POWERS AND ROOTS OF COMPLEX NUMBERS DR SHILDNECK
POWERS AND ROOTS OF COMPLEX NUMBERS DR. SHILDNECK
MULTIPLYING COMPLEX NUMBERS •
DEMOIVRE’S THEOREM •
EXAMPLE •
ROOTS OF COMPLEX NUMBERS Remember from the Fundamental Theorem of Algebra that polynomials of degree n have exactly n zeros (roots) in the complex number system. This means that x 4 = 256, which we can re-write as x 4 – 256 = 0, has exactly FOUR roots. This indicates that the number 256 has exactly four 4 th roots in the complex numbers (4, -4, 4 i, -4 i). In general, all nonzero complex numbers have exactly n distinct nth roots. That is, they have two square roots, three cube roots, fourth roots, etc.
DEMOIVRE’S THEOREM (FOR ROOTS) •
EXAMPLE •
EXAMPLE •
ROOTS OF COMPLEX NUMBERS Observations about the roots of complex numbers The roots of complex numbers all have the same modulus (which can be thought of as the radius of a circle). When plotting the roots, you will notice that the roots are equally spaced around that circle.
ROOTS OF UNITY (SPECIAL CASE) Finding the pth roots of 1 When written in polar form, one is written as r = 1. Thus, the modulus is 1, which means the pth roots of 1 lie on the unit circle. Like all nonzero complex numbers, 1 has p distinct pth roots in the complex number system.
EXAMPLE •
ASSIGNMENT Handout # 53 -75 odd
- Slides: 12