Fast facts about complex numbers

1. i ² = -1. Thus i ⁴ = 1

2. (-i)² = -1. Thus (-i)⁴ = 1

3. (a + ib)+(c + id) = (a + c) + i(b + d)

4. (a + ib)⋅(c + id) = (ac-bd) + i(ad+bc)
   (since bdi ²=-bd)

5. (cos x + i sin x)⋅(cos y + i sin y)=cos(x+y) + i sin(x+y)
   (by (4) and the formulas for sums of angles)

6. (cos x + i sin x)^k = cos kx + i sin kx
   (using (5) repeatedly)

7. (cos 2π/n + i sin 2π/n)ⁿ = 1, for all positive integers n
   (using (6), and that cos 2πk = 1 and sin 2πk = 0)

8. For a fixed positive integer n, let ω= ω¹ = cos 2π/n + i sin 2π/n. Then
   1. ω^k = cos 2πk/n + i sin 2πk/n
      (using (6))

   2. ωⁿ = 1
      (this is just (7), restated)

   3. ω^{k + n}= ω^k. Also, ω^{k + nr} = ω^k for any r ∊ ℤ
      (since ω^{k + n} = ω^k⋅ωⁿ and ω^{k + nr} = ω^k⋅(ωⁿ)^r = ω^k⋅1^r = ω^k)

   4. (ω^k)ⁿ = 1
      (since (ω^k)ⁿ = ω^kn = ω^nk = (ωⁿ)^k = 1^k = 1)

   5. If n is even, ω^n/2 = -1, and ω^{k+ n/2} = -ω^k
      (by (8), ω^n/2 = cos 2π(n/2)/n + i sin 2π(n/2)/n = cos π + i sin π = -1,
      and then ω^{k+ n/2} = ω^kω^n/2 = ω^k(-1))

   By (8.4), any power of ω is an nth root of 1. However only n of these powers are different, as can be seen from (6). It turns out that these are the only nth roots of 1. However some of these roots are kth roots of 1 for smaller values of k than n.

   We say z is a primitive nth root of 1 iff zⁿ = 1 ∧ z^r ≠ 1 for all r < n.

9. The 8 8th roots of 1 are 1, -1, i, -i and ±√2/2 ± i⋅√2/2. The first four are not primitive 8th roots of 1; the others are.

10. If ω= (cos 2π/8 + i sin 2π/8) = √2/2 + i⋅√2/2, then
    1. ω³ = -√2/2 + i⋅√2/2
    2. ω⁵ = -√2/2 - i⋅√2/2
    3. ω⁷ = √2/2 - i⋅√2/2
    4. ω + ω⁷ = √2
    5. ω³ + ω⁵ = -√2
    6. ω + ω³ = i√2
    7. ω³ + ω⁷ = -i√2
    These facts are relatively easy to remember by noting that 2π/8 = 45°, and using (11).

11. The nth roots of 1 are at the vertices of a regular n-gon if a+ib is identified with the point (a,b) in ℝ².

12. Any algebraic equation remains true if -i replaces i.

13. If ω is an nth root of 1, ω ≠ 1, and the sum is taken as k ranges from 0 through n-1, then Σ ω^k = 0
    (by the formula for the sum of a geometric series)