Contents
Operations with Complex Numbers
Complex numbers can be added, subtracted, multiplied, and divided from each other using simple mathematical properties, namely the associative, commutative, and distributive ones. It is also important to remember that $ j^2 = -1 $.
Some Examples:
Addition/Subtraction
$ \,\! (a + b*j) + (c + d*j) = (a + c) + (b + d)*j $
$ \,\! (a + b*j) - (c + d*j) = (a - c) + (b - d)*j $
Multiplication/Division
Multiplication and division are a little more complicated than addition and subtraction, mostly because when $ j $ is multiplied by itself, it results in a $ -1 $.
$ \,\! (a + b*j)*(c + d*j) = a*c + a*d*j + b*c*j + b*d*j^2 $
$ \,\! = a*c + (a*d + b*c)*j + (-1)*b*d $
$ \,\! = (a*c - b*d) + (a*d + b*c)j $
When dividing, it is important to know that $ j $ should never be in the denominator
$ \,\! \frac{(a + b*j)}{(c + d*j)} = \frac{(a*c + b*d)}{(c^2 + d^2)} + \frac{(b*c - a*d)}{(c^2 + d^2)}*j $
