De Moivre and complex roots
Use De Moivre’s theorem to find powers and all the roots of a complex number.
For integer n, [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ). Keep the angular unit consistent. Convert to rectangular form only after applying the power rule.
The nth roots of a nonzero number have modulus r^(1/n) and arguments (θ + 2kπ)/n for k = 0,…,n − 1. They are equally spaced on a circle. Taking only θ/n misses the remaining roots.
Worked example
Find the cube roots of 8.
- Write 8 with modulus 8 and argument 0.
- Each root has modulus 2.
- The arguments are 0, 2π/3 and 4π/3.
Answer: 2, −1 + √3 i, −1 − √3 i
Revise first: Modulus and argument.
Finding every root, without counting one twice
Write z = r(cos θ + i sin θ) before taking powers or roots. Powers multiply the argument; taking an nth root divides the argument but also introduces n choices. An argument describes a direction, so adding a full turn leaves the same complex number.
For zⁿ = R(cos α + i sin α), the roots have modulus R^(1/n) and arguments (α + 2kπ)/n, with k = 0, …, n − 1. Taking just the principal argument loses roots. Taking k = n repeats the first root.
The cube roots of −1
- The modulus is 1. Take α = π, so the arguments are (π + 2kπ)/3.
- For k = 0, 1, 2 this gives π/3, π and 5π/3.
- The roots are 1/2 + (√3/2)i, −1 and 1/2 − (√3/2)i. Each cubes to −1.
- Their sum is zero, agreeing with the missing z² coefficient in z³ + 1.
Why does k = 3 add no new root?
Its argument is 7π/3, the same direction as π/3. There are exactly three distinct roots.
Take the idea further
Rotate all the roots by π/6. What equation do the rotated roots satisfy? Since w = ze^(iπ/6), w³ = −i.
Practise de moivre and complex roots
Course mapping
These specification references show where the topic occurs. The questions cover only some parts of each topic.
- 9FM0 · core · 2 Complex numbers: De Moivre and roots
- 7367 · core · B Complex numbers: De Moivre and roots
- H245 · core · 2 Complex numbers: De Moivre and roots
- H645 · core · Complex numbers: De Moivre and roots
- 9FM0 · core · 2 Complex numbers: Powers of a complex number
- 7367 · core · B Complex numbers: Powers of a complex number
- H245 · core · 2 Complex numbers: Powers of a complex number
- H645 · core · Complex numbers: Powers of a complex number
- 9FM0 · core · 2 Complex numbers: All arguments of complex roots
- 7367 · core · B Complex numbers: All arguments of complex roots
- H245 · core · 2 Complex numbers: All arguments of complex roots
- H645 · core · Complex numbers: All arguments of complex roots
Next practice: Complex loci.