In this Trigonomtery and PreCalculus section, we will be exploring complex and imaginary numbers. Complex numbers are made up of a real number part and an imaginary number part. 

Video Transcription

My goal on this course is to teach you everything you need to know so that when you set foot into calculus one day that you will be prepared and need to know all the things you need to do well in that class. Now, up to this point, the only assumption that I have made is that you a little bit of algebra and a few algebra videos on the market that are excellent primers for everything that you need to know with regards to algebra. And what you will find is typically what happens is you will go through algebra in school and then one day, you will take calculus, and there is this gap of knowledge that needs to happen in there and I am going to cover that gap here. Usually, that class is covered by a course in either precalculus or trigonometry; either course really will get you I think pretty much ready for calculus. My goal is to teach you the content of that course so that you are ready to go when you get into class. So let us begin. We are going to start talking about something called complex numbers. So I am going to write that on the board. Complex numbers give people a hard time a lot of times. And I think that if I had to pick one name in math that someone is made up for some concepts and it totally confused people for centuries afterwards, it will be this one. Forget about the name. Complex has nothing to do with it. Another word for this thing that you will hear that we will talk about is imaginary numbers that is almost even worst. Imaginary makes you think that it does not even exists, it is totally useless. I know you may have heard these terms before. My goal is to you what they are. Recall for a minute the following. Recall that if we had something like x square is equal to five, and when I say recall, I just mean think back the algebra. How would you solve an equation like that? Well, in order to get rid of that square term, what you need to do is take a square root of both sides. So what you will have is square root of x squared equals square root of five, right. But notice that every time you do something like this, you will need to put a plus or minus in front of your square root. And that is just something that you should just pick up from algebra. So what you will have is x is equal to plus or minus the square root of five. Or maybe, a different example might be if you have x squared is equal to 25, x would be plus or minus five, okay. That means that there are two solutions to this equation. One of them is positive five; one of them is negative five. The reason is, you can take positive five put it in here. Positive five times positive five will give you 25. You can also take negative five and you can put it in there, and negative five times negative five again will give you positive 25. All of this has really not much to do with complex numbers but it is a good intro for what we are going to need, okay. So, let us say that we are working with the problem kind of like this; x squared is equal to 25, okay. What is you had x squared is equal to negative 25. How would you solve the problem like that? Well, you might be tempted to put a square root over this and a square root over this, and just kind of leave it like that circle on your paper. Well, that is not going to cut it. That is not correct. Let us just see what would happen if we just say that x is equal to a plus or minus five here. Let us just pretend that we just assume this was the answer. We already said that five times five is positive 25 so, positive five is not a solution to this at all. Because we are looking for x squared is equal to negative 25. Let us say we take negative five, we put that in here. Negative five times negative five again, gives us positive 25. So it looks on the surface, that there is no number that you can multiply by itself. Either positive or negative it looks like in the surface, there is no number that I can multiply by itself, to give me negative 25 and that looks like a total show stopper. And in fact, there is a number, there are set of numbers that will satisfy this equation. They are called imaginary numbers. Let m