Complex Numbers
A complex number is an expression of the form x + iy where x and y are real numbers.
Either x or y may be equal to zero: real numbers and imaginary numbers are special cases of complex numbers.
Given a complex number z = x + iy the real number x is known as the real part of z and the real number y is known as the imaginary part of z.
The notation Re is used for real part and Im is used for imaginary part.
Note that the imaginary part of z is a real number y (NOT iy).
Adding and Subtracting Complex Numbers
To add or subtract complex numbers, we add or subtract their real and imaginary parts separately:
Examples:
1) (2 + 3i) + (4 + i) = 6 + 4i
2) (3  5i)  7i = 3  12i
3) (5  4i) + (3 + 2i)  (8 + i) = 3i
4) 12  4i + (3 + 4i) = 15
Multiplying complex numbers
To multiply complex numbers, we use the usual rules and the identity i^{2} = 1:
Examples:
1) (2 + 3i)(4 + 5i) 
= (2 Ã— 4) + (3i
Ã— 4) + (2
Ã— 5i) + (3i Ã— 5i) 

= 8 12i + 10i  15 

= 7 + 22i 
2) (2 + 3i)^{3} 
= (2 + 3i)(2 + 3i)(2 + 3i) 

= (5 + 12i)(2 + 3i) 

= 46 + 9i 
The complex conjugate
Given a complex number z = x + iy, the complex conjugate is given by z^{*}
= x  iy (pronounced â€œz starâ€):
To find the complex conjugate of any expression, replace i by â€“i.
Examples:
1) (3 + 4i)^{*} = 3  4i
2) (5  2i)^{*} = 5 + 2i
3) (6i)^{*} = 6i
4) [(2 + 4i)^{*}]^{*} = [2  4i]^{*} = 2 + 4i
The last expression demonstrates that
A complex number and its complex conjugate have the property that:
Examples:
1) (3 + 4i) + (3 + 4i)^{*} = 3 + 4i + 3  4i = 6
2) (3 + 4i)  (3 + 4i)^{*} = 3 + 4i  3 + 4i = 8i
3) 5i + (5i)^{*} = 5i  5i = 0
4) 5i  (5i)^{*} = 10i
Complex conjugate of a product
The modulus of a complex number
A complex number z = x + iy multiplied by its complex conjugate is a real number:
The modulus of a complex number z = x + iy is written
 z  and is equal to the positive square root of the sum of the squares of its real and imaginary parts:
Examples:
The modulus is a real number
The quotient of complex numbers
In order to simplify an expression with a complex number in the denominator, we multiply both the numerator and denominator by the complex conjugate:
In this way, we get a real number in the denominator (the modulus squared of
a + bi).
Examples:
In general:
