Mathematics

My Goals

By the end of this unit, I will be able to:

࿌ Identify a real part and imaginary part of a complex

number.

࿌ Convert a complex number from one form to another.

࿌ Represent a complex number on Argand diagram.

࿌ State De Moivre’s formula and Euler’s formulae.

࿌ Apply the properties of complex numbers to perform

operations on complex numbers in algebraic form, in

polar form or in exponential form.

࿌ Find the modulus and the nth roots of a complex

number.

࿌ Solve in the set of complex numbers a linear or

quadratic equation.

࿌ Use the properties of complex numbers to factorise

a polynomial and to solve a polynomial equation in

the set of complex numbers.

࿌ Apply complex numbers in trigonometry and

alternating current problems.

Introduction 

Until now, you know several sets of numbers, including  . In each case, the numbers correspond to points on the number line. You already know how to find the square root of a positive real number, but the problem occurs when finding square root of negative numbers.

1.1. Concepts of complex numbers

A complex number can be visually represented as a pair of numbers (a ,b) forming a vector from the origin or point on a diagram called Argand diagram (or Argand plane), named after Jean-Robert Argand, representing the complex plane. This plane is also called Gauss plane. The x axis − is called the real axis and is denoted by Re while the y axis − is known as the imaginary axis; denoted Im as illustrated in fig.1.2.

Remark

If b ≠ 0 , from activity 1.11, the sign cannot be taken

arbitrary because the product xy has sign of b . Then,

࿌ If b > 0 , we take the same sign.

࿌ If b < 0 , we interchange signs.

In each case, the complex number has two roots.

1.2.6. Equations in the set of complex numbers

Linear equations

Activity 1.12

Find the value of z in;

1. z i + −= 3 40         2.4 43 −+ = − i iz z i

3. (1 4 + += iiz i )   4. (1 2 − =+ iz i )

An alternative way of defining points in the complex plane, other than using the x and y coordinates, is to use the distance of a point P from the origin together with the angle between the line through P and O and the positive part of the real axis. This idea leads to the polar form of complex numbers. The argument or phase θ (or amplitude) of z is the angle that the radius r makes with the positive real axis, as illustrated in figure 1.5, and is written as arg ( z) .

Notice 

Addition and subtraction in polar form of complex number is not possible directly as it is the case in multiplication and division. For addition and subtraction of complex numbers to be possible, each complex number has to be converted in to Cartesian form first.