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 “
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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
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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
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Some Application of Complex Numbers
- Solution for quadratic equations
- Advanced Calculus
- Signal processing use for wireless transmission
- Electronics
- Electrical