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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 – imaginary numbers.

Let’s start by considering the solution of an arbitrary quadratic equation:

You have probably seen the case when the discriminant b2 - 4ac is positive:

  x2 - 5x + 6 = 0
(x - 2)(x - 3) = 0
x = 3 or


= 2

But what happens if the discriminant b2 - 4ac is negative?

If the discriminant b2 - 4ac is negative, we have to take the square root of a negative number to solve the equation:

To do this, we use “imaginary numbers” 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 i2 = -1.

Examples of imaginary numbers:

Addition and subtraction:


the usual rules apply, but we use i2 = -1, too:

We use i2 = -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 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:

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