Complex numbers

The need for complex numbers:

Consider the quadratic equation ax2+bx+c=0.

The two roots of this equation will be obtained from the formula

If b²-4ac > 0, then the roots are real and distinct.

If b²-4ac = 0, then the roots are real and equal.

If b²-4ac < 0, then the roots are not real.

For the equation x²-4=0 the solutions are

x²=4

x= ± 2

Consider the equation x²+ 4=0

In this case x²=-4

x= ± √−4

x= ± 2√−1

This leads to a set numbers known as complex numbers.

The complex number is a number z of the form z= a+ib where a and b are real numbers, and i is the another number such that i²=-1

In the above example the value of x is ±2i

In z = a+ib, a is called the real part of z, and b is called the imaginary part of z.

Examples:

3+2i

6-5i

-7=-7+0i

2i=0+2i

Four operations:

z₁=a+ib and z₂=c+id

Then z₁+z₂= (a+c)+ i(b+d)

z_1-z₂= (a-c)+ i(b-d)

z₁z₂= (a+ib)(c+id)=ac + iad + ibc + i²bd = ac + i(ad + bc) – bd   , (because i²=-1)

                = (ac - bd) + i(ad + bc)

Here c-id is the complex conjugate of c+id.

If z is a complex number, its conjugate is  

Properties:

• z z‾ = (a+ib) (a-ib)=a²+b²
• 
• If z is real then z=z‾. Conversely if z= z‾ , z is real. i.e. a+ib = a-ib then b=0
• z+z‾ = 2a Therefore, a= Re(z)=
  Im(z) =
• 
• 
• 
• ,where z₂ ≠ 0
• 
• Let z = a+ib. Then the absolute value or modulus value of z denoted by |z|=
• |z1+z2| ≤ z1+z2 which is known as triangle inequality

Geometrical Representation

Let z=a+ib

|z|=

Polar form of a complex number:

Let (r,θ) be the polar coordinates of the point z.

Z= a+ib=r( cos θ+isin θ)=r

where  r = is called modulus or absolute value of z and  θ= tan^-1 is called amplitude or argument of z.

General rule for determining the argument θ:

Let z= a+ ib

θ= ∏-α ; 

θ= α ;

θ = -∏+α ;

θ= -α

Take      α= tan^-1