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okay there's one more feature of complex numbers that I want to share with you and we'll do that down here so our definition of J is J squared equals -1 and now what I want to do is a sequence of multiplications by J this is a this is a really important property of this this imaginary unit we knew powers of J and I'm going to plot them while we're doing that I want to plot them on the imaginary or on the complex plane so this is the real axis and this is the imaginary axis so I'm going to take powers of J so first one is J to the 0 and anything to the 0 is 1 so J to the 0 is 1 and if we plot that on the imaginary axis here's here's 1 and there's no real part so it's just right on the the real axis okay let's do J to the one that's J times itself one time and so that is equal to J if I plot that number that's up here that's up on the imaginary axis right there there's J alright this has no real part it's all imaginary ok let's keep going let's do J squared and what is that equal to well I wrote it down up here J squared is equal to minus 1 so J squared is equal to minus 1 where's that in the complex plane that's over here at minus 1 on the real axis no imaginary part let's do the next one let's do J cubed what is that equal to J to the third power is equal to J squared times J and we have those two right here J squared is minus 1 and J to the 1 is J so that equals minus J and where do we plot that we plot that down on the imaginary axis in the complex plane right here - J okay now we've got four answers let's go one or two more okay J to the fourth is equal to what it's equal to J squared times J squared let's look at up it's minus 1 times minus 1 and what is that equal to that equals 1 so let's go plot that well we've already plotted it so that's that's this answer right here alright so we already have that let's do one more J to the fifth what is that equal to that equals J to the fourth times J and J to the fourth is right here it's 1 times J equals J let's go plot that one well we've already plotted it it's already right here okay so you can see there's a pattern here one J minus 1 minus J 1 J it's going to be minus 1 and minus J it keeps repeating here's what's interesting about this ok if we draw this as vectors if I draw these imaginary numbers as vectors when I multiplied by J when I multiplied 1 by J it rotated it 90 degrees and then that was the first step right here when I multiplied 1 times J I got J now when we went to J squared we ended up at minus 1 so multiplying by J again cause the vector to go down here like that and that was another 90 degrees and if I take it again the next one went this way and the final one went this way so this is the property of J this is the key property of the imaginary unit multiplying by the imaginary unit is the nature of it is this 90 degree rotation there's this idea of a number that causes other numbers to rotate and that's the feature of je that makes it super important and it's the reason that we use imaginary numbers in electrical engineering so the key idea here is that J rotates that's the point that's what we love about J so the last thing I want to mention is the negative powers of J what happens if we have J to the minus 1 let's figure out what that means so that of course is 1 over J and if I multiply this by J over J and anything over itself is 1 so I haven't changed the value of this and that equals J on top over J times J or J squared and what is J squared equal to and we wrote it down right here J squared is minus 1 so this equals J over minus 1 or equals minus J so whenever we see J in a fraction J to the minus 1 basically it introduces a minus sign and the J comes up to the top out of the fraction so J to the minus 1 equals minus J and we'll use that occasionally to help us in our in our math so this was a really quick review of complex numbers and if any of this was new to you I really encourage you to go back and watch sales videos on complex numbers and if this is something you've seen before I hope that knocked off some of the rust and we're ready to go and use these numbers