1. 1Complex numbersV-01 Reference: Croft & Davision, Chapter 14, p.621 http://www.math.utep.edu/sosmath 1. Introduction Extended the set of real numbers to find solutions of greater range of equations. Let j be a root of the equation Then and COMPLEX NUMBERS

2. 2Complex numbersV-01 The number, and are called imaginary number Example 1.1 Write down (a), (b), (c) Solution: (a) (b) (c)

3. 3Complex numbersV-01 Example 1.2 Simplify (a), (b) Solution: (a) (b) or

4. 4Complex numbersV-01 is a real part and (or ) is an imaginary part Example 1.3 Solve Solution The solution are known as complex number.

5. 5Complex numbersV-01 Complex Number 16th Century Italian Mathematician – Cardano z = a + bj (Rectangular Form) –a: real number; real part of the complex number –b: real number; imaginary part of the complex number –bj: imaginary number a b Re Im a + bj

6. 6Complex numbersV-01 2. Algebra of Complex Numbers Addition and Subtraction of Complex Numbers If and then

7. 7Complex numbersV-01 Example 2.1 If and, find and Solution:

8. 8Complex numbersV-01 If and then Multiplication of complex numbers

9. 9Complex numbersV-01 Example 2.2 Find if and. Solution:

10. 10Complex numbersV-01 Example 2.3 Find if and. Solution:

11. 11Complex numbersV-01 Conjugate If then the complex conjugate of z is = and. Example 2.4 If, find and. Solution: a b Re Im z = a + bj = a - bj θ θ -b

12. 12Complex numbersV-01 i.e. Division of two complex numbers

13. 13Complex numbersV-01 Example 2.5 Simplify Solution:

14. 14Complex numbersV-01 Simplify Solution: Example 2.6

15. 15Complex numbersV-01 Simplify Solution: Example 2.7

16. 16Complex numbersV-01 Example 2.7 Find the values of x and y if. Solution: If then and Equality of Complex Numbers

17. 17Complex numbersV-01 3. Argand Diagram, Modulus and Argument The representation of complex numbers by points in a plane is called an Argand diagram. Example 3.1 Represent, and on an Argand diagram.

18. 18Complex numbersV-01 Remark: Solution: 12 3 1 2 3 -2 -3 -2 -3 Real axis Imaginary axis a b Re Im a + bj θ The angle θ is called the argument

19. 19Complex numbersV-01 Example 3.2 Find the modulus and argument of (a), (b), (c) and (d). Wrong ! Solution:

20. 20Complex numbersV-01 Important Note !!! Argument of a complex number The argument of a complex number is the angle between the positive x-axis and the line representing the complex number on an Argand diagram. It is denoted arg (z).

21. 21Complex numbersV-01 4. Polar Form, Product and Quotient in Polar Form which is the polar form expression a b Re Im a + bj θ

22. 22Complex numbersV-01 Express the complex number in polar form. Solution: Example 4.1 4 -3 θ Wrong! α

23. 23Complex numbersV-01 Express in true polar form. Solution: Example 4.2 330 0 30 0 A C -30 0

24. 24Complex numbersV-01 Let and.

25. 25Complex numbersV-01 Example 4.3 If and find and. Solution:

26. 26Complex numbersV-01 Example 4.4 Express the conjugate of in true polar form. Solution:

27. 27Complex numbersV-01 5. Exponential Form To derive the exponential form we shall need to refer to the power series expansions of cos x, sin x …

28. 28Complex numbersV-01 Euler’s Formula !!!

29. 29Complex numbersV-01 Define and  == == which is the exponential form expression and  is in radian.

30. 30Complex numbersV-01 Express the complex number and its conjugate in exponential form. Solution: Example 5.1 a b Re Im a + bj θ which is the exponential form expression and  is in radian. 

31. 31Complex numbersV-01 Find (a) and (b). Solution: and which is the exponential form expression and  is in radian. Example 5.2

32. 32Complex numbersV-01 Example 6.1 Use De Moivre’s theorem to write in an alternative form. Solution 6.De Moivre’s Theorem

33. 33Complex numbersV-01 Example 6.2 If z = r(cos  + j sin  ), find z 4 and use De Movire’s theorem to write your result in an alternative form. Solution: z 4 = r 4 (cos  + j sin  ) 4 = r 4 (cos 4  + j sin 4  )

34. 34Complex numbersV-01 More concise form: If z = r  then z n = r n  n  For example, z 4 = r 4  4  Example 6.3 (a)If z = 2  /8 write down z 4. (b)Express your answer in both polar and Cartesian form. Solution (a) z 4 = 2 4  (4)(  /8) = 16  /2 (b)i) z 4 = 16 (cos  /2 + j sin  /2) ii) a = 16 cos  /2 = 0 b = 16 sin  /2 = 16

35. 35Complex numbersV-01 Example 6.4 If z = 3 (cos  /12 + j sin  /12) find z 3 in Cartesian form Solution z 3 = 3 3 (cos(3)(  /12) + j sin(3)(  /12)) = 27(cos  /4 + j sin  /4) a = 27cos  /4 = 19.09 b = 27sin  /4 = 19.09

36. 36Complex numbersV-01 Example 6.5 (a) Express z = 3 + 4j in polar form. (b) Hence, find (3 + 4j) 10, leaving your answer in polar form. Solution