What are Complex Numbers?

We all know that the square of a real number is always non-negative. e.g. square of 5 = ($5^2$) = 25.
However, square root of a number can be positive or negative. $\sqrt{25} = ±5$

What about the square root of a negative number? Does it exist?
Certainly a negative number can not have a real square root. So we need to go beyond the realm of real numbers.
These are imaginary numbers.

Square root of a negative number is an imaginary number.
e.g. $\sqrt{-25}$ = $\sqrt{-1}$ $\sqrt{25}$ = $i5$

The symbol $i$ = $\sqrt{-1}$

A number of the form $x + iy$, where $x$ and $y$ are real numbers and $i = \sqrt{-1}$, is called a complex number

$x$ is called the real part of $x + iy$ and is written as $R(x + iy)$ and
$y$ is called the imaginary part of $x + iy$ and is written as $I(x + iy)$

If the imaginary part of a complex number is zero, the complex number is known as pure real number.
If the real part of a complex number is zero, the complex number is known as pure imaginary number.

Properties of Complex Numbers

1. If $x$₁ + $iy$₁ = $x$₂ + $iy$₂, then $x$₁ - $iy$₁ = $x$₂ - $iy$₂

2. Two complex numbers $x$₁ + $iy$₁ and $x$₂ + $iy$₂ are said to be equal when
      R($x$₁ + $iy$₁) = R($x$₂ + $iy$₂) i.e. $x$₁ = $x$₂ and
      I($x$₁ + $iy$₁) = I($x$₂ + $iy$₂) i.e. $y$₁ = $y$₂

3. Sum, difference, product and quotient of any two complex numbers is itself a complex number.

4. For two complex numbers $z$₁ = $x$₁ + $iy$₁ and $z$₂ = $x$₂ + $iy$₂,
      $z$₁ . $z$₂ = ($x$₁$x$₂ - $y$₁$y$₂) + $i$($x$₁$y$₂ + $x$₂$y$₁)

5. Every complex number $x + iy$ can be expressed in the form $r(cos θ + i sin θ)$ i.e.
      $x = r cos θ$
      $y = r sin θ$


      Let us square and add $x$ and $y$ using above values
      ⇒ $x^2 + y^2 = (r cos θ)^2 + (r sin θ)^2$
      As $sin^2 θ + cos^2 θ = 1$, we get
      ⇒ $x^2 + y^2 = r^2$
      Taking positive square root only, we get $ r = \sqrt(x^2 + y^2)$

      Also, if we divide $y$ by $x$, we get $\dfrac{y}{x} = \dfrac{r sin θ}{r cos θ} = tan θ$
      or θ = tan^-1$(\dfrac{y}{x})$

      Thus, $x + iy$ = $r(cos θ + i sin θ)$
      where $ r = \sqrt(x^2 + y^2)$ and θ = tan^-1$(\dfrac{y}{x})$

Definition:
   • The number $ r = + \sqrt(x^2 + y^2)$ is called the modulus of $x + iy$ and is written as mod($x + iy$) or |$x + iy$|
   • The angle θ = tan^-1$(\dfrac{y}{x})$ is called the amplitude or argument of $x + iy$
   • A pair of complex numbers $x + iy$ and $x - iy$ are said to be conjugate of each other.

6. If we represent a complex number by z and its conjugate by z, then
   zz = ($x + iy$)($x - iy$) = $x^2$ + $y^2$

7. Conjugate of a conjugate is the complex number itself.
    z = z

Geometric representation of imaginary numbers

Complex numbers are represented on the complex plane or z-plane.
Complex plane can be thought of as a modified Cartesian plan by replacing x-axis by real axis and y-axis by imaginary axis.

Refer figure below. All the real numbers are represented by points on X'OX axis and all imaginary numbers are represented on Y'OY axis.
Point P represents complex number $z = (x + iy)$ and point Q represents its conjugate z= $(x - iy)$

Arithmatic operations on complex numbers

To add or subtract two complex numbers, simply add real and imaginary parts of the first number with the real and imaginary parts of the second number respectively.
e.g. The sum of $1 + 2i$ and $2 + 1i$ is $3 + 3i$

This addition can be represented graphically on the complex plane.


In the right hand side figure, the blue line represents complex number $1 + 2i$ and the red line represents complex number $2 + 1i$
Now, if we assume the blue and the red lines as two adjacent sides of a parallelogram and complete the parallelogram, the diagonal represented by green line would represent the summation of two complex numbers i.e. $3 + 3i$.