answer. 2, −1 + i√3 and −1 − i√3 — three roots at the vertices of an equilateral triangle of radius 2.
Where marks get dropped
These are the specific errors that cost credit on nth roots & roots of unity questions — QED's rubric penalises each of them separately.
- Giving only the real root. Every non-zero complex number has exactly n distinct nth roots, and the question wants all of them.
- Omitting the 2πk term, which collapses all n roots into one.
- Continuing past k = n−1. Those values repeat roots already found.
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nth roots & roots of unity — frequently asked questions
What are the roots of unity?
The n solutions of zⁿ = 1, namely e^(2πik/n). They form a regular n-gon on the unit circle with one vertex at 1, and they sum to zero for n ≥ 2.
Why do the roots sum to zero?
They are the roots of zⁿ − 1 = 0, whose z^(n−1) coefficient is 0, so by Vieta’s formula the sum of roots is 0. Geometrically, symmetric vectors around a circle cancel.
How do I find roots of a general complex number?
Identical method: take the real nth root of the modulus and divide the argument (plus 2πk) by n. Only the starting angle differs from the roots of unity.
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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