Complex exponential magnitude

in this video we're going to talk a bunch about this fantastic number e to the J Omega T and one of the coolest thing is going to happen here we're going to bring together what we know about complex numbers and this exponential form of complex numbers and sines and cosines as a function of time and what we're going to end up with is the idea of a number that spins I think this is really one of the coolest things in electronics and it's really that the essence of all signal processing theory so this number here e to the J Omega T this is based on Euler's formula just as a reminder Euler's formula is e to the J we'll use theta as our variable equals cosine theta plus J times sine of theta that's one form of Euler's formula and the other form is with a negative up in the exponent we say e to the minus J theta equals cosine theta minus J sine theta now if I go and plot this what it looks like is this so if I plot this on a complex plane and that's a plane that has a real axis and an imaginary axis now we remember that J is the is the variable that we use for the imaginary unit J squared is equal to minus 1 and we use that an electrical engineering instead of AI one way to express this number is by plotting it on this complex plane and so if I pick out a location for for some complex number and I say oh okay this is the x-coordinate is cosine of theta and what's this coordinate here on the J axis on the J axis that's sine theta and if I draw a line right through our number this is the angle theta so this is one of the representations of complex numbers is this Euler's formula or the exponential form and we can represent it this way here this notation is challenging I can't help but every time I look at this I start to do e to the complex to something and everything I know about exponential taking exponents and things like that it kind of confuses me in my head but what I've done over time is basically say e to the J anything that whole thing is a complex number and this is what this complex number looks like right there so let's take a look at some of the properties of this complex number all right one of the things we can ask is what is the magnitude of e to the J theta if I put magnitude absolute value or magnitude bars around that and what that says is what is this value here for R and we can figure that out using the Pythagorean theorem right what we know is this squared equals x value squared which is cosine plus the Y value squared which is sine theta so it equals cosine squared of theta plus sine squared of theta okay that's just we just applied the Pythagorean theorem to this this right triangle right here this this right triangle now from trigonometry from trigonometry we know what what's the value of this cosine squared plus sine squared for any angle equals one so that tells us that e to the J theta magnitude squared is equal to 1 or the e to the J theta magnitude is also 1 and so we write down e to the J theta the magnitude of that is equal to 1 let's go back to our friend up here that says that the length of this vector this is a complex number that is distance 1 away from the origin so we'll tuck that away we know that the magnitude of e to the J theta is 1 and I can actually go over here and draw now a circle on here like this if I put the circle right through there there's the unit circle and as a radius of one so I know that for any value of theta that might my complex number e to the J theta it's going to be somewhere on this yellow circle so e to the J theta is somewhere on this circle and the angle is what the angle is right here it's whatever is multiplied by J up in that exponent anything that multiplies by J that's an angle so what if I wanted something that was not on the unit circle what if I wanted something that was farther away from the origin than that well what I would do is I would say I would take a sum some amplitude e to the J theta and this amplitude would expand the length of that vector so I would say if I wants to know how far away it is well the magnitude of e to the J theta is 1 and the magnitude of a is a so this equals a and if I was to sketching a circle for that let's say hey was a little bit bigger than 1 it would be a little bit bigger out and this would have a value here that would be a value of a the radius would be a pretty flexible notation so with this notation we can represent any number in the complex plane with with this kind of format here we'll take a little break here and in the next video we'll put in for theta we'll put in an argument that has to do with time and we'll see what happens to this complex exponential