In this section, we'll provide a basic introduction to complex numbers. What is the difference between an imaginary number and a complex number? An imaginary number is just i, or a multiple of i (like 5i, or -4i). A complex number is a real number and an imaginary number combined with addition or subtraction. Here are a couple examples:

1 + 2i
3 - 4i
+ i

That last one might look a bit gross to you, but trust me; some day (probably when you're taking Pre-Calculus) you'll look at that and say, "Hey! I recognize those numbers - that's a very cool complex number!" Okay, maybe you won't, but your math teacher probably thinks it's cool, even if you don't.

It's important to note that real numbers and imaginary numbers are not like terms, which means you can't combine them, just like you couldn't combine the two terms in 3 + 2x. So even though i is not a variable (remember, it's the square root of negative one!), in some ways it functions like a variable, and you can manipulate it just like you would manipulate a variable. With the exception that if you multiply it by itself, instead of getting i², it's actually -1!

Complex numbers are usually written in a very specific form, called standard form, or "ay plus bee eye form" (or, "a + bi form"). Math teachers can be real sticklers for writing it in the proper form, which is good, because if everyone writes complex numbers in different ways, it can get very confusing.

Here's an example of how confusing things can get: take a look at the two complex numbers below:

, - + i

Do those two complex numbers look at all alike? No, they don't. And yet, they are exactly the same number! (In a later section, you'll find out how we know that.) But if one scientist writes and another writes - + i, they won't realize they have the same answer, and they'll be stuck while they try to figure out where they went wrong. So we train students early on to always write complex numbers in exactly the same form.

Standard form requires that the real part of the complex number be written first, and the imaginary part be written second. 

Example One
Write -3i + 2 in standard form.

Solution
2 - 3i

Example Two
Write + in standard form

Solution
This might look like it's in standard form already, but standard form technically requires that the fraction be written as a coefficient of i:

+ i  (Not everybody is that much of a stickler for form, so you should check with your teacher to find out if this matters to them!)

Even though real and imaginary terms are not like terms, all imaginary terms are like terms, so they can be combined:

Example Three
Simplify 3 + 5i - 2 - i

Solution
We can combine the real parts: 3 - 2 = 1, and we can combine the imaginary parts: 5i - i = 4i

Now we can write the answer in a + bi form: 1 + 4i

Example Four
Simplify i² + 6i + 3

Solution
Remember that i² = -1, so this is the same as:

-1 + 6i + 3

Combining the real parts gives 2 + 6i

1.

Identify the real part in the complex number 17 - 3i

2.

Identify the imaginary part in the complex number 17 - 3i

3.

Write 5i + 2 in standard form.

4.

Explain what a complex number is.

5.

Simplify the following: 12 +4i - 3i - 2

6.

Simplify the following: 3i³ - 2i +7 +i⁴