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.y can 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 Express -1 + i.√(3) in polar form Obtain Hence, Hence, Thus

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 We can also deduce that Then the conjugate is written Therefore Hence, we can deduce that This is called De Moivre’s Theorem Also,

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. 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 • U:\1st Year Share\Mathematics for Engineers\Course materials from semester 1\Revision Material\Complex Numbers Reading • 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