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# Intro to the imaginary numbers

CCSS.Math:

## Video transcript

in this video I want to introduce you to the number I which is sometimes called the imaginary imaginary unit and we're going to see here and it might be a little bit difficult to fully appreciate right from the get-go that it's a more bizarre number than some of the other wacky numbers we learn in mathematics like pi or e it's more bizarre because it doesn't have a a tangible value in the sense that we're normally used to defining numbers I is defined as the number whose square is equal to negative one this is the definition this is the definition of I and it leads to all sorts of interesting things now some places you will see I define this way I as being equal to the principal square root of negative one I want you to just point out to you that this is not wrong and it might make sense to you you know if something squared is negative one then maybe it's the print it's the principal square root of negative one and so these seem to be almost the same statement but I just wanna make you a little bit careful when you do this some people will even go as far as saying 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 principal square root of a negative number and it being defined for imaginary and as we'll learn in the future complex numbers but it for your understanding right now you don't have to differentiate them you don't have to split hairs between any of these definitions now with this definition let's just think about what the 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 should give us all sorts of weird things and we'll see is that the powers of I are kind of neat because they kind of cycle or they do cycle through a whole through through a set of values so I could start with let's start with I to the 0th power and so you might say look anything that 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 straightforward anything to the zeroth power including I is one then you say okay what is I to the first power well anything to the first power is just that number times itself once so that's just going to be 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 is equal to negative one let's try I to the third power I'll do this in a color I haven't used I to the third power I to the third power well that's going to be 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 it's negative one times I let make it clear this is the same thing as this which is the same thing as that I squared is negative one so when you multiply it out negative one times I will write as negative I now what happens if we take I to the fourth power I'll do it 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 but 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 give you negative one but you have a negative out here so it's I times I is negative one and then you have a negative that gives you positive one that gives you pause let me write it down so 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 we're just multiplying a bunch of numbers we can switch the order this is the same thing as negative 1 times I times I I times I by definition is negative 1 negative 1 times negative 1 is equal to 1 so I to the fourth is the same thing as I to the 0th power now let's try I to the fifth power I to the fifth power well that's just going to be I to the fourth times I and we know what I to the fourth is it is 1 so it's 1 times I or it is just I again and so once again it is exactly the same thing as I to the first power let's try and just to see the pattern keep going let's try I to the seventh power I'm sorry I to the sixth power 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 it's just I so it's I times I it is equal to by definition I times I is negative one and then let's finish off well we could keep going on this way we can keep putting higher and higher powers of I here and 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 let's just verify this cycle keeps going I to the 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 8th once again it'll be one I the ninth will be I again so on and so forth