Complex Numbers Shorts Notes and Formula for JEE Main and Advanced

Complex Numbers

Complex numbers are the numbers of the form $a+ ib$ where $i= \sqrt {-1}$  and a and b are real numbers

Algebra for i

$i= \sqrt {-1}$
$i^2= -1$
$i^3 =i \times i^2= -i$
$i^4= i^2 \times i^2 = 1$
$i^{4n} =1$
$i^n + i^{n+1} + i^{n+2} + i^{n+3} =i^n( 1 + i +i^2 + i^3 ) = i^b(1 + i -1 -i)=0$

Addition of Complex Numbers

Let z₁ = x₁ + iy_{1  ­} and z₂ = x₂ + iy₂  z₁ +z₂ = (x₁ + x₂) + i(y₁ + y₂)

Subtraction of Complex Numbers

Let z₁ = x₁ + iy_{1  ­} and z₂ = x₂ + iy₂  z₁ -z₂ = (x₁ – x₂) + i(y₁ – y₂)

Multiplication of Complex numbers

Let z₁ = x₁ + iy_{1  ­} and z₂ = x₂ + iy₂ 

z₁ z₂ = (x₁ + iy₁ ) (x₂ + iy₂  ) =(x₁x₂ – y₁y₂) +i(x₁y₂ +x₂y₁)

Multiplicative Inverse

Let z= a+ ib , Then Multiplicative inverse is given as
$\frac {1}{z}=z^{-1} = \frac {1}{a+ ib} = \frac {a}{a^2 + b^2} – \frac {ib}{a^2 + b^2}$

Division of Complex Numbers

Let z₁ = x₁ + iy_{1  ­} and z₂ = x₂ + iy₂ 
z₁ /z₂ = (x₁ + iy₁ ) /(x₂ + iy₂  ) =z₁ z₂^-1

Conjugate of Complex Number

If z= x+iy then Conjugate is given by $\bar{z} = x-iy$

Properties of Conjugate of Complex Number

If z= x + iy

• $\bar {\left( \bar{z} \right)}= z$
• $ z + \bar{z} = 2 Re(z) = 2x$
• $ z – \bar{z} = 2 Im(z) = 2iy$
• If $ z + \bar{z} = =0$, then $z=-\bar{z} $ and z is purely Imaginary
• $\overline {z_1 + z_2} = \bar{z_1} + \bar{z_2}$
• $\overline {z_1 – z_2} = \bar{z_1} – \bar{z_2}$$
• $\overline {z_1 z_2} = \bar{z_1} \bar{z_2}$
• $\overline {z^n} = \left( \bar{z} \right)^n$

Modulus of complex Number

If z= z + iy, then modulus is defined

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

Properties of Modulus of complex Number

• $|z| \geq 0$
• If |z| = 0 , then z=0
• |z| > 0, then $z \ne 0$
• $ -|z| \leq Re(z) \leq |z|$
• $ -|z| \leq Img(z) \leq |z|$
• $z \bar{z} =|z|^2$
• $|z|=|-z|$
• $|z_1 z_2| = |z_1||z_2|$
• $|\frac {z_1}{z_2}| = \frac {|z_1|}{|z_2|}$
• $|z_1 + z_2| \leq |z_1| + |z_2|$
• $|z_1 + z_2| \geq ||z_1| – |z_2||$
• $ ||z_1| – |z_2|| \leq |z_1 + z_2| \leq |z_1| + |z_2|$
• $|z_1 – z_2| \leq |z_1| – |z_2|$
• $|z^n| = |z|^n$
• $|z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 + (z_1 \bar{z_2} + z_2 \bar{z_1})$
• $|z_1 – z_2|^2 = |z_1|^2 + |z_2|^2 – (z_1 \bar{z_2} + z_2 \bar{z_1})$
• $|z_1 + z_2|^2 + |z_1 – z_2|^2 =2 |z_1|^2 +2 |z_2|^2$
• $|az_1 +b z_2|^2 + |bz_1 -a z_2|^2 =(a^2 + b^2)( |z_1|^2 + |z_2|^2)$

Argand Diagram

Complex number $z=a +ib$ can be represented by a point P (a, b) on a x-y plane. The x -axis represent the real part while the y-axis represent the imaginary part

Important points
(1) The length OP is called the module of z and is denoted by $|z| =\sqrt {x^2 + y^2}$
(2) Purely real number lies on X-axis while purely imaginary number lies on y-axis
(3) The Line OP makes an angle $\theta$ with the positive direction of x-axis in anti-clockwise sense is called the argument or amplitude of z
It is given by
$arg (z) = \tan^{-1} \frac {y}{x}$
The unique value of $\theta$ such that $-\pi < \theta \leq \pi$  is called the principal value of the amplitude or principal argument
(4) The principal argument of the complex number is find using the below steps
Step 1) for z=a+ib , find the acute angle value of $\theta =tan^{?1} \frac {|y|}{|x|}$
Step 2) Look for the values of a ,b
if (a, b) lies in First quadrant then Argument=$\theta$
if (a, b) lies in second quadrant then Argument =$\pi ?\theta$
if (a, b) lies in third quadrant then Argument =??+?
if (a, b) lies in Fourth quadrant then Argument =??

Properties of Argument of Complex Numbers

• $arg (z_1z_2)= arg(z_1) + arg (z_2) + 2k \pi$
• $arg (\frac {z_1}{z_2})= arg(z_1) – arg (z_2) + 2k \pi$
• $arg (\frac {z}{\bar{z}})= 2 arg(z) + 2k \pi$
• $arg (\bar{z})= – arg(z) $
• $arg (z^n)= n arg(z) + 2k\pi $

Where k is a integer such that Arg lies between $-\pi$ and $\pi$