Advanced Level Pure Mathematics

Complex Numbers

Algebra

Chapter 10Complex Numbers

10.1Introduction2

10.2Geometrical Representation of a Complex Number4

10.3Polar Form of a Complex Number4

10.4Complex conjugate8

10.5Geometrical Applications11

10.6Transformation19

10.7DeMoivre's Theorem and nth Roots of a Complex Number 22

10.1Introduction

AFundamental Concepts

(1)A complex number is a number of the form where a, b are real numbers and .

(2)The set C of all complex numbers is defined by

C

wherea is called the real part of z and and

b is called the imaginary partof z and .

(3) is said to be purely imaginary if and only if and .

(4)When , the complex number z is real.

N.B. .

ExampleSolve in terms of .

Solution

BOperations On Complex Numbers

Let and . Then

(1)

(2)

(3)=

(4)==, where .

N.B.(i) ;

(ii).

ExampleIf and , find

(a)(b)(c)(d)

Solution

ExampleExpress the following in the form of , where are real numbers.

(a) where

(b)

(c)

Solution

ExampleFind the square roots of the complex number and .

Solution

10.2Geometrical Representation of a Complex Number

From the definition of complex numbers, a complex number is defined by the two real numbers and . Hence, if we consider the real part as the in the rectangular coordinates system and the imaginary part as the then the complex number can be represented by the point on the plane. This plane is called the complex plane or the Argand diagram. On this plane, real numbers are represented by points on which is called the real axis; imaginary numbers are represented by points on the which is called the imaginary axis. The number is represented by the origin O.

Any point on this plane can be used to represent a complex number .

For example, as shown in Figure, the represents respectively the complex numbers.

10.3Polar Form of a Complex Number

A.Polar Form

A complex number can be represented by a vector as shown in Figure.

The length of the vector , , is called the modulus of the complex number , and it is denoted by . The angle between the vector and the positive real axis is defined to be the argument or amplitude of and is denoted by or amp.

is infinitely many-valued, that is,

, .

If lies in the interval , we call this value the principal value.

Theorem

ExampleExpress the complex number and in polar form.

Solution

B.Use of Polar Form in Multiplication and Division

TheoremLet ,

(1),.

(2),

proof

ExampleLet and .

(a)By expressing and in polar form, find and .

(b)Find the modulus and the principal value of the argument of

Solution

ExampleProve that if and , then is purely imaginary.

Solution

ExampleShow that and are both real , then either and are both real or .

Solution

10.4Complex conjugate

A.Complex Conjugate

DefinitionLet , where . The complex conjugate of , denoted by is defined as

TheoremProperties of Complex Conjugate

Let be a complex number. Then

(1) is real if and only if .

(2)(3)

(4)(5)

(6)(7)

proof

TheoremProperties of Complex Conjugate(Continued)

Let and be two complex numbers. Then

(1)

(2)

(3)

ExampleProve that, for any complex numbers and .

Solution

ExampleLet . Find .

Solution

ExampleProve that if and , then is purely imaginary.

Solution

ExampleLet and be complex numbers with modulus equal to 1.

Show that if , then .

Solution

B.Roots of Polynomials with Real Coefficients Occurs in Conjugate Pairs

Let () be a polynomial with real coefficients and degree(). If () is a root of this polynomial, then is also a root.

10.5Geometrical Applications

Vectors / Complex Numbers
Addition /
= /
=
Subtraction /
= /
=
Scalar multiplication / /

In the set C of all complex numbers, if is regarded as a vector ; then as far as the above three operations are concerned complex numbers behave similar to those of vectors.

Geometrical Meaning of the Difference of Two complex Numbers

Suppose the complex numbers representing the points and on the Argand diagram be and representing the points and on the Argand diagram respectively.

(1)The complex number represents the vector ;

(2)The modulus represents the length of ;

(3) represents the angle between the vector and the positive x-axis.

Usually, if the point on the Argand diagram is represented by the complex number , we use to denote it. Therefore, for any four points , , and on the Argand diagram, the angle between the vectors and , as shown in Figure, is given by

TheoremAngle between Two Line Segments

Let and be four points on the Argand diagram. If is the angle between the line segments and , then

is considered to be positive if it is obtained by rotating anti-clockwise the vector representing the denominator to the vector representing the numerator.

Collinear

TheoremLet be three distinct points in the Argand diagram representing respectively the complex numbers . Then are collinear if and only if ,

Where is a non-zero real number.

Proof

Equation of a Circle

Let be a point in the complex plane and the complex number corresponding to it be . The equation of

the circle with as centre and radius is given by

where is the complex number corresponding to any point on the circle.

This equation is then rewritten as

This equation is in the form , where is a real constant.

ExampleFind the centre and radius of the circle with equation in the complex plane.

Solution

TheoremGiven that and are three points on the Argand diagram. Then

(1)

(2)The three points are collinear if and only if is real.

(3) and are mutually perpendicular if and only if is purely imaginary.

ExampleLet be a non-zero complex number and . If the points and are respectively represented by the complex numbers and , show that is an equilateral triangle.

Solution

ExampleLet and be two non-zero complex numbers. Prove that if , then , where is a non-negative integer.

Solution

ExampleSuppose the vertices and of an equilateral triangle represent the complex numbers and respectively.

(a)Show that .

(b)If and are the roots of the equation show that

Solution

TheoremMore Properties on Moduli

Let and be complex numbers. Then

(1)

(2)(Triangle Inequality)

Corollary

ProofThis property can be proved by using mathematical induction on .

ExampleLet and be complex numbers.

Prove that

Solution

Loci

When a variable complex number has to satisfy some specific conditions, there is a set of points in the Argand diagram representing all the possible values of . The graph of this set of points is called the locus of the complex number .

ExampleInterpret the following loci in the Argand diagram.

(a)

(b)

(c)

Solution

ExampleIf is pure imaginary, interpret the locus of in the Argand diagram.

Solution

ExampleLet be a complex constant and a real constant.

Show that the equation represents a straight line.

Solution

10.6Transformation

Translation or Displacement

DefinitionLet be a fixed complex number. The function is called a translation.

ExampleGiven a translation defined by

(a)Plot the point .

(b)Sketch the image of the set under .

Solution

Enlargement

DefinitionLet be a fixed real number, the function is called an enlargement.

ExampleGiven two enlargements defined by and .

(a)Plot the point .

(b)Sketch the image of the (i)ellipse in Figure A under .

(ii)triangle (T) in Figure B under .

Solution

Rotation

DefinitionLet be a fixed real number. The function is called a rotation and is the angle of rotation.

ExampleGiven a rotation

(a)Plot the point .

(b)Sketch the image of region in figure under .

Solution

ExampleLet . If the locus of on the is a unit circle centred at the origin, i.e. , show that the locus of the points represented by on the is an ellipse.

Solution

ExampleThe complex numbers and are represented by points and in an Argand diagram. If and describes the line , prove that describes a circle whose centre is at the origin.

Solution

10.7DeMoivre's Theorem and nth Roots of a Complex Number

For any real number ,

In particular, if , we have .

For any positive integer , by induction on , the result may be generalized as

and this is known as the DeMoivre's Theorem for integral index.

For any negative integer , we may let . Then

Hence, also holds for negative integers .

For any rational number . Put , where and no loss of generality if is taken as to be

positive. Then

In general, for any real number , positive number and rational number , we have

The nth roots of a complex number are the values of which satisfy the equation . If we

write and assuming that the equation is satisfied by , then

By equating the real parts and imaginary parts on both sides, we have

where .

For , since

We obtain the distinct complex roots for with the values of obtained in .

For or , the root obtained is equal to one of the roots mentioned above. Hence, the equation

has only distinct complex roots.

TheoremDeMoivre's Theorem for Rational Index

Let be a positive integer and be a real number. Then

, where

Using the DeMoivre's Theorem, we will have the following properties.

(i)

(ii)

(iii)

(iv)

Application of DeMoivre's Theorem to Trigonometry

ADirect application of DeMoivre's Theorem and the binomial theorem, we are able to express

(i)multiple angles such as and in terms of and , and

(ii)powers of and back again into multiple angles.

ExampleVerify that

< Express in terms of powers of and

Solution

Example(a)Show that

(b)Prove .

For what values of that the result is not true?

Solution

ExampleProve that

Hence show that the roots of the equation are, and , and deduce that

Solution

BIf , we have

;

As and

ExampleExpress and in terms of functions of multiple angles.

Solution

ExampleProve

Solution

Example(a)Prove that .

(b)Prove that .

Solution

ExampleBy expanding show that

Solution

Example(a)Show that .

(b)Using (a), or otherwise, solve for values of between

0 and . Hence find the value of .

Solution

The nth roots of a Complex Number

If , then , .

ExampleFind the three cube roots of and locate them in the complex plane.

Solution

ExampleFind the fifth roots of .

Also, interpret the result in the Argand diagram.

Solution

nth of Unity

Theoremnth of Unity of Their Properties

Let be a positive integer. Then the equation has distinct roots given by

These roots are called the roots of unity. If we denote one of them by , then we have

Proof

ExampleLet be a positive integer and , find the values of

(a),

(b),

(c)

(d) where .

Solution

Solution of Equations

ExampleSolve the equation When show that the roots occur in

Conjugate pairs.

Solution

ExampleLet be a positive integer. By solving the equation , show that

Hence deduce that

and

Solution

Prepared by K. F. Ngai

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