Complex Roots

1. Complex Roots • Solve for z given that z4=81cis60°

2. De Moivre’s Theorem can be used to solve equations involving a complex numbers & powers If zn= r cis , then one solution will be z= r1/n cis /n However, the Fundamental Theorem of Algebra states that there must be n roots… and these are equally spaced around the argand diagram

3. Any complex number can be written in a more general form as z= rcis(Ө+2πk) or z= rcis(Ө+360k) • Eg: 81cis60 could also be written as: • 81cis420, 81cis780, 81cis1140 etc • Or more generally, 81cis(60+360k) • Applying DeMoivres theorem, • z4=81cis(60°+360k) Substitute in k values:

4. Notice that the roots are always symmetrical around the origin 3cis(105) They are spread at angles of 360/n (in this case 90°) 3cis(15) 3cis(-165) 3cis(-75)

5. To find ALL complex roots we must apply De Moivre’s Thm to the general expression r cis ( + k2) or r cis ( + 360k) e.g. Solve z4 = 4 +3i Step 1: Write in polar form Step 2: Express in general form Step 3: Use De Moivre’s Theorum Step 4: Substitute values of k up to n to generate solutions. Step 5: Give your solutions in the same form they were asked (in this case, rectangular).

6. More practice: 32.4 p.299