Roots — equally spaced on a circle
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Flip to reveal answersHow many nth-roots does a non-zero complex number have?
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Question
How many nth-roots does a non-zero complex number have?
Answer
Exactly n distinct nth-roots.
Question
Where do the nth-roots sit on an Argand diagram?
Answer
On a circle of radius R^(1/n), equally spaced 2π/n apart (a regular n-gon).
Question
Formula for the nth-roots of R cisφ?
Answer
z_k = R^(1/n) cis((φ + 2πk)/n) for k = 0, 1, …, n − 1.
Question
How do you get all the roots once you have one?
Answer
Keep adding 2π/n to the argument until you have n of them.
Question
The three cube roots of 1?
Answer
1, cis(2π/3) = −½ + (√3/2)i, cis(4π/3) = −½ − (√3/2)i.
Question
Cube roots of 8?
Answer
2, −1 + √3 i, −1 − √3 i (modulus 2, spaced 120°).
Question
What modulus do all the nth-roots share?
Answer
R^(1/n), where R is the modulus of the original number.
Question
Why do the roots form a regular polygon?
Answer
They share the same modulus (so lie on a circle) and are equally spaced 2π/n apart.
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Full study notes for Roots — equally spaced on a circle
Topic 1.14 hub
De Moivre & roots (HL only)
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