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