We have to expand the numbers system one more time, to include complex numbers. Unfortunately it is not easy to motivate this expansion as compellingly as the previous expansions. Complex numbers have a tremendous range of applications, that will become apparent as you move on in Mathematics, Science, and Engineering, but they may be hard to appreciate at this stage.
For our purposes we start with the observation that there is no real number such that
We now take a leap of faith the kind of which sometimes gives mathematics a bad name. We don't know of a number satisfying the above equation, so we stipulate that there is a solution and we give it a name. We call it .
There are only two things about we need to know and appreciate. One is the fact that
The second fact is that we define the arithmetic operations with such that the ordinary rules of arithmetic continue to hold.
Let's consider some consequences of this fact. We obtain in particular
Let's take stock and introduce some language. The symbol is called the imaginary unit. A complex number is an expression that can be written in the form where and are real numbers (and multiplies ). That particular form is sometimes called the standard form of a complex number. is called the real part of , and is its imaginary part. If the real part of is zero, and the imaginary part non-zero, then is called an imaginary number. (If the imaginary part is zero, then is a real number.) If you want to emphasize that the real part is zero you can call the number pure imaginary or purely imaginary.
Note that the imaginary part of a complex number is in fact a real number (and not, as you might reasonably expect, an imaginary number).
Consider, for example, the complex number . Its real part is , and its imaginary part is . The number is also a complex number. Its standard form is , its real part , and its imaginary part . The number is imaginary, the number is real. Both numbers are complex. In fact, all real numbers and all imaginary numbers are complex. On the other hand, some complex numbers are real, some are imaginary, and some are neither.
(A small aside: The textbook defines a complex number to be imaginary if its imaginary part is non-zero. According to that definition an imaginary number may have a non-zero real part. For example, the number would be imaginary. That definition is non-standard. I expect it's just a mistake. In any case you should not adopt that definition, it is not useful.)
The fact that some numbers are imaginary does not mean that they are inferior or mysterious, or somehow less than real. It just means they are complex numbers whose real part is zero. The word real in this context emerged in the course of history, but its modern meaning in our context is quite different from the meaning of "realistic". As far as a mathematician is concerned, real, imaginary, and complex numbers are all equally "realistic".
Here is one of the principles that make understanding mathematics possible:
You work with complex numbers as you would with ordinary algebraic expressions containing the variable , except that you replace with wherever it occurs.
Let's look at the four basic operations. For the sake of illustration we'll combine the complex numbers and . We'll also give the general formulas but you don't have to memorize them since the principle given in the previous paragraph tells you (almost) everything you need to know.
We now return to the problem of dividing two complex numbers. We are bothered by the complex number in the denominator and we get rid of it by multiplying with its conjugate complex in the numerator and denominator. Hence
In general we have