Complex Numbers
Imaginary numbers
The use of real positive numbers to describe the magnitude of actual quantities in the world (e.g. money) gives them an immediate significance.
This is a bit more difficult for negative numbers, though the concept of debt, say, makes their meaning clear.
Now we are going to consider a new type of number â€“ imaginary numbers.
Letâ€™s start by considering the solution of an arbitrary quadratic equation:
You have probably seen the case when the discriminant b^{2}  4ac is
positive:

x^{2}  5x + 6 
= 0 
→ 
(x  2)(x  3) 
= 0 
→ 
x = 3 
or 
x 
= 2 
But what happens if the discriminant b^{2}  4ac is negative?
If the discriminant b^{2}  4ac is negative, we have to take the square root of a negative number to solve the equation:
To do this, we use â€œimaginary numbersâ€ by introducing some new notation:
So the solution of the quadratic equation is written in terms of the imaginary number
i:
An imaginary number has the form iy where y is a real number and i^{2} = 1.
Examples of imaginary numbers:
Addition and subtraction:
Multiplication:
the usual rules apply, but we use i^{2} = 1, too:
We use i^{2} = 1 to multiply two different imaginary numbers, so that for a and
b real:
and for multiplication of a real and an imaginary number, we get an imaginary number:
Division:
Division of an imaginary number by a real number:
Division of an imaginary number by another imaginary number:
Division of a real number by an imaginary number:
Useful identitiy:
