If you are studying this unit, you should already have a basic understanding of radicals, and particularly you should understand square roots. If you see the following: , you should be able to quickly tell what the answer is: it's 3. Why? Because 3 x 3 = 9, and that's what the radical (square root) means. Of course, (-3)(-3) also equals 9, but in mathematics, we've defined the radical operation to be the positive square root of a number. That's why, if you enter in your calculator, it doesn't give you two answers; there is (by definition) only one answer.

You also can probably tell me what is, and . You might need to use your calculator to find = 1.414 (approximately).

But if you put into your calculator, it will most likely give you an error message that looks something like this: "Domain Error." This error message simply means that you're not allowed to put -1 into the square root function. (If you've studied functions, you know that this is because -1 is not in the domain of the square root function, which is why it's called a domain error.)

But what if we could take the square root of -1? There is no real number that, when you multiply it by itself, results in a negative number, but what if there was a number outside the Reals that, when multiplied by itself, gave a result of -1?

Since it's not a member of the Real numbers, we don't have a symbol for it yet. So we need to decide, and agree upon, a symbol to use to represent this not-real number that is the square root of negative one. Rene DesCartes, thinking that this was a pretty silly idea, referred to this non-real number as "imaginary," so we will use the symbol i to represent it.

Thus, we have our fundamental definition of i: i = .

From this, we can develop some more equations fairly easily:

i² = -1
i³ = i²i = -1i = -i
i⁴ = i³i = -ii = -i²= 1
i⁵ = i⁴i = 1i = i
i⁶ = i⁵i = ii = i² = -1
i⁷ = i⁶i = -1i = -i
i⁸ = i⁷i = -ii = ...

Wait a minute...it looks like we're going in circles here! We seem to have a repeating pattern, which we could turn into a rule:

i^r = 1 if r is a multiple of 4
i^r = i if r is one more than a multiple of 4
i^r = -1 if r is two more than a multiple 4
i^r = -i if r is three more than a multiple of 4

Another way of saying this is, for every r, i^r = i^{r - 4}.

We can use these rules to find the value of i to any integer exponent:

i²⁰ = 1, because 20 is a multiple of 4
i¹³ = i, because 13 is one more than a multiple of 4
i⁹⁹ = -i because 99 is three more than a multiple of 4

Having this little "imaginary number" allows us to figure out some other things that we couldn't figure out before. For example, what's ?

Well, = •  = 5i

We don't necessarily have a good sense of what 5i means, but at least we have a way of writing , and that's a good start. 5i is also called an imaginary number.

It might seem strange to be talking about "imaginary numbers," and you might be thinking, "Why are we studying something that doesn't even exist?" If you feel that way, you're in good company with some historical mathematicians like Rene DesCartes, who didn't think there was much point to these strange numbers. However, despite the fact that they are called imaginary, there are actually some real-world applications for these numbers. 

Some of these applications are in the fields of electronics and quantum mechanics (no wonder DesCartes was skeptical about them; these fields weren't exactly popular in his day!). And since most students don't hit these topics in high school, you may get through high school without ever seeing a practical application for imaginary numbers. But trust me; they exist! And if you work through this unit, you'll have the rudimentary understanding you'll need someday if you study in these fields.

1.

Simplify .

2.

Simplify (Give an answer rounded to three decimal places)

3.

Simplify i¹¹

4.

Simplify i²⁰

5.

Simplify i⁷i⁹

6.

Simplify i²⁴ + i²⁶