Introduction

1. Introduction • We will cover 5 topics today • 1. Complex Numbers Definitions and Rules • 2. The Argand Diagram • 3. Complex Numbers and Polar Coordinates • 4. Complex Numbers in Exponential Form • 5. Applications of Complex Numbers

2. Complex Numbers Equality if and A complex number is defined as Sum What is i3, i4, i5 and i6? Subtraction Multiplication Conjugate The standard form of a complex number is Division Real part Imaginary part

3. Complex Numbers Find Properties of the conjugate If What is the conjugate of

4. Argand Diagram The polar angle θ is called the argument of z and is written ‘arg z’. Polar angles differing by an angle of 2π are equivalent A complex number in the form z = x + i.ycan be represented by a pair of real numbers (x,y) known as an ordered pair. This pair of numbers can be expressed on a Cartesian axis and this is called an Argand diagram. r is called the modulus of z (or mod z) and is written |z| Imaginary Axis, y Properties of the modulus r Real axis, x

5. Argand Diagram Imaginary Axis, y b (0,2) Let z = 1 + i. Plot the following complex numbers on an Argand Diagram a) b) c) d) d (0,1) Real axis, x a (1,-1) c (0,-2)

6. Complex Numbers and Polar Coordinates Recall the following diagram i.e. r is equal to the modulus of z Imaginary Axis, y θ = Arg z The Principal Value of the Argument r The pair of equations Real axis, x has exactly one solution for θ within this range. By substitution we can say The complex number can be specified in terms of the polar coordinates ‘r’ and ‘θ’.

7. Complex Numbers and Polar Coordinates Obtain Express -1 + i.√(3) in polar form Hence, Thus Hence,

8. Complex Numbers in Exponential Form Consider the function Hence, we can conclude that If we take the Taylor expansion of both terms we find that and Where |z| = r and θ = Arg(z) We also know that

9. Complex Numbers in Exponential Form If Then Then the conjugate is written This is called De Moivre’s Theorem Hence, we can deduce that Also, Hence, We can also deduce that because

10. Complex Numbers in Exponential Form Express the following complex numbers in exponential form using principal values of the arguments (2) -5i, Thus (1) i, In each case put r.Cos(θ) equal to the real part and r.Sin(θ) equal to the imaginary part. (3) -3, Thus Now we use an Argand diagram to calculate the principal value of the angle θ thus

11. Complex Numbers in Exponential Form (4) 3 - 4i, α = -0.927 radians

12. Complex Numbers Examples Find all of the solutions to the equation Therefore We first express 4 - 4.i in polar form, thus i.e. Hence, Five successive values of n give distinct solutions; other values of n merely duplicate existing solutions. And by using an Argand diagram The five solutions are Let hence

13. Complex Numbers Examples Expand Cos6(θ) in terms of θ We know n = 1 hence, (using a Binomial Expansion) By De Moivre’s theorem and Hence By adding these together we get

14. Complex Number Exam Questions Use the binomial theorem, or otherwise, to expand the expression De Moivre’s theorem gives that for any positive integer n. Taking n = 4 use the above to show that And obtain an analogous expression for Sin(4θ)

15. Complex Number Exam Questions We know that

16. Complex Number Exam Questions Now we can collect the real and imaginary terms together Equate real parts to give Equate imaginary parts to give Hence

17. Complex Number Exam Questions It is given that the polynomial can be written in the form where q(x) is a quadratic a) Obtain q(x) b) Calculate the roots of q(x) c) Plot the roots of q(x) on an Argand diagram

18. Complex Number Exam Questions a) Obtain q(x) If then Equate coefficients of x x4 : 1 = a i.e. a = 1 x3 : -2 = b i.e. b = -2 x2 : -3 = c – 9a i.e. c = 9a – 3 = 6 Hence

19. Complex Number Exam Questions b) The roots of q(x) are given by the quadratic formula c) Argand Diagram Imaginary Axis, y (1,√5) Real axis, x (1,-√5)

20. Complex Number Exam Questions Obtain the modulus and argument (in radians) of the complex numbers z1 = 2 + 3i and z2 = -1 – i and write z1 and z2 in the polar form r.ei.θ Hence or otherwise, determine the polar form of the complex numbers a) b)

21. Complex Number Exam Questions Given With Hence With However from the Argand diagram we can determine that Hence the correct root is -3π/4 Therefore

22. Complex Number Exam Questions a) b)

23. Conclusion Today we have looked at 1. Complex Numbers Definitions and Rules 2. The Argand Diagram 3. Complex Numbers and Polar Coordinates 4. Complex Numbers in Exponential Form 5. Applications of Complex Numbers • Essential reading for next week • HELM Workbook 10.1: Complex Arithmetic • HELM Workbook 10.2: Argand Diagrams and the Polar Form • HELM Workbook 10.3: The Exponential Form of a Complex Number • HELM Workbook 10.4: De Moivre’s Theorem • OR • CHAPTER 11 ofMathematics for Engineers (Croft & Davidson)