This theorem gives us an efficient way to calculate powers of complex numbers. From the above, we can see that if n is any integer, then repeated multiplication of a complex number must be defined in polar form as:
Example,zn = rn[cos(θ) + sin(θ)i]n = rn[cos(nθ) + sin(nθ)i]
Another example,z6 = [cos(7π/8) + sin(7π/8)i]6 = cos(42π/8) + sin(42π/8)i
z4 = {2[cos(π/3) + sin(π/3)i]}4
= 24[cos(π/3) + sin(π/3)i]4
= 16 [cos(4π/3) + sin(4π/3)i]
= -8 - 8√3i
DeMoivre's theorem is not only true for the integers but can be extended to fractions. That is
zp/q = rp/q[cos(θ) + sin(θ)i]p/q
= rp/q[cos(pθ/q) + sin(pθ/q)i]
This gives us a general method to calculate all the root of a real number.
Repeating the example from above, using three equivalent arguments
θ, θ + 2π, θ + 4π
Then we can find all the roots:1 = cos(0) + sin(0)i = cos(2π) + sin(2π)i = cos(4π) + sin(4π)i
11/3 = cos(0/3) + sin(0/3)i = 1
= cos(2π/3) + sin(2π/3)i = -0.5(1 - √3i)
= cos(4π/3) + sin(4π/3)i = -0.5(1 + √3i)
This is the same result we found above.