MathsTips.com

Maths Help, Free Tutorials And Useful Mathematics Resources

  • Home
  • Algebra
    • Matrices
  • Geometry
  • Trigonometry
  • Calculus
  • Business Maths
  • Arithmetic
  • Statistics
Home » Algebra » More on Complex Numbers

More on Complex Numbers

The earlier chapter on Complex Numbers explained what we mean by a complex number together with a few properties of complex numbers. Basic mathematical operations of complex numbers were introduced. So, the previous chapter may be considered as the preliminary base on which this chapter aims to build further. In this chapter we shall cover a few advanced topics under complex numbers.

 Amplitude(or Argument) of a complex number:

Let z=x+iy where x,y are real, i=\sqrt{-1} and x^2+y^2\neq 0 ; then the value of \theta for which the equations:

x=\lvert z \rvert \cos\theta …(1) and

y=\lvert z \rvert \sin\theta …(2)

are simultaneously satisfied is called the Argument(or Amplitude) of z and is denoted by Arg z (or, \: Ampz) .

Clearly, equations (1) and (2) are satisfied for infinite values of \theta ; any of these values of \theta is the value of Amp z . However, the unique value of \theta lying in the interval -\pi < \theta \leq \pi and satisfying equations (1) and (2) is called the principal value of Arg z and we denote this principal value by arg z or amp z .Unless otherwise mentioned, by argument of a complex number we mean its principal value.

Since, cos(2n\pi+\theta)=cos\theta and sin(2n\pi+\theta)=sin\theta (where n=any integer), it follows that,

Ampz=2n\pi+ampz where -\pi < ampz \leq \pi

Geometrical representation of modulus and amplitude:

graphing-complex-numbers

Let us assume that the point P(x,y) represents the complex number z=x+iy . We draw perpendicular \overline{PM} on \overrightarrow{OX} and join \overline{OP} . If \overline{OP}=r and \angle{XOP}=\theta , then from the right-angled triangle POM we get,

x=r\cos\theta, y=r\sin\theta

\tan\theta=\dfrac{\overline{PM}}{\overline{OM}}=\dfrac{y}{x} \Rightarrow \theta=\tan^{-1}\dfrac{y}{x}

and r^2=OP^2=OM^2+PM^2=x^2+y^2 \Rightarrow r=\overline{OP}=\sqrt{x^2+y^2}

\therefore z+x+iy=r\cos\theta+ir\sin\theta=r(\cos\theta+\sin\theta)

where, r=\sqrt{x^2+y^2}=\lvert z\rvert\

and \theta=\tan^{-1}\dfrac{y}{x}=Argz

This form of representation, z=r(\cos\theta+i\sin\theta) where, r=\lvert z\rvert\ and \theta=Argz of the complex number z , is called the polar form or the modulus-amplitude form of z.

If the point P(x,y) represents the complex number z=x+iy and argz=\theta then:

a) 0< \theta < \dfrac{\pi}{2} when P lies in the first quadrant

b) \dfrac{\pi}{2}< \theta < \pi when P lies in the second quadrant

c) -\pi < \theta < -\dfrac{\pi}{2} when P lies in the third quadrant

d) -\dfrac{\pi}{2} < \theta < 0 when P lies in the fourth quadrant

Note:

1) The horizontal axis or the X -axis is also called the real axis. The vertical axis or the Y -axis is called the imaginary axis.

In particular,

when P lies on the positive real axis, principal value of \theta is 0.

when P lies on the negative real axis, principal value of \theta is \pi .

when P lies on the positive imaginary axis, principal value of \theta is \dfrac{\pi}{2} .

when P lies on the negative imaginary axis, principal value of \theta is -\dfrac{\pi}{2} .

2) The argument of z=0 i.e., z=0+i. i.e., the origin O, is not defined.

3) It must be kept in mind that argz is the principal value of \theta while Argz is just one of the many values of \theta which satisfy the equations,

x=\lvert z \rvert \cos\theta …(1) and

y=\lvert z \rvert \sin\theta …(2)

Example: Find the amplitude of \dfrac{i}{1-i}

Solution: \dfrac{i}{1-i}=\dfrac{i(1+i)}{(1-i)(1+i)}=\dfrac{i+i^2}{1-i^2}=\dfrac{i-1}{2}=-\dfrac{1}{2}+i.\dfrac{1}{2}

Clearly, in the z-plane, the point z=-\dfrac{1}{2}+i.\dfrac{1}{2}=(-\dfrac{1}{2},\dfrac{1}{2}) lies in the second quadrant.

Hence, if ampz=\theta then, \tan\theta= \dfrac{\dfrac{1}{2}}{-\dfrac{1}{2}}=-1 where, \dfrac{\pi}{2} < \theta \leq \pi

\therefore \tan\theta=-1 =\tan(\pi-\dfrac{\pi}{4})=\tan\dfrac{3\pi}{4}

\therefore \theta=\dfrac{3\pi}{4}

\therefore the reqiured amplitude of \dfrac{i}{1-i}=\dfrac{3\pi}{4}

Example: Find the amplitude of \sqrt{12} +6(\dfrac{1-i}{1+i})

Solution: \dfrac{1-i}{1+i}=\dfrac{(1-i)^2}{(1+i)(1-i)}=\dfrac{1+i^2-2i}{1-i^2}=\dfrac{1+(-1)-2i}{1-(-1)}=-\dfrac{2i}{2}=-i

\therefore z=\sqrt{12} +6(\dfrac{1-i}{1+i})=2\sqrt{3} -6i

Clearly, in the z-plane, the point z=2\sqrt{3} -6i=(2\sqrt{3}, -6) lies in the fourth quadrant.

Hence, if ampz=\theta then,

\tan\theta=-\dfrac{6}{2\sqrt{3}}=-\sqrt{3} where, -\dfrac{\pi}{2} < \theta < 0

\therefore \tan\theta=-\sqrt{3}=-\tan\dfrac{\pi}{3}=\tan(-\dfrac{\pi}{3})

\therefore \theta=-\dfrac{\pi}{3} , i.e., the required amplitude of z is (-\dfrac{\pi}{3})

Algebra of Complex Numbers:

Let z_{1}=a+ib and z_{2}=c+id

Now, in the earlier chapter “Complex Numbers” we have shown that:

For addition: z_{1} +z_{2}=(a+c)+i(b+d)=A+iB where:

A=a+c and B=b+d and both are real.

For subtraction: z_{1} -z_{2}=a+ib -(-c-id)=(a-c)+i(b-d)=A+iB where:

A=a-c $ and B=b-d and both are real.

For multiplication: z_{1}z_{2}=(a+ib)(c+id)=ac+iad+ibc+i^2bd=(ac-bd) +i(bc+ad)=A+iB where:

A=ac-bd and B=bc+ad and both are real.

Note: Proceeding similarly, the product of more than two complex numbers can be expressed in the form A+iB

For division: \dfrac{z_{1}}{z_{2}}=\dfrac{a+ib}{c+id}=\dfrac{(ac+bd)-i(ad-bc)}{c^2+d^2}=\dfrac{ac+bd}{c^2+d^2}+\dfrac{bc-ad}{c^2+d^2}=A+iB where:

A=\dfrac{ac+bd}{c^2+d^2} and B=\dfrac{bc-ad}{c^2+d^2} and both are real.

So, it is very clearly that when the four fundamental mathematical operations, viz., addition, subtraction, multiplication and division carried out between two complex numbers, the result is also a complex number of the form A+iB where both A and B are real.

Now, we consider two more algebraic operations:

1) Any integral power of a complex number is a complex number:

Let z=x+iy be a complex number where x, y are real and let n be any integer.

Case I: If n is a positive integer then, z^n=z.z.z....upto \: n \: factors=(x+iy)(x+iy)...upto \: n \: factors=A+iB

[\because  the product of more than two complex numbers can be expressed in the form A+iB ] where, A, B are real.

Case II: If n is a negative integer w eassume n=-m where m is taken to be any positive integer.

Then, z^n=z^{-m}=\dfrac{1}{z^m}=\dfrac{1}{C+iD} where C, D are real.

=\dfrac{C-iD}{(C+iD)(C-iD)}=\dfrac{C}{C^2+D^2}+i(-\dfrac{D}{C^2+D^2})=A+iB

where, A=\dfrac{C}{C^2+D^2} and B=-\dfrac{D}{C^2+D^2}

\therefore any integral power of a complex number is a complex number.

2) Any root of a complex number is a complex number:

Let z=x+iy be a complex number where x \neq 0, y \neq 0 are real and let n be any positive integer.

If the nth root of z be a then,

\sqrt[n]{z}=a \: or, \: \sqrt[n]{x+iy}=a \\    or, \: x+iy=a^n...(1)

By hypothesis, x \neq 0, y \neq 0 Hence, a^n is an imaginary quantity.

Now, a^n can be an imaginary quantity if and only if a is an imaginary number and equation (1) will be satisfies if and only if a is an imaginary number of the form A+iB where A\neq 0, B \neq 0 are real.

\therefore any root of a complex number is a complex number.

Properties of Complex Numbers:

1) If x, y are real and x+iy=o then x=0, y=0

2) If x, y, p, q are real and x+iy=p+iq then x=p and y=q

3) The set of complex numbers satisfies commutative, associative and distributive laws, i.e., if z_{1}, z_{2} \: and \: z_{3} be three complex numbers then,

i) z_{1}+z_{2}=z_{2}+z_{1} [commutative law for addition] and

z_{1}.z_{2}=z_{2}.z_{1} [commutative law for multiplication].

ii) (z_{1}+z_{2})+z_{3}=z_{1}+(z_{2}+z_{3}) [associative law for addition] and

(z_{1}z_{2})z_{3}=z_{1}(z_{2}z_{3}) [associative law for multiplication].

iii) z_{1}(z_2{}+z_{3})=z_{1}z_{2}+z_{1}z_{3} [distributive law]

4) The sum and product two conjugate complex numbers are both real.

Proof: Let z=x+iy be a complex number where x, y are real.

Then, conjugate of z=\overline{z}=x-iy

Now, z+\overline{z}=x+iy+x-iy=2x which is real.

Again, z.\overline{z}=(x+iy)(x-iy)=x^2-y^2 which is real.

Note: If z=x+iy then, \lvert z\rvert\ =\sqrt{x^2+y^2}

Again, z.\overline{z}=x^2+y^2

\therefore \sqrt{z.\overline{z}}=\sqrt{x^2+y^2}

Hence, \lvert z\rvert\ =\sqrt{z.\overline{z}}

\Rightarrow modulus of a complex quantity =the positive square root of the product of the complex quantity and its conjugate.

5) If the sum and product of two complex numbers are both real, then the complex numbers are conjugate to each other.

Proof: Let z_{1}=a+ib and z_{2}=c+id be two complex numbers where a, b, c, d are al real and b \neq 0, d \neq 0

It is given that the sum of z_{1} and z_{2} is real.

\therefore z_{1}+z_{2}=(a+c)+i(b+d) is real.

\Rightarrow b+d=0 \Rightarrow d=-b

Also, z_{1}z_{2}=(ac-bd)+i(ad+bc) is real.

\Rightarrow ad+bc=0 \Rightarrow -ab+bc=0 [\because d=-b ]

\Rightarrow b(c-a)=0 \Rightarrow c=a [\because b \neq 0 ]

\therefore z_{2}=c+id=a+i(-b)=a-ib=\overline{z_{1}} which shows that z_{1} and z_{2} are conjugate to each other.

6) For two complex numbers z_{1} and z_{2} ,

\lvert z_{1}+z_{2}\rvert\ \leq \lvert z_{1}\rvert\ +\lvert z_{2}\rvert\

Proof: Let z_{1}=r_{1}(\cos\theta_{1}+i\sin\theta_{2}) and z_{2}=r_{2}(\cos\theta_{2}+i\sin\theta_{2})

\therefore \lvert z_{1}\rvert =r_{1} and \lvert z_{2}\rvert =r_{2}

Again, z_{1}+z_{2}=r_{1}\cos\theta_{1}+r_{1}i\sin\theta_{1}+r_{2}\cos\theta_{2}+r_{2}i\sin\theta_{2}

=(r_{1}\cos\theta_{1}+r_{2}\cos\theta_{2})+i(r_{1}\sin\theta_{1}+r_{2}\sin\theta_{2})

\therefore \lvert z_{1}+z_{2}\rvert\ =\sqrt{[(r_{1}\cos\theta_{1}+r_{2}\cos\theta_{2})^2+i(r_{1}\sin\theta_{1}+r_{2}\sin\theta_{2})^2]}

=\sqrt{[r_{1}^2(\cos^2\theta_{1}+\sin^2\theta_{1})+r_{2}^2(\cos^2\theta_{2}+\sin^2\theta_{2})+2r_{1}r_{2}(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2})]}

=\sqrt{r_{1}^2+r_{2}^2+2r_{1}r_{2}\cos(\theta_{1}-\theta_{2})}

Now, \lvert \cos(\theta_{1}-\theta_{2})\rvert\ \leq 1

\therefore \lvert z_{1}+z_{2}\rvert\ \leq \sqrt{r_{1}^2+r_{2}^2+2r_{1}r_{2}}

\Rightarrow \lvert z_{1}+z_{2}\rvert\ \leq r_{1}+r_{2}

\Rightarrow \lvert z_{1}+z_{2}\rvert\ \leq \lvert z_{1}\rvert\ +\lvert z_{2}\rvert\

Note:

1) For three complex numbers z_{1}, z_{2}, z_{3} we have,

Proof:

\lvert z_{1}+z_{2}+z_{3}\rvert\ = \lvert (z_{1}+z_{2})+z_{3}\rvert\ \leq \lvert z_{1}+z_{2}\rvert\ +\lvert z_{3}\rvert\

\Rightarrow \lvert z_{1}+z_{2}+z_{3}\rvert\ \leq \lvert z_{1}\rvert\ +\lvert z_{2}\rvert\ +\lvert z_{3}\rvert\

In general, for n complex numbers z_{1}, z_{2},...,z_{n} we have,

\lvert z_{1}+z_{2}+...+z_{n}\rvert\ \leq \lvert z_{1}\rvert\ +\lvert z_{2}\rvert\ +... +\lvert z_{n}\rvert\

2) For two complex numbers z_{1} and z_{2} , we have,

i) \lvert z_{1}-z_{2}\rvert\ \leq \lvert z_{1}\rvert\ +\lvert z_{2}\rvert\

ii) \lvert z_{1}-z_{2}\rvert\ \geq \lvert \lvert z_{1}\rvert\ -\lvert z_{2}\rvert\ \rvert\

Proof:

i) \lvert z_{1}-z_{2}\rvert\ =\lvert z_{1}+(-z_{2})\rvert\ \leq \lvert z_{1}\rvert\ +\lvert -z_{2}\rvert\

\Rightarrow \lvert z_{1}-z_{2}\rvert\ \leq \lvert z_{1}\rvert\ +\lvert z_{2}\rvert\ [Proved]

ii) \lvert z_{1}\rvert\ =\lvert (z_{1}-z_{2})+z_{2}\rvert\ \leq \lvert z_{1}-z_{2}\rvert\ +\lvert z_{2}\rvert\

\Rightarrow \lvert z_{1}-z_{2}\rvert\ \geq \lvert z_{1}\rvert\ -\lvert z_{2}\rvert\ ...(1)

Similarly, \lvert z_{1} -z_{2}\rvert\ \geq \lvert z_{2}\rvert\ -\lvert z_{1}\rvert\ ...(2)

From (1) and (2) we get,

\lvert z_{1} -z_{2}\rvert\ \geq \lvert \lvert z_{1}\rvert\ -\lvert z_{2}\rvert\ \rvert\ [Proved]

Exercise:

1) Express in the form A+iB where A, B are real:

a) \: (\dfrac{1+i}{1-i})^3

b) \: \dfrac{i}{2+i} +\dfrac{3}{1+4i}

2) Find the amplitude of the following complex numbers:

a) \: \dfrac{\sqrt{3}+i}{-1-i\sqrt{3}}

b) \: (1+i)(\sqrt{3}+i)

3) If z_{1}=-3+4i and z_{2}=12-5i , show that,

a) \: \lvert z_{1}+z_{2}\rvert\ < \lvert z_{1}\rvert\ +\lvert z_{2}\rvert\

b) \: \lvert z_{1}z_{2}\rvert\ =\lvert z_{1}\rvert\ \lvert z_{2}\rvert\

c) \: \lvert \dfrac{z_{1}}{z_{2}}\rvert =\dfrac{\lvert z_{1}\rvert}{\lvert z_{2}\rvert}

4) Express (\sqrt{3}-i) in modulus-amplitude form.

5) If \lvert z_{1}+z_{2}\rvert^2 +\lvert z_{1}-z_{2}\rvert^2 =2[\lvert z_{1}\rvert^2 +\lvert z_{2}\rvert^2]

« Combination
Classification of Numbers »


Filed Under: Algebra Tagged With: Amplitude of Complex Number, Argument of Complex Number, Complex Numbers

Comments

  1. Janardhan Reddy says

    March 2, 2020 at 8:32 am

    Good

    Reply

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Table of Content

  • Complex Numbers
  • Quadratic Equations
  • Logarithm
  • Permutation
  • Combination
  • More on Complex Numbers
  • Classification of Numbers
  • Positive and Negative Quantities
  • Understanding Simple Algebraic Formulas With Examples
  • Integers
  • Linear Inequalities
  • An Introduction to Fundamental Algebra
  • Basic Number Properties – Commutative, Associative and Distributive
  • Algebraic Multiplication and Division
  • Simple Equations in One Variable
  • Simple Formulae and their Application
  • Rational and Irrational Numbers
  • Problems Leading to Simple Equations
  • Simultaneous Equations
  • Mathematical Induction
  • Different Type of Sets
  • Indices
  • Framing Formulas
  • Sequences
  • Introduction to Matrices
  • Addition Of Matrices
  • Subtraction Of Matrix
  • Multiplication of Matrices
  • Determinant of Matrices
  • Co-factor of Matrices
  • Minor of Matrices
  • Transpose and Adjoint of Matrices
  • Inverse of a Matrix
  • System of Linear Equations in Matrices
  • Introduction to Polynomials
  • Classification of Polynomials
  • Addition and Subtraction of Polynomials
  • Multiplication of Polynomials
  • Factoring Polynomials
  • Zeroes of Polynomial
  • Remainder Theorem of Polynomials
  • Factor Theorem of Polynomial
  • Simplifying Polynomial Fractions
  • Roots of a Polynomial
  • Addition of Polynomial Fractions
  • Subtraction of Polynomial Fractions
  • Multiplying polynomial fractions
  • Division of Polynomial Fractions

© MathsTips.com 2013 - 2023. All Rights Reserved.