Like many stories, the story of complex numbers begins with a lie. A lie which you were told many years ago, by your math teachers, no less. (On behalf of all math teachers everywhere, I sincerely apologize.) And the lie was this: you can’t find the square root of a negative number.
Well, OK, it’s not exactly a lie. It just depends on your point of view.
The problem, of course, is that any positive real number multiplied by itself yields a positive real number — but any negative real number multiplied by itself yields a positive real number, too. So there are no real numbers which can be the square root of any negative real numbers.
But here is where, as mathematicians, we decide that we are undaunted by this fact, and that we will invent a new sort of number so that negative real numbers can have square roots after all. (The strange thing is, usually when we “invent” something in this way, it ends up corresponding so nicely with so many other things that we begin to wonder if we really invented it after all…)
So, we define the imaginary number to be the square root of negative one:
Now we can express the square root of any negative number in terms of , since
So, for example,
In general, we call any such multiple of (like ) an imaginary number.
Now then, a complex number is simply a number of the form , that is, a real number plus an imaginary number. For example, or . One neat thing about complex numbers is that they are closed under the four normal arithmetic operations. In other words, if you take any two complex numbers and add, subtract, multiply, or divide them, you get another complex number. Here’s how:
- To add or subtract two complex numbers, just collect like terms; that is, add or subtract the real parts separately and add or subtract the imaginary parts separately. Like this:
- To multiply two complex numbers, use FOIL to multiply them like any two binomials, keeping in mind the fact that since , it is the case that . Like this: