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.