# High School: Number and Quantity

### The Complex Number System HSN-CN.A.1

1. Know there is a complex number i such that i2 = -1, and every complex number as the form a + bi with a and b real.

It's a little-known fact that in Disney World, in the Journey into the Imagination Pavilion, lives a purple dragon named Figment. (No, Figment is not Barney. They're not even related. Figment is a dragon, not a dinosaur, and he doesn't have that annoying voice or theme song.) Figment is quite a rule breaker—he does things that others tell him he simply can't.

Before the ride was rehabbed, there was a wall toward the end of the ride. It pictured all sorts of things imaginary—pigs that flew and three headed cows and the expression "i2= -1."

What on Earth are we talking about? Well, what the Imagineers at Disney remembered from high school is that there is a field of numbers based on something imaginary. We call this field of numbers "complex numbers" (since "imaginary" sounds a tad too mythical) and its most basic unit is the number i. Yes, the number. Not the letter.

What's the big deal about i? Well, the big deal is that i = .

Yeah, we know. Square roots and negative numbers just don't go together. Well that was then, and this is now.

Once your students get past the idea that -1 can have a square root, they can have lots of fun with imaginary numbers. The complex number system is composed of numbers in the form "a + bi," where both a and b are real numbers. (That means we can have numbers like 2 + 5i or 7 – 12i.) Eventually, they can even do all sorts of operations with complex numbers.

We'll take it one step at a time, though.

#### Drills

1. What's the big deal about ? Why does is need to be imaginary?

Because a negative squared is positive and a positive squared is positive

Think about it. There's no "normal" way to find the square root of -1. Using -1 won't work (since -1 × -1 = 1) and using 1 won't work (since 1 × 1 = 1). The only way to make the problem work is to invent a number, i. While all the other choices are true, they don't explain the need for the number i.

2. If i = , what does  equal?

2i

Our first venture into the world of imaginary numbers! We can rewrite  as . The first radical simplifies to 2 and the second to i. Since they're products, we just multiply the two together. So the answer is 2i.

3. What does equal?

We can rewrite  as , and then simplify it. The first radical simplifies to i, although "simplify" might not be the best word to use since we're making things imaginary and all. The second doesn't simplify. So the right answer is (C).

4. What does 4 + 2i equal?

4 + 2i

We should treat imaginary numbers as you would treat any other radical. (Imaginary friends get the same treatment as friends do. Why shouldn't imaginary numbers get the same treatment numbers do?) Unfortunately, they're imaginary, so we'll treat them like they're irrational and only combine like terms.

5. What does i2 equal?

-1

We said that , right? If all we do is square both sides of that equation, we'll have our answer. The square root on the  cancels out when we square it, and we're left with i2 = ­-1. That's (B), and it's kind of the whole idea behind the complex number system.

6. What does 3 + 7 + 8i equal?

10 + 8i

Here, we have two real numbers (3 and 7) and one imaginary number (8i). We can add real numbers, but real numbers and imaginary ones can only take the form a + bi. We can't do more than that. So in this case, we can add 3 and 7, but that 8i is all by itself. A match.com profile might help it be less lonely, though. In any case, our answer is (B).

7. What is ?

7i

If we split up what's under the radical, we can have , which will give us 7 × . Since we know that , we should have no problem arriving at (A) as the correct answer.

8. What is ?

10 + 5i

This question still takes the format a + bi, as it should. In this case, our a is , or 10 and our bi is , or 5i. The two are added together, and since a + bi is the final formatting on complex numbers, the most simplified answer we can give is (C).

9. What does equal?

Before we even look at imaginary numbers, we have two of the exact same number. Instead of writing , we can simply write . Now, let's take a look at what i has to offer us. If we pull out an i from under the radical, we get . Since that's the simplest we can make it, our answer is (D).

10. Are there any uses for imaginary numbers (aside from rides at Disney World)?