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Roots of Complex NumbersSection 9.3
nth root
• If z and w are complex numbers and if n ≥ 2 is an integer, then z is an nth root of w iff zn = w
Example 1
• Write the 3 cube roots of 1331 in trigonometric form.
First, z3 = 1331z = [r, θ] by De Moivre’sz3 = [r3, 3θ] = [1331, 0]So r3 = 1331 so, r = 113θ = 0 + 2π or θ = 0 + 2/3π
Example 1
3θ = 0 + 2π or θ = 0 + 2/3πSo, we have
[11, 0] 11 (cos (0) + isin (o)) 11[ 11, 2/3 π] 11 (cos (2π/3)+ isin
(2π/3))[11, 4/3 π] 11 (cos (4π/3)+ isin
(4π/3)
nth root of complex numbers theorem• The n nth roots of [r, θ] are
where k = 0, 1, 2, .., n-1
The n nth roots of r(cos θ + isin θ)
Where k = 0, 1, 2, …, n-1
]2
,[n
kn
rn
))2
sin()2
(cos(n
kn
in
kn
rn
Example 2Express the sixth roots of [729, 60˚]
Nth roots of complex numbers theorem
[3, 10˚]
]6
360
6
60,729[6 k
[3, 70˚]
[3, 250˚]
[3, 130˚]
[3, 190˚]
[3, 310˚]
Example 3
Express the square roots of i in polar, trigonometric, and binomial.
Rectangular: (0,1)Polar: [1, π/2]Square roots in polar: ]
2
2
2
2/,1[2
k
]4,1[
]4
5,1[
Example 3
to trigonometric:
to binomial:
to trigonometric:
to binomial:
]4,1[
]4
5,1[
)4
sin4
(cos1
i
2
2
2
2i
)4
5sin
4
5(cos1
i
2
2
2
2i
Example 4There are exactly two real nth roots of a positive
number if n is even and exactly one real nth root if n is odd. The other nth roots are vertices of a regular n-gon centered at the origin.
Example: z5 = 51 What figure does this make?How many real and nonreal solutions does this
make?
Regular pentagon
Nonreal: 4Real: 1
Homework
Pages 539 – 5402, 4 – 10, 13