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# Complex exponentials spin

## Video transcript

in the last video we did a quick review of the exponential and what it means and then we looked and figured out what the magnitude of an exponential is the magnitude is equal to 1 now we're going to look closely at this this complex exponential as it represents a cosine a part of a cosine now we're going to keep combining some of our ideas from the last couple of videos and you remember if we one of the things we did was we used Oilers formula and turn it inside out and developed an expression for cosine so if I say cosine of theta I can say that equals 1/2 times e to the plus J theta plus e to the minus J theta this is cosine of theta expressed as two separate Exponential's and now we're going to take a really special step I'm going to put in an argument right here of time I'm going to say cosine of Omega T T is time and this shape here this symbol here is the lowercase Omega from the Greek alphabet and that's the frequency so time is in units of seconds and Omega is a frequency so it's in in the units of 1 over seconds that's the units of frequency or per seconds and when these multiply together we get dimensionless number right here and we can take the cosine of a dimensionless number so what is this equal to this equals 1/2 times e to the plus J Omega T plus e to the minus J Omega T when we make T the argument of the cosine here T is this stuff that keeps going up and up and up the number T gets just gets bigger all the time and so we ended up with a cosine wave form and I'll just make a bad cosine looking thing here that's what a cosine looks like and it keeps going and going and going so we have an idea of what a cosine wave looks like the frequency determines how fast this goes up and down or how often it goes up and down but now what I want to do is I want to look at a very special thing I want to look at what is this thing right here what is this thing e to the plus J Omega T and what we see is this cosine here is made of two of these things so whatever these things are I can make a cosine out of them so now we're going to look really carefully at e to the J Omega T what we just reviewed was that this is a complex number let's let's draw that complex number you so we're going to put a number out here we know it falls on the unit circle we know it's angle is whatever is multiplying the J up in this exponent whatever it's up in the exponent is the angle of this thing so this angle right here is Omega T and we know the magnitude of this is as we decided before the magnitude is 1 that's why it falls on the on the unit circle okay so now look at this here's this number T that's determining the angle and that means what that means that the angle is increasing with time if time is equal to zero the point is right here at time equals zero because angle is zero as time proceeds the angle keet starts growing and growing and it basically keeps growing and it keeps going comes back to here after Omega T equals to PI and then what happens it goes keeps going around it again and this basically goes along for as long as time goes along so here's this number here's this complex number moving along the unit circle in time over and over and over again so this is a number that is rotating the number is rotating so I can write here e to the J Omega T and I know that that because times up here I know it's rotating in time all right now I'm going to put a different number on there let's put it over here say well let's actually start this number at zero and I'm going to call this number e to the minus J Omega T what does that number look like that's this guy here that's this one here will make him orange so at time equals zero it's e to the zero or one just as we would expect now as time gets bigger the angle the thing multiplying J is minus Omega T and so the angle is becoming more and more negative so after a little bit of time it's here and after a little bit more time it's here and what we notice is it keeps it rotates this way this is what happens when you have e to the minus J Omega T you rotate in this direction and it keeps going and going and going so these two numbers are pretty similar in behavior except one rotates counterclockwise and the other rotates clockwise in our coordinate system which is the complex plane so in summary if you see either of these shapes e to the minus plus J Omega T or e to the minus J Omega T what pops into your head is a number that spins so for me the simple idea is I have a number here and I have a number here they both spin in the complex space and to represent those in in mathematical notation I need this kind of notation here which is a little bit awkward but as I get used to it e to the J Omega T it's a spinning number e to the minus J Omega T is a spinning number this is an amazingly powerful idea and will be able to describe every signal that happens using these kind of terms