What is an Complex Number / Imaginary number?

We know that real value of square roots of negative numbers does not exist.
 So we imagine a value “i”  for    -1  so we can easily  evaluate square root of  negative numbers

ie
Square root of a negative number will be a imaginary number or square of a imaginary number is a negative number.

Imaginary numbers are represented by  ” a + i b “




Graphical Representation




Mathematical Operations of imaginary number

 
Mathematical Operations of a imaginary number is almost same as Operations between one variable polynomials
 
If z = a + i b  &   z’ = x + i y

Addition of Complex Numbers

adding real part and imaginary part separately
 
ie,
 z + z’ = ( a + i b ) + ( x + i y ) = ( a + x ) + i ( b + y )
 
Example:
( 3 + 4 i ) + ( 2 + 2 i ) = 5 + 6 i
 

Subtraction of complex number

It is similar to addition, subtract imaginary part and real part separately
ie,
z – z’ = ( a + i b ) – ( x + i y ) = ( a – x ) + i ( b – y )
 
Example:
( 3 + 4 i ) – ( 2 + 2 i ) = 1 + 2 i
 

Multiplication of complex number

z z’ = ( a x – b y ) + i ( a y + b x )
 
Proof
z z’ = ( a + i b )( x + i y )
      = a x + i a y + i b x + i² b y
      = ( a x – b y ) + i ( a y + b x )
Example:
( 3 + 4 i ) ( 2 + 2 i ) = 6 + 6 i + 8 i – 8 = – 2 + 14 i
 
 

Division of complex number

z ÷ z’ = ( a x + b y – i ( a y – b x )) / ( x² + y² )
Proof
z ÷ z’ = ( a + i b ) / ( x + i y ) 
        = (( a + i b )( x – i y )) / (( x + i y )( x – i y ))
        = ( a x – i a y + i b x + b y ) / ( x² + y² )
        = ( a x + b y – i ( a y  – b x )) / ( x² + y² )
 
Example:
( 3 + 4 i )/( 2 + 2 i) = (( 3 + 4 i )( 2 – 2 i )) / (( 2 + 2 i )( 2 – 2 i ))
                        = ( 6 – 6 i + 8 i+8) / 8
                        = ( 14 + 2 i ) / 8
                        = ( 7 + i ) / 4
 

Representation Of a Complex Number

The polar form of a complex Number is another way to represent a complex number. commonly we are using Rectangular form

Rectangular Coordinate Representation of complex number

Commonly we are using rectangular form.
 
z = a + i b  is rectangular coordinate form
Rectangular Coordinate Form of a Complex Number

Polar Coodinate Representation of complex number

 z =  r (cos Ѳ + i sin Ѳ) is polar Form
Where r² = a²+b²  , sin Ѳ = b/r, cos Ѳ = a/r
Polar Representation Of Complex Number

Some Application of Complex Numbers

  • Solution for quadratic equations
  • Advanced Calculus
  • Signal processing use for wireless transmission
  • Electronics
  • Electrical