1. Complex Numbers

3. Simplify: Answer:

4. Simplify: Answer: Simplify: Answer:

5. If z = 1 – 2i, simplify the following: a) b) Answer:

6. We usually use the letter z to represent a complex number. The Real portion of z is given as Re(z) = x. The Imaginary portion of z is given as Im(z) = y.

7. what is the modulus of z, orIf is the modulus and is always positive or If what is is the complex conjugate of z.

8. Graphing Complex Numbers use the Argand Diagram or the Complex Plane. x-axis would be the real numbers y-axis would be the imaginary numbers

9. Graph the complex number z = 2 + 3i Graph the complex number z = -4i

10. If z = x + yi then modulus is: and the argument is:

11. If z = x + yi notice also thatand therefore this is called the modulus-argument form of a complex number.

12. Notice also that: What if r = 1 and This is known as Euler’s Formula and was discovered in 1748. It is considered by many to be the most beautiful of all formulas since it combined so many different numbers together into one simple formula.

13. Write the following in modulus-argument form:

15. If and what is To find the product of two complex numbers, you multiply their moduli and add their arguments.

16. If and what is To find the quotient of two complex numbers, you divide their moduli and subtract their arguments.

17. Given the following complex numbers below, find the following in modulus-argument form: a) b)

18. Given the following complex numbers below, find the following in modulus-argument form: a) b)

19. Given the following complex numbers below, find the following in modulus-argument form: a) b)

20. If: then, what is?

21. This is known as DeMoivre’s Theorem

22. If z = 1 + i, find Answer: Find the value of Answer:

23. Find the value of Answer:

24. If z = x + yi, find x and y if: Find the modulus and argument of: x = -3, y = -1

25. Given that, where b is real and positive, find the exact value of b when Answer: Letandbe complex numbers. Solve the simultaneous equations: Give your answers in the formwhere Answer:

26. Given: a. Expand using the Binomial Theorem. b. Expand using de Moivre’s Theorem. c. By equating the real and imaginary parts, show that: and find the values of a, b, and c. Answer: a = 32, b = -32 and c = 6