Complex Numbers in Polar Form: Geometric and Algebraic Proofs

1. MAT 3730Complex Variables Section 1.3 Vectors and Polar Forms http://myhome.spu.edu/lauw

2. Preview • More on Vector Representation of complex numbers • Triangle Inequalities • Polar form of complex numbers • (Need to begin 1.4,may be?)

3. Recall We can identifyz as the position vector

4. Recall We can identifyz as the position vector

5. Triangle Inequality

6. Geometric Proof of the 1st Form

7. Geometric Proof of the 1st Form

8. (Classwork)Algebraic Proof of the 1st Form

9. Geometric Proof of the 2st Form

10. 2nd Form from the 1st Form

11. Polar Form of Complex Numbers

12. Recall We can identifyz as the ordered pair (x,y).

13. Polar Form of Complex Numbers We can also use the polar coordinate

14. Polar Form of Complex Numbers We can also use the polar coordinate Note that is undefined if z=0.

15. Polar Form of Complex Numbers We can also use the polar coordinate

16. Example 1

17. Problems 1. 2.

18. Property of Arguments • The argument of a complex number z is not unique. •  is called the Principal Argument if • Notation:

19. Example 1 (Remedy)

20. Example 1

21. Polar Form of Complex Numbers We can also use the polar coordinate

22. Product of Complex Numbers in Polar Form

23. Next Class • Read Section 1.4 • We will introduce the Complex Exponential and Euler Formula • Review Maclaurin Series (Stewart section 12.10?)