2.4 Complex Numbers

You may have learned in previous math classes that there is no taking the square root of a negative number. But your other teachers kind of lied to you about that. In fact, you can take the square root of a negative number. But your answer is going to be some quantity of the imaginary unit i.

In this same vein, the square root of -36 would be 6i.

The standard form of i is a + bi. a is some real number, and bi is just a quantity of i. This is a type of complex number. But be aware that any real number can be a complex number if you assume that b = 0!

This brings us to our next order of business - working with i. In adding and subtracting, you can really just treat i as any old variable. i can be negative.

(5 + 2i) - (6 - 7i) = ?
5 + 2i - 6 + 7i
5 - 6 +2i + 7i
=
-1 + 9i

Multiplication and division is slightly different. Use the same steps until you get to the end. Then look at your i term.

Now, this will be true not only for these four exponents, but for any exponent of i. But because there are only four options, you can figure out which applies to your exponents.
Take your exponent and divide it by 4 (the number of options we have). Go until you have a remainder. This remainder should be either 1, 2, 3, or 4; that number represents which of the original four (illustrated above) applies.

One more thing. Sometimes when you're dividing or just when you're being asked to put a number into standard form, you'll have to deal with an ugly fraction. That's where this magical thing called a conjugate comes in. It's pretty easy to use. Take your denominator and get that into standard a + bi form if it's not already. Then swap the plus sign for a minus sign (or, if you've got a minus sign, swap it for a plus sign). Do NOT make changes to the positivity or negativity of anything else. Put your new standard form in both the numerator and the denominator of a new fraction and multiply across with your original ugly fraction. When your fraction reduces, you'll get your standard form.