Class 11: Complex Numbers – Lecture Notes – ICSE / ISC / CBSE Mathematics Portal for K12 Students

Euler was the first mathematician to introduce the symbol $i$ (iota) for $\sqrt{- 1}$ i.e. a solution of $x^2 + 1 = 0$ with the property $i^2 = -1$ . He also called this symbol as the imaginary unit.

There are two basic ways to represent a complex number $z$ algebraically:

Cartesian form: $\boldsymbol{ z = x + iy }$

polar form: $\boldsymbol{z = r e^{i \theta} = r ( \cos \theta + i \sin \theta) }$ with $\boldsymbol{r \geq 0}$

Notice that, in the polar form $\boldsymbol{= re^{i \theta}}$ , we use the identity

$\boldsymbol{e^{i \theta} = \cos \theta + i \sin \theta}$ .

Here we consider $\cos \theta + i \sin \theta$ as the definition of the expression $e^{i \theta}$ . Anyway, there is a name for this identity: Euler’s formula. One of the most amazing things in complex numbers is the following so–called addition formula:

$\boldsymbol{e^{i (\alpha + \beta)} = e^{i \alpha} e^{i \beta}}$

Using Euler’s formula, we can rewrite it as

$\boldsymbol{\cos (\alpha + \beta) + i \sin(\alpha + \beta) = ( \cos \alpha + i \sin \alpha)( \cos \beta + i \sin \beta)}$

Multiplying out the right hand side and comparing the real and imaginary parts of both sides, we have arrived at

$\boldsymbol{\cos ( \alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta }$

$\boldsymbol{\sin ( \alpha + \beta) = \cos \alpha \sin \beta + \sin \alpha \cos \beta }$

which are well-known (but by no means obvious) identities in trigonometry

Geometrically, $z$ is represented as a vector with $(x, y)$ as its coordinates in a Cartesian plane, called complex plane.

The magnitude of this vector is called the absolute value or the modulus of $z$ and is denoted by $|z|$ . It is equal to r given in its polar form:

$\boldsymbol{|z| = r = \sqrt{x^2+y^2}}$

The angle between this vector and the $x-$ axis is given by $\theta$ .

When a complex number is represented in Cartesian form, say $z = x + iy$ , the real number $x$ is called the real part of $z$ and the real number $y$ is called the imaginary part of $z$ . We write

$\boldsymbol{x = Re(z)}$ , $\boldsymbol{y = Im (z)}$

Purely real and purely imaginary complex numbers: A complex number $z$ is purely real if its imaginary part is zero i.e. $Im (z) = 0$ and purely imaginary if its real part is zero i.e. $Re (z) = 0$ .

Set of complex numbers : The set of all complex numbers is denoted by $C$ i.e. $\boldsymbol{C= \{ a+ib : a,b \in R \}}$

Since a real number $'a'$ can be written as $a+0i$ . Therefore, every real number is a complex number. Hence, $R \subset C$ , where $R$ is the set of all real numbers.

Two complex numbers are equal if and only if both of their real parts and their imaginary parts are equal.

Thus For $z = x + iy$ and $z' = x' + iy'$ , we have $z = z' \leftrightarrow x = x'$ and $y = y'$

Using the Cartesian form, addition and multiplication of complex numbers are straightforward, as long as we keep in mind that $i^2 = -1$ :

$\boldsymbol{(a + bi) + (c + di) = (a + c) + (b + d)i }$

$\boldsymbol{(a + bi)(c + di) = (ac - bd) + (ad + bc)i }$

The usual algebraic identities still hold for complex numbers, such as

$(z + w)(z - w) = z^2 - w^2$ and $(z + w)^2 = z^2 + 2zw + w^2$

The relation between the Cartesian form and the polar form is given by

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

If we write $P$ for the point in the plane with Cartesian coordinates $(x, y)$ , then $r$ is the length of the line segment $\overline{OP}$ and $\theta$ is the angle between the line $OP$ and the $x-$ axis. The angle $\theta$ in the polar form $z = re^{i \theta}$ is called the argument of $z$ .

Every complex number $z$ has a “twin sister” $\overline{z}$ , called the complex conjugate of $z$ . The twins $z$ and $\overline{z}$ do not quite have exactly the same look. They are more like mirror images to each other. We put the pair $z$ and $\overline{z}$ together in the Cartesian form and the polar form as follows:

$\boldsymbol{z = x + iy }$ $\boldsymbol{\overline{z} = x - iy }$

$\boldsymbol{z = re^{i \theta} = r( \cos \theta + i \sin \theta) }$

$\boldsymbol{\overline{z} = re^{-i \theta} = r ( \cos (- \theta) + i \sin (-\theta)) }$

It is clear from the above identities that the complex conjugate of $\overline{z}$ is $z$ . Also, $z$ is a real number if and only if $z = \overline{z}$ .

One of the most useful identities about complex numbers is the following:

$\boldsymbol{|z|^2 = z \overline{z} }$

The proof of this is simple: writing $z = x + iy$ , we have

$z \overline{z} =( x +iy)(x-iy) = x^2 - i^2 y^2 = x^2 + y^2 = |z|^2$

Integral Powers of $i$

To find the value of in for $n \in Z$ , we may follow the following steps:

If $n=0$ , then write $i^n = 1$

$i^n = \begin{cases} i, & \text{if} \ n = 1 \\ -1, & \text{if} \ n = 2 \\ -i, & \text{if} \ n = 3 \\ 1, & \text{if} \ n = 4 \\ i^r, & \text{if} \ n > 4 \ \text{where} \ r \ \text{ is the remainder when n is divided by } 4 \end{cases}$

Imaginary Quantities

The square root of a negative real number is called an imaginary quantity or an imaginary number.

Theorem: If $a, b$ are positive real numbers, then $\sqrt{-a} \times \sqrt{-b} = - \sqrt{ab}$

Proof:

LHS $= \sqrt{-a} \times \sqrt{-b} = \sqrt{-1} \times \sqrt{a} \times \sqrt{-1} \times \sqrt{b} = -1 \times \sqrt{ab} =$ RHS. Hence proved.

Note: For any two real numbers $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ it true only when at least one of $a$ and $b$ is either positive or zero. In other words $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ is not valid if $a$ and $b$ both are negative.

Note: For any positive real number $a$ , we have $\sqrt{-a} = i \sqrt{a}$

Equality of Complex Numbers

Two complex numbers $z_1= a_1 + ib_1$ and $z_2 = a_2 + ib_2$ are equal if $a_1=a_2$ , and $b_1 = b_2$ i.e. $Re (z_1) =Re (z_2)$ and $Im (z_1) =Im (z_2)$ .

Thus, $z_1= z_2 \Leftrightarrow Re (z_1) =Re (z_2)$ and $Im (z_1) =Im (z_2)$

Addition of Complex Numbers

Let $z_1 = a_1 + ib_1$ and $z_2 = a_2 + ib_2$ be two complex numbers. Then their sum $z_1 + z_2$ is defined as the complex number $(a_1 + a_2) + i(b_1 + b_2)$

It follows from this definition that the sum $z_1 + z_2$ is a complex number such that $Re (z_1 + z_2) = Re(z_1) + Re (z_2)$ and, $Im (z_1 + z_2) =Im (z_1) +Im (z_2)$

Properties of addition of complex numbers

i) Addition is Commutative: For any two complex numbers, $z_1 \ \& \ z_2$

$\boldsymbol{ z_1+z_2 = z_2+z_1 }$

ii) Addition is Associative: For any three complex numbers $z_1, z_2 \ \& \ z_3$ ,

$\boldsymbol{ (z_1+z_2)+z_3 = z_1+(z_2+z_3) }$

iii) Existence of Additive Identity: The complex number $0 = 0 + i 0$ is the identity element for addition: i.e. $\boldsymbol{z + 0 =z = 0 + z}$ for all $z \in C$ .

iv) Existence of Additive Inverse: For any complex number $z = a + ib$ , there exists $-z =(- a) + i(-b)$ such that $\boldsymbol{z + (-z) =0 = (-z) + z}$ . The complex number $-z$ is called the additive inverse of $z$ .

Subtraction of Complex Numbers

Let $z_1 = a_1 + ib_1$ and $z_2 = a_2 + ib_2$ be two complex numbers. Then the subtraction of $z_2$ from $z_1$ is denoted by $z_1 - z_2$ and is defined as the addition of $z_1$ and $- z_2$ . Thus,

$\boldsymbol{z_1 -z_2 = z_1 + ( -z_2) = (a_1-a_2) + i( b_1-b_2)}$

Multiplication of Complex Numbers

Let $z_1 = a_1 + ib_1$ and $z_2 = a_2 + ib_2$ be two complex numbers. Then the multiplication of $z_1$ with $z _2$ is denoted $z_1z_2$ and is defined as the complex number

$\boldsymbol{z_1z_2 = (a_1a_2-b_1b_2)+ i ( a_1b_2+a_2b_1)}$

Properties of Multiplication:

i) Multiplication is commutative: For any two complex numbers $z_1$ and $\boldsymbol{ z_2, z_1z_2 = z_2z_1}$ for all $z_1,z_2 \in C$ .

ii) Multiplication is associative: For any three complex numbers $z_1,z_2, z_3$ , then $\boldsymbol{(z_1 z_2) z_3 = z_1(z_2 z_3)}$ for all $z_1,z_2, z_3 \in C$

iii) Existence of identity element for multiplication: The complex number $1 = 1 + i0$ is the identity element for multiplication i.e. for every complex number $z, \ \ \boldsymbol{z \cdot 1 = z = 1 \cdot z }$ .

iv) Existence of multiplicative inverse: Corresponding to every non-zero complex number $z = a + ib$ , there exists a complex number $z_1= x + iy$ such that $\boldsymbol{z \cdot z_1 = 1 = z_1 \cdot z}$ .

The multiplicative inverse of $z$ is denoted by $z^{-1}$ or $\displaystyle \frac{1}{z}$

v) Multiplication of complex numbers is distributive over addition of complex numbers : For any three complex numbers $z_1, z_2$ and $z_3$ .

$z_1(z_2+z_3) = z_1 z_2 + z_1 z_3 \hspace{2.0cm} \text{(Left distributivity)}$
$(z_2+z_3) z_1 = z_2z_1+z_3z_1 \hspace{2.0cm} \text{(Right distributivity)}$

Division of Complex Numbers

The division of a complex number $z_1$ by a non-zero complex number $z_2$ is defined as the multiplication of $z_1$ by the multiplicative inverse of $z_2$ and is denoted by $\displaystyle \frac{z_1}{z_2}$ .

Let $z_1 = a_1 + ib_1$ and $z_2 = a_2 + ib_2$ be two complex numbers.

Therefore $\displaystyle \frac{z_1}{z_2} = \Big( \frac{a_1a_2+b_1b_2}{{a_2}^2+ {b_2}^2} \Big) + i \Big( \frac{a_2b_1-a_1b_2}{{a_2}^2+ {b_2}^2} \Big)$

Properties of Conjugate

If $z_1, z_2$ and $z_3$ are complex numbers, then

i) $\overline {( \overline{z}) } = z$

ii) $z + \overline{z} = 2 Re(z)$

iii) $z - \overline{z} = 2 Im(z)$

iv) $z = \overline{z} \Leftrightarrow z$ is purely real

v) $z + \overline{z} = 0 \Rightarrow z$ is purely imaginary

vi) $z \overline{z} = \{ Re(z) \}^2 + \{ Im(z) \}^2$

vii) $\overline{ z_1+z_2} = \overline{z_1} + \overline{z_2}$

viii) $\overline{ z_1-z_2} = \overline{z_1} - \overline{z_2}$

ix) $\overline{ z_1\cdot z_2} = \overline{z_1} \cdot \overline{z_2}$

x) $\displaystyle \overline{ \Big( \frac{z_1}{z_2} \Big) } = \frac{\overline{z_1}}{\overline{z_2}} , z_2 \neq 0$

Modulus of Complex Number

The modulus of a complex number $z = a + ib$ is denoted $|z|$ and is defined as

$|z| = \sqrt{a^2+b^2}= \sqrt{\{ Re(z) \}^2 + \{ Im(z) \}^2}$

Clearly $|z| > 0$ for all $z \in C$ .

Properties of modulus:

i) $|z| = 0 \Leftrightarrow z=0 \ i.e. \ Re(z) = Im (z) = 0$

ii) $|z| = |\overline{z}| = |-z|$

iii) $- |z| \leq Re(z) \leq |z| ; - |z| \leq Im(z) \leq |z|$

iv) $z \overline{z} = {|z|}^2$

v) $|z_1z_2| = |z_1| |z_2|$

vi) $\displaystyle \Big| \frac{z_1}{z_2} \Big| = \frac{|z_1|}{|z_2|} ; z_2 \neq 0$

vii) $|z_1+z_2|^2= |z_1|^2+|z_2|^2+ 2 Re(z_1 \overline{z_2})$

viii) $|z_1-z_2|^2= |z_1|^2+|z_2|^2 - 2 Re(z_1 \overline{z_2})$

ix) $|z_1+z_2|^2 + |z_1-z_2|^2 =2 \big(|z_1|^2+|z_2|^2 \big)$

x) $|az_1-bz_2|^2 + |bz_1+az_2|^2 =(a^2+b^2) \big(|z_1|^2+|z_2|^2 \big)$ , where $a, b \in R$

Reciprocal of a Complex Number

Let $z = a + ib$ be a non-zero complex number. Then,

$\displaystyle \frac{1}{z} = \frac{1}{a + ib} = \frac{1}{a + ib} \times \frac{a - ib}{a - ib}$

$\displaystyle \Rightarrow \frac{1}{z} = \frac{a-ib}{a^2-i^2b^2} = \frac{a-ib}{a^2+2b^2}$

$\displaystyle \Rightarrow \frac{1}{z} = \frac{a}{a^2+2b^2} +i x \frac{(-b)}{a^2+2b^2}$

Clearly $\displaystyle \frac{1}{z}$ , is equal to the multiplicative inverse of $z$

$\displaystyle \frac{1}{z} = \frac{a-ib}{a^2+2b^2} = \frac{\overline{z}}{|z|^2}$

Thus, the multiplicative inverse of a non-zero complex number z is same as its reciprocal and is given by

$\displaystyle \frac{Re(z)}{|z|^2} + i \frac{-Im(z)}{|z|^2} = \frac{\overline{z}}{|z|^2}$

Square Roots of Complex Numbers

Let $a +ib$ be a complex number such that $\sqrt{a+ib} =x+iy$ where $x$ and $y$ are real numbers. Then,

$\sqrt{a+ib} =x+iy$

$\Rightarrow a+ib = (x+iy)^2$

$\Rightarrow a+ib = (x^2-y^2)+ 2i xy$

On equating real and imaginary parts, we get

$x^2 - y^2 = a$ … … … … … i)

and $2xy = b$ … … … … … ii)

Now, $(x^2+y^2)^2 = ( x^2 - y^2)^2 + 4 x^2 y^2$

$\Rightarrow (x^2+y^2)^2 = a^2 + b^2$

$\Rightarrow (x^2+y^2) = \sqrt{a^2 + b^2}$ … … … … … iii)

Solving the equations (i) and (ii), we get

$\displaystyle x^2 = \frac{1}{2} \Big\{ \sqrt{a^2 + b^2} + a \Big\} \Rightarrow x = \pm \sqrt{\frac{1}{2} \Big\{ \sqrt{a^2 + b^2} + a \Big\}}$

$\displaystyle y^2 = \frac{1}{2} \Big\{ \sqrt{a^2 + b^2} -a \Big\} \Rightarrow x = \pm \sqrt{\frac{1}{2} \Big\{ \sqrt{a^2 + b^2} - a \Big\}}$

If $b$ is positive, then $x$ and $y$ are of the same sign

$\displaystyle \sqrt{a+ib} = \pm \Big[ \sqrt{\frac{1}{2} \Big\{ \sqrt{a^2 + b^2} + a \Big\}} +i \sqrt{\frac{1}{2} \Big\{ \sqrt{a^2 + b^2} - a \Big\}} \Big]$

If $b$ is negative, then $x$ and $y$ are of different sign

$\displaystyle \sqrt{a+ib} = \pm \Big[ \sqrt{\frac{1}{2} \Big\{ \sqrt{a^2 + b^2} + a \Big\}} -i \sqrt{\frac{1}{2} \Big\{ \sqrt{a^2 + b^2} - a \Big\}} \Big]$

Representation of Complex Numbers

A complex number can be represented in the following forms: (i) Geometrical form (ii) Vectorial form (iii) Trigonometrical form or, Polar form.

Geometrical Representation of Complex Numbers

The plane in which we represent a complex number geometrically is known as the complex plane or Argand plane or the Gaussian plane. The point P, plotted on the Argand plane, is called the Argand diagram.

Let us represent $z = x+ i y$ on Argand Plane

If a complex number is purely real, then its imaginary part is zero. Therefore, a purely real number is represented by a point on x-axis. A purely imaginary complex number is represented by a point on y-axis. That is why x-axis is known as the real axis and y-axis, as the imaginary axis.

The length of the line segment $OP$ is called the modulus of $z$ and is denoted by $|z|$ .

$|z| = \sqrt{x^2+y^2} = \sqrt{ \{ Re(z) \}^2 + \{ Im(z) \}^2 }$

The angle $\theta$ which $OP$ makes with positive direction of x-axis in anticlockwise sense is called the argument or amplitude of $z$ and is denoted by arg(z) or amp(z).

$\displaystyle \tan \theta = \frac{PM}{OP} = \frac{y}{x} = \frac{Im(z)}{Re(z)} \Rightarrow \theta = \tan^{-1} \Big( \frac{Im(z)}{Re(z)} \Big)$

Polar or Trigonometrical from of Complex Numbers

Let $z = x + i y$ be a complex number represented by a point $P(x, y )$ in the Argand plane. Then by the geometrical representation of $z = x + i y$ , we obtain

$OP = |z|$ and $\angle POX = \theta = arg(z)$

In $\triangle POM$ , we obtain

$\displaystyle \cos \theta = \frac{OM}{OP} = \frac{x}{|x|} \Rightarrow x = |z| \cos \theta$

$\text{and } \displaystyle \sin \theta = \frac{PM}{OP} = \frac{y}{|y|} \Rightarrow y = |z| \sin \theta$

$\therefore z = x + iy$

$\Rightarrow z = |z| \cos \theta + i |z| \sin \theta$

$\Rightarrow z = |z| ( \cos \theta + \sin \theta)$

$\Rightarrow z = r ( \cos \theta + \sin \theta)$ , where $r = |z|$ and $\theta = arg(z)$

This form of $z$ is called a polar form of $z$ . If we use the general value of the argument of $\theta$ then the polar form of $z$ is given by

$z = r [ \cos (2n \pi + \theta) + i \sin (2n \pi + \theta )]$ , where $r =| z |, \theta = arg (z)$ and $n$ is an integer.

Case 1: Polar form of $z = x + iy$ when $x > 0$ and $y > 0$ :

In this case, we have $\theta = \alpha$ .

So, the polar form of $z = x + i y$ is $z = r ( \cos \alpha + i \sin \alpha)$

Case 2: Polar form of $z = x + iy$ when $x < 0$ and $y > 0$

In this case, we have $\theta = \pi - \alpha$ .

So, the polar form of $z = x + i y$ is $z = r [ \cos (\pi - \alpha)+i \sin (\pi- \alpha)] = r(- \cos \alpha + i \sin \alpha)$

Case 3: Polar form of $z = x + iy$ when $x <0$ and $y <0$

ln this case, we have $\theta = - ( \pi - \alpha)$ .

So, the polar form of $z$ is given by $z = r [ \cos (\pi - \alpha) + i \sin(-( \pi - \alpha ) ) ] = r(- \cos \alpha -i \sin \alpha)$

Case 4: Polar form of $z = x + iy$ when $x > 0$ and $y < 0$

In this case, we have $\theta = - \alpha$ .

So, the polar form of $z$ is $z = r [ \cos (- \alpha) + i \sin (- \alpha)] = r ( \cos \alpha - i \sin \alpha)$