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# Complex numbers

## Video transcript

this video is going to be a quick review of complex numbers if you studied complex numbers in the past this will knock off some of the rust and it'll help explain why we use complex numbers in electrical engineering if complex numbers are new to you I highly recommend you go look on the Khan Academy videos that Sal's done on complex numbers and those are in the algebra 2 section so let's get started the the complex numbers are based on the concept of the imaginary J the number J in electrical engineering we use the number J instead of I and J squared is defined to be minus 1 so that's that's the definition of J and that's referred to as an imaginary number I don't really like the name imaginary but that's what we call it it's a really useful concept in electrical engineering so with that definition we define a complex number and the usual variable we often use for that is a Z and a complex number has a real part we'll call that X and it has an imaginary part that we're going to call J Y so J is explicit out here this is the imaginary part of the number this is the real part of Z so based on what this number looks like this suggests that we can maybe plot this on a two-dimensional plot and we'll call this the complex plane and the complex plane looks like this we can plot two parts will have a real part over here on what is usually the x-axis and we'll have an imaginary part which is the vertical axis so this is this is referred to as the complex plane and if I have a complex number Z I could represent it on this plane by basically going over X like this going over a distance X and up a distance Y that will give me an imaginary number and that's Z so Z is a location in this complex space and that's one representation of a complex number so the other common way to represent a complex number is by drawing a line from the origin here and going right through Z like that and then we basically have some radius R from the origin to distance out to Z and it's measured by some angle like that that angle will be theta so in the orange is are in theta and in the blue here we have X&Y and those are two different ways to represent exactly the same number Z so over here I can say I can say Z equals R at some angle this is angle symbol of theta now I can go over here and I can work out how do we convert between the two how do I convert from R to Y it and X and how do I go the other way so one thing I notice is I just use some simple trigonometry so this distance here if I know R say I know R this distance here X is equal to the cosine of theta times the distance R R cosine theta so I can say x equals R cosine of theta so if I want to figure out the Y distance here and I know our already let me just move here's here's the Y distance right here I can say y equals R times the sine of theta that's this distance here okay so if I know R and theta this is how I get X&Y now let's go the other way suppose I know x and y and I want to know R and theta so R this is a right triangle here there's our right triangle so I use the Pythagorean theorem so to convert from x and y to R I use the Pythagorean theorem R squared equals x squared plus y squared and now if I want to find theta I use another little bit of trigonometry tangent is opposite over adjacent opposite over adjacent is y over X so tangent of theta equals y over X so if you're going to do this on your calculator you would say that theta equals the inverse tangent of Y over X so there's two conversions between two different forms of the of the of the complex number we want to be able to use these conversions and we want to be able to use either of these two representations freely and go back and forth between them now there's a third representation that's also going to be really useful to us now what I'm going to do is I'm going to take this X&Y expression here and I'm going to put it back into this way this this rectangular way of writing Z what that looks like is Z equals x is R times cosine theta and Y is equal to R I'll put the are out front here R sine theta with a J in front of it so I can write plus J sine theta now if you look closely at this expression right here we recognize this we recognize this as one side of Oilers formula and the other side of boilers formula I can rewrite Z as R times e to the J theta and this is called the exponential form of a complex number all this means what does this mean here what is this thing this means exactly the same thing as this and this is one of the two ways we can write complex numbers so this are e to the J theta that is Z right here that means a complex number sitting out here at at radius R from the origin at angle theta that's what you think of when you see e to the J something written down it's it's just a representation of a complex number and this this form is going to be particularly useful because if you remember when we were solving all those differential equations we always liked exponential solutions so I want to put some squares around these guys these are the three ways that we can represent a complex number and they're all equivalent and I'll go over here just as a reminder note I'll write down boilers formula that's where that comes from and let me write that down over here Oilers formula is e to the J theta equals cosine theta plus J sine theta the other form has a negative exponent e to the minus J theta equals cosine theta minus J sine theta so that's Euler's formula and sal has videos on how to derive this equation and you would search on this term here in Khan Academy