![]() Now there's a third representation that's also gonna be really useful to us. Use these conversions and we want to be able to use either of these two representations freely and go back and forth between them. So there's two conversions between two different forms of the of the complex number. This on your calculator you would say that theta equals the inverse tangent of y over x. ![]() And now if I want toįind 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 to convert from x and y to r I use the Pythagorean theorem, r squared equals x squared plus y squared. So r, this is a right triangle here, there's our right triangle, so I use the Pythagorean theorem. Out the y distance here and I know r already, let me just move, here's 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 one thing I notice is I just used some simple trigonometry. Now I can go over here and I can work out how we convert between the two, how do I convert from r to y and x and how do I go the other way. So over here I can say, I can say z equals r at some angle, that's the angle symbol of theta. So in the orange is r and theta and in the blue here we have x and y and those are twoĭifferent ways to represent exactly the same number z. 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. Represent a complex number is by drawing a line from the origin here and going right through z, like that. So z is a location in this complex space. That will give me an imaginary number and that's z. 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. So this is referred toĪs the complex plane. And we'll have an imaginary part which is the vertical axis. We can plot two parts, we'll have a real part over here on what is usually the x axis. So based on what this number looks like, this suggests that we can maybe plot this on a two dimensional plot. This is the imaginary part of the number. It has an imaginary part that we're gonna call jy. 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 I don't really like the name imaginary but that's what we call it. ![]() And that's referred toĪs an imaginary number. And j squared is defined to be minus one. The complex numbers areīased on the concept of the imaginary j, the number j, 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 thoseĪre in the Algebra II section. Why we use complex numbers in electrical engineering. If you studied complex numbers in the past this will knock off some of the rust and it will help explain This video's going to be a quick review of complex numbers. There's a property that emerges from multiplication that gives us numbers that rotate (or even spin!) However, when you multiply Complex numbers it gets really exciting and different. To be perfectly clear that we are plotting complex numbers we label the axes Re and Im, not x and y.Ĭomplex numbers add together just like Cartesian numbers. Complex numbers have that pesky little j in the imaginary term. Cartesian number pairs are usually plotted with x-axis and y-axis. They bear a resemblance to another kind of 2-part number used in Cartesian coordinate system (horizontal part, vertical part. The Complex Plane a horizontal axis called the real axis (often labeled "Re") and the vertical axis is the "imaginary" axis (often labeled "Im" or "j").Ĭomplex numbers are 2-part numbers (real part, imaginary part). We can plot this number z on a 2-dimensional coordinate system if we invent the "complex plane". If this idea is new for you check out Sal's complex number videos in the Algebra 2 section of KA.Ĭomplex numbers, "z", have the form z = a + jb, where "a" is the real part and "jb" is the imaginary part. This video is intended as a review of complex numbers.
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