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Intro to the imaginary numbers


Video transcript

In this video, I want to introduce you to the number i, which is sometimes called the imaginary, imaginary unit What you're gonna see here, and it might be a little bit difficult, to fully appreciate, is that its a more bizzare number than some of the other wacky numbers we learn in mathematics, like pi, or e. And its more bizzare because it doesnt have a tangible value in the sense that we normally, or are used to defining numbers. "i" is defined as the number whose square is equal to negative 1. This is the definition of "i", and it leads to all sorts of interesting things. Now some places you will see "i" defined this way; "i" as being equal to the principle square root of negative one. I want to just point out to you that this is not wrong, it might make sense to you, you know something squared is negative one, then maybe its the principle square root of negative one. And so these seem to be almost the same statement, but I just want to make you a little bit careful, when you do this some people will even go so far as to say this is wrong, and it actually turns out that they are wrong to say that this is wrong. But, when you do this you have to be a little bit careful about what it means to take a principle square root of a negative number, and it being defined for imaginary, and we'll learn in the future, complex numbers. But for your understanding right now, you dont have to differentiate them, you don't have to split hairs between any of these definitions. Now with this definition, let us think about what these different powers of "i" are. because you can imagine, if something squared is negative one, if I take it to all sorts of powers, maybe that will give us weird things. And what we'll see is that the powers of "i" are kind of neat, because they kind of cycle, where they do cycle, through a whole set of values. So I could start with, lets start with "i" to the zeroth power. And so you might say, anything to the zeroth power is one, so "i" to the zeroth power is one, and that is true. And you could actually derive that even from this definition, but this is pretty straight forward; anything to the zeroth power, including "i" is one. Then you say, ok, what is "i" to the first power, well anything to the first power is just that number times itself once. So that's justgoing to be "i". Really by the definition of what it means to take an exponent, so that completely makes sense. And then you have "i" to the second power. "i" to the second power, well by definition, "i" to the second power is equal to negative one. Lets try "i" to the third power ill do this in a color i haven't used. "i" to the third power, well that's going to be "i" to the second power times "i" And we know that "i" to the second power is negative one, so its negative one times "i" let me make that clear. This is the same thing as this, which is the same thing as that, "i" squared is negative one. So you multiply it out, negative one times "i" equals negative "i". Now what happens when you take "i" to the fourth power, I'll do it up here. "i" to the fourth power. Well once again this is going to be "i" times "i" to the third power. So that's "i" times "i" to the third power. "i" times "i" to the third power Well what was "i" to the third power? "i" to the third power was negative "i" This over here is negative "i". And so "i" times "i" would get negative one, but you have a negative out here, so its "i" times "i" is negative one, and you have a negative, that gives you positive one. Let me write it out. This is the same thing as, so this is "i" times negative "i", which is the same thing as negative one times, remember multiplication is commutative, if you're multiplying a bunch of numbers you can just switch the order. This is the same thing as negative one times "i" times "i". "i" times "i", by definition, is negative one. Negative one times negative one is equal to positive one. So "i" to the fourth is the same thing as "i" to the zeroth power. Now lets try "i" to the fifth. "i" to the fifth power. Well that's just going to be "i" to to the fourth times "i". And we know what "i" to the fourth is. It is one. So its one times "i", or it is one times "i", or it is just "i" again. So once again it is exactly the same thing as "i" to the first power. Lets try again just to see the pattern keep going. Lets try "i" to the seventh power. Sorry, "i" to the sixth power. Well that's "i" times "i" to the fifth power, that's "i" times "i" to the fifth, "i" to the fifth we already established as just "i", so its "i" times "i", it is equal to, by definition,"i" times "i" is negative one. And then lets finish off, well we could keep going on this way We can keep putting high and higher powers of "i" here. An we'll see that it keeps cycling back. In the next video I'll teach you how taking an arbitrarily high power of "i", how you can figure out what that's going to be. But lets just verify that this cycle keeps going. "i" to the seventh power is equal to "i" times "i" to the sixth power. "i" to the sixth power is negative one. "i" times negative one is negative "i". And if you take "i" to the eighth, once again it'll be one, "i" to the ninth will be "i" again, and so on and so forth.