1Complex numbersV-01 Reference: Croft & Davision, Chapter 14, p 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

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

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

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.

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

6Complex numbersV Algebra of Complex Numbers Addition and Subtraction of Complex Numbers If and then

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

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

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

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

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

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

13Complex numbersV-01 Example 2.5 Simplify Solution:

14Complex numbersV-01 Simplify Solution: Example 2.6

15Complex numbersV-01 Simplify Solution: Example 2.7

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

17Complex numbersV 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.

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

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

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).

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

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

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

24Complex numbersV-01 Let and.

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

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

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

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

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

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. 

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

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

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  )

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

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 = b = 27sin  /4 = 19.09

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