Download Powers of Complex Numbers in Polar Form

Powers of Complex Numbers in Polar Form
(2+i)2 = (2+i)12 = Powers of Complex Numbers in Polar Form
Let z = r(cosθ +isinθ)
z2 = r(cosθ +isinθ)r(cosθ +isinθ)
= r2(cos2θ +isin2θ)
z3 = r2(cos2θ +isin2θ)r(cosθ +isinθ)
= r3(cos3θ +isin3θ)
z4 = r3(cos3θ +isin3θ)r(cosθ +isinθ)
= r4(cos4θ +isin4θ)
The formula for the nth power of a complex number in polar form is known as DeMoivre's Theorem (in honor of the French mathematician Abraham DeMoivre (1667‐1754).
DeMoivre's Theorem
Let z = r(cosθ+isinθ) be a complex number in polar form. If n is a positive integer, z to the nth power, zn, is
zn = [r(cosθ+isinθ)]n = rn[cos(nθ)+isin(nθ)] = rnCiS(nθ)
Ex1. Try. Ex2. Try. Roots of a Complex Number in Polar Form
Yesterday we saw that [2(cos20o +isin20o)]6 = 64(cos120o + isin120o)
Thinking backwards we could note that:
A sixth root of 64(cos120o + isin120o) is 2(cos20o +isin20o)
It is one of SIX distinct 6th roots of 64cis120o.
Nth Root of a Complex Number
In general, if a complex number, z, satisfies the equation zn = w, then we can say that z is a complex nth root of w.
Also, z is one of n complex nth roots of w.
Given the fact that roots are actually powers (remember x1/2 = √x), DeMoivre's Theorem can also be applied to roots...
DeMoivre's Theorem for Finding Complex Roots
Let w = rcisθ be a complex number in polar form. If w ≠ 0, then w has n distinct complex nth roots given by the formula:
(radians)
or
(degrees)
where k = 0, 1, 2, ..., n‐1.
Important Notes about roots:
1. There are two distinct square roots
There are three distinct cube roots
There are four distinct fourth roots, and so on.
2. Each one of the distinct nth roots has the same modulus, .
3. Successive roots have arguments (angles) that differ by the same amount.
4. Successive roots are equally spaced around a circle centered about the pole.
Ex1. Find all complex 4th roots of 16cis120o.
TRY. Find all complex 5th roots of 243cis(80o).
Ex2. Use DeMoivre's Theorem to find all cube roots of 8.
TRY. Use DeMoivre's Theorem to find all cube roots of 27.
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