Unit 1: Complex Numbers
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.
