1. INTRODUCTION OPERATIONS OF COMPLEX NUMBER THE COMPLEX PLANE THE MODULUS & ARGUMENT THE POLAR FORM

2. COMPLEX NUMBERS (C) COMPLEX NUMBERS (C) REAL NUMBERS (R) REAL NUMBERS (R) INTEGERS (Z) INTEGERS (Z) RATIONAL NUMBERS (Q) RATIONAL NUMBERS (Q) IRRATIONAL NUMBERS (Q) IRRATIONAL NUMBERS (Q) WHOLE NUMBERS (W) WHOLE NUMBERS (W) NATURAL NUMBERS (N) NATURAL NUMBERS (N)

3. To solve algebraic equations that do not have real solutions. To solve Complex number: Since, Real solution No real solution

4. Example 1 : Solve

5. Example : Solution

6. Definition 1.1 If z is a complex number, then the standard equation of Complex number denoted by: where a, b R a – Real part of z (Re z) b – Imaginary part of z (Im z)

7. Example 1.2 : Express in the standard form, z:

8. Re(z) = 2, Im (z) = -2 Example 1.2 : Solution: Re(z) = 3, Im (z) = 2√10

9. Definition 1.2 2 complex numbers are said equal if and only if they have the same real and imaginary parts: Iff a = c and b = d

10. Example 1.3 : Find x and y if z 1 = z 2 :

11. Definition 1.3 If z 1 = a + bi and z 2 = c + di, then:

12. Example 1.4 : Given z 1 = 2+4i and z 2 = 1-2i

13. Definition 1.4 The conjugate of z = a + bi can be defined as: ***the conjugate of a complex number changes the sign of the imaginary part only!!!

14. Example 1.5 : Find the conjugate of

15. The Properties of Conjugate Complex Numbers

16. Definition 1.5 (Division of Complex Numbers) If z 1 = a + bi and z 2 = c + di then: Multiply with the conjugate of denominator

17. Example 1.6 : Simplify and write in standard form, z:

18. The complex number z = a + bi is plotted as a point with coordinates (a,b). Re (z) x – axis Im (z)y – axis Im(z) Re(z) O(0,0) z(a,b) a b

19. Definition 1.6 (Modulus of Complex Numbers) The modulus of z is defined by Im(z) Re(z) O(0,0) z(a,b) a b r

20. Example 1.7 : Find the modulus of z:

21. The Properties of Modulus

22. Definition 1.7 (Argument of Complex Numbers) The argument of the complex number z = a + bi is defined as 1 st QUADRANT2 nd QUADRANT 4 th QUADRANT 3 rd QUADRANT

23. Example 1.8 : Find the arguments of z:

24. Based on figure above: b a (a,b) r Re(z) Im(z)

25. The polar form is defined by: Example 1.9: Represent the following complex number in polar form:

26. Answer 1.9 : Polar form of z:

27. Example 1.10 : Express the following in standard form of complex number:

28. Answer 1.10 : Standard form:

29. Theorem 1: If z 1 and z 2 are 2 complex numbers in polar form where then,

30. Example 1.11 : a) If z1 = 4(cos30+isin30) and z2 = 2(cos90+isin90). Find : b) If z1 = cos45+isin45 and z2 = 3(cos135+isin135). Find :

31. Answer 1.11 :

32. Think of Adam and Eve like an imaginary number, like the square root of minus one: you can never see any concrete proof that it exists, but if you include it in your equations, you can calculate all manner of things that couldn't be imagined without it. Philip Pullman Philip Pullman In The Golden Compass (1995, 2001), 372-373.