10.6 Roots of Complex Numbers

Notice these numerical statements. These are true! But I would like to write them a bit differently. 32 = 2 5 –125 = (–5) 3 16 = 2 4 Keeping this “change in appearance” in mind, let’s extend this to the complex plane. 3i is a fourth root of 81 because (3i) 4 = 81 In general … For complex numbers r and z and for any positive integer n, r is an nth root of z iff r n = z. We will utilize DeMoivre’s Theorem to verify.

Ex 1) Find the four fourth roots of Let w = r cis θ represent a fourth root. Then but there are lots of angles that terminate at we must consider multiples Four 4 th roots:

Complex Roots Theorem For any positive integer n and any complex number z = r cis θ, the n distinct nth roots of z are the complex numbers for k = 0, 1, 2, …, n – 1 Now, what does this mean?? Explain what “to do” in plain words! Ex 2) Find the cube roots of 1000i  i think! (graph in head)

Ex 3) Graph the five fifth roots of 32.

Ex 5) You can use these answers to find the cube root of 8 *multiply #4 answers by The various nth roots of 1 are called the roots of unity. 1 in polar is: 1 cis 0 so nth roots of unity are of the form for k = 0, 1, 2, …, n – 1 n is nth root Ex 4) Find the three cube roots of unity and locate them on complex plane. r = 1 and they are spaced rad apart

The complex roots theorem provides a connection between the roots of a complex number and the zeros of a polynomial. Ex 6) Find all solutions of the equation x = 2i x 3 = –2 + 2i (aka find 3 roots of –2 + 2i) polar

Homework #1007 Pg 526 #1, 2, 7, 8, 12, 13, 16, 17, 19, 23, 27, 42, 43, 44