The Real World Uses of Imaginary Numbers

what is e to the pi times I this is one of my favorite things to ask students who have just learned what E is because they know what all those are but at the same time this is really hard to guess because it should be just some really random number but as many of you probably know the answer is negative one which is very odd because it's such a normal number now some mathematicians call this the most beautiful equation in all of mathematics and there's even books written about it for those who need a little background this is saying that e an infinitely long non repeating decimal that has to do with continuous compounding interest raised to the power of Pi which has to do with circles times I or the square root of negative one which is known as an imaginary number is negative one now this expression itself does not apply to the real world so much but Euler's formula which is where this expression comes from does in high school we learned that I is the square root of negative 1 and I squared is therefore negative 1 and then the cycles and patterns of for expressions that you see here you might learn a few other concepts but overall it's pretty dry and with a name like imaginary numbers that begs the question who even cares well once I got to college level courses I saw how yes these actually have real-world applications in physics and engineering as well as math and other sciences imaginary numbers help us model how electric circuits behave how electromagnetic waves travel through air and space which what allows us to listen to the radio talk on the phone and so on to help us model fluid flow and also the physics of the quantum world in engineering they show up a lot specifically in signal analysis and control theory as well now those are a lot of the main applications actually but I want you to know how and why this works and then I'll show more examples as an electrical engineer once I got to my second circuit analysis class in college we took a break from all engineering work and really focused on the math behind imaginary numbers and that was are we ready to use them to analyze circuits and signals in the weeks and classes to come the beginning of that was Oilers formula so let's see where that comes from what I'm about to show is calculus to information that anyone with some algebra 2 or pre calc knowledge should be able to follow so let's pull up desmos and we're gonna graph sine of X if you haven't seen this before well now you have then on top of that we're gonna graph the line just y equals x now are these graphs the same of course not but they do have some similarities for one they cross right at the origin they share a point right here and the line y equals x is also tangent to two y equals sine of X for those who don't know what that term is yet don't worry about it now what I'm going to do is add another term to this X and it's gonna be minus X to the third over 3 factorial now what do we see these graphs are again not the same but they're definitely more closely related for those who have taken calculus this blue curve here has the same point at x equals 0 as sine of X but also the same first second and third derivative of sine of X at x equals 0 now let's add another term of X to the fifth divided by 5 factorial and again what you'll notice is these aren't the same but they are a little more closely related in terms of looks if we keep running out more and more terms using that same pattern this equation becomes identical to sine of X so as you can see the blue curve is looking very very similar to our graph of sine of X at least up to a certain point at large values of X yes it does deviate a lot but zoom in and they look almost the exact same this equation you see here if you keep writing out more and more terms is again identical to sine of X and it's known as the Maclaurin series of sine of X if you plug in a number for X here and 1 for X and all these other terms you'll get the same number on both sides at least if you do the infinite stuff then we can do the same thing for cosine of X the Maclaurin series looks similar but this one starts with y equals 1 then we're going to add a minus X to the second over 2 factorial term then plus X to the 4th over 4 factorial and the pattern just continues right now I'm just showing that these Maclaurin series actually match the respective grass I honestly could just show the equations but I think it's better to see it graphically then the last thing we're going to do is e to the X this Maclaurin series starts with 1 and then we add an X then X to the second over 2 factorial and again that pattern just continues ok so now I have three of our key functions in all their Maclaurin series which remember if you plug in anything for X here and that same thing for all these other X's the two sides will be equal now instead of the e to the X we're gonna do e to the i-x that will equal that same Maclaurin series but with all the X is replaced with IX you're gonna see how every I is to some integer exponent well using that pattern of I to some power from before we can make the equation look like this I won't show how it gets to this but it's not too bad if you want to try it on your own now do you notice anything this right here is cosine X it's the Maclaurin series from above and this is sine of X which is x I and thus we have e to the i-x equals cosine of X plus I times sine of X which is Euler's formula I showed before and hey if you put in PI for X here here and here you get e to the pi times I equals cosine of PI plus I times sine of Pi cosine of PI is negative 1 sine of pi is 0 so that just goes away and we get that identity from before so now you know where that comes from and there's an overview of a proof for Euler's formula but now why would we actually need to use this this next example is going to be pretty much word-for-word identical to what I've done before so if you have seen just skip to this part of the video so to put simply the basics of Y complex numbers are important is phase as in the phase of a trig function the only difference between these two functions is their phase or basically a left or right shift when looking at a graph now if you've taken any trig you may be thinking wait I've done phase before and it does not require using imaginary numbers the fascias tells you how offset the graph will be which is correct but what happens when you want to add two trig functions of different phases now complex numbers come in the question is why well let's put up Oilers formula again now let's plug in something like t plus 15 for X then we have e to the I times T plus 15 degrees equals cosine of those same terms plus I times sine of those same terms so look at that the real part of this is just this a normal looking cosine function with some phase offset now if you want to add that to another cosine function with the same frequency of a different phase then we can just rewrite that using Euler's formula and the part we want to add is again also the real part so essentially if you add these up here and take the real part of it you would find what these two added up are which is what I said we wanted to do so if I just rewrite this as adding the exponential parts you see on the left what I can do is distribute the ID of both of the terms then we have an exponent with two separate terms separated by addition meaning if you remember your rules of exponents you can separate that into multiplication each with E as the base and the two separate components in each of the exponents you can do that for the other one as well and since they both have an e to the I T here you can factor that out and the inside is something you can solve using Euler's formula but just plugging in numbers now e to the I times 15 is cosine of 15 plus I times sine of 15 or 0.9 6 6 plus I times 0.25 9 it's just oil errs formula with an actual number now and we can do that with a calculator then e to the 50 I is about point 6 4 3 + I times 0.766 add those up and we get 1.0 6 9 + 1.0 - 5i now I'm not going to show how I get this next part but I can turn this back into an exponential which is 1.9 oh 8 e to the 32.5 i if you don't believe this just use Euler's formula again and you'll see you can go from this to this ok let's take a sec to just review where we are for those who are lost the original question was what does this simplify to that's it that's all we're doing right now try and turn this in the one then using Euler's formula we show that the original question is the same as the real part of this so that's what we really want just the real part the part that does not have an I multiplied by it if that can be expressed as one term we'll have our answer then this last part was just solving for the e to the 50 I plus e to the 15 I so we can get a simplified answer there now that we have it we can throw it in right here now I have two Exponential's multiplied so I get 1.90 a times e to the I T plus 32.5 I since I'm allowed to add exponents like that then I can factor out an I and I get this expression now that we have this we can again use Euler's formula and turn it into a cosine and sine function and finally remember how we said once we add up everything the real part of our answer will be the solution well that real part is 1.9 oh 8 cosine of T plus 32.5 and now we know that the original question simplifies to this one cosine equation these are the exact same thing now one of the most well-known applications of all this is for AC circuits or circuits with an alternating current this is where I really applied complex numbers for the first time for example if two voltages with different phases are put into a circuit you need to combine them into one before doing analysis which you now know the basics of whenever we were given a sinusoidal voltage of some amplitude and phase we always had to change that to exponential form with the amplitude in front and the angle up top multiplied by I we did not need to include the time variable do the math here actually but we usually wrote it in this notation instead just focus on the amplitude and phase well that's nothing to worry about though then we learned how throwing in a capacitor or inductor to a circuit will shift the voltage and current depending on their values the math represent that is all complex numbers inductors and capacitors have something called impedance sort of like resistance for a resistor but impedance also tells us how the voltage and current will be shifted not just reduced an inductor might have an impedance of 10 I yes it is represented by an imaginary number and you'll see why 10 I is the same as 10 e to the I times 90 degrees if you don't see how this is true just use Euler's formula which I'll here if you just want to pause the video to find current in a circuit you just do voltage over and peanuts you'd write both of those in exponential form where the voltage has an amplitude of 20 and a phase offset of 30 degrees in this case and then we of course know the inductors impedance which goes down here then you solve this with some basic algebra and subtract the exponents to get negative 60 degrees up top then simply divide the coefficients to get 2 in front you can then turn this into its real component which is all we cared about and we have the current in this way to simplistic circuit is 2 cosine of T minus 60 degrees which yes is a different phase compared to the input voltage and that's why we use imaginary numbers to determine shifts in these circuits before I show a few more applications hopefully you're seeing something nothing in the real world is really quantified by imaginary numbers what I mean by that is there's not 5i amps of current flowing through your electronics it's all real numbers like 10 amps 2 volts and all that but these complex numbers simply make the math work to manipulate equations and simplify the process as we need that's it you cannot physically hold or measure I of anything moving on to some more applications one topic I've talked about many times is the Fourier transform which basically says any function can be made up of a bunch of sine and cosine functions added up this is the main thing I learned about my first signals class because when looking at a messy signal if you can break it up into the frequencies that make it up aka the frequencies of those cosine and sine functions you can do much more to analyze it and manipulate it for the purposes of audio processing speech recognition radar etc and the equation for the Fourier transform can be shown here and in it you'll see an imaginary number as well as something that looks very similar to that e to the i-x we've been seeing now sort of a random example that I'm going to tie to physics let's say we have a function that is very narrow with this one spike the Fourier transform of this would be mostly a sinusoidal function but don't even worry about how or why those who know the Dirac Delta function that's really what I'm going for while avoiding having to explain it now in quantum mechanics as you may know from my physics video you cannot know the position and velocity of a particle at the same time there's uncertainty to it the more you know about one the less you know about the other well mathematically this graph might represent a wave function describing the position of a particle it's localized and shows that we have a very well-defined position here not much uncertainty the Fourier transform is a non localized function oscillating up and down which reveals a velocity or really momentum that is not well-defined I can accurately determine one thing and therefore I am uncertain about the other and now you know a little about the math behind that as well although this is way oversimplified if you watch the MIT lectures on quantum mechanics you would see imaginary numbers come up in lectures three and four and five six seven and you get the idea it's everywhere in quantum mechanics and like I said imaginary numbers come up a good amount in control theory which is big for mechanical aerospace or electrical engineers and mechatronics majors just to name a few for example these are two graphs taken from my controls class textbook you see that J right here labeling the y-axis that's just I the imaginary unit electrical engineers use J instead of I because I is typically used for current on this plot they actually label the imaginary axis though these X's and zeros and where they're located on the real and imaginary axes tell us things like the stability of the system for example and how certain parameters change with respect to individual components of the control system these control systems are found in rockets fighter jets robots autonomous vehicles and more but I'm not going to go into a more technical detail than that and before I end this video I just want to note that you don't need imaginary numbers for these real world applications they just help make things simpler Leonard Euler who was alive during the 1700s made a lot of progress on our understanding of complex numbers and it was in 1892 when a man named Charles Steinmetz joined General Electric and soon after published a paper showing how to use imaginary numbers to greatly simplify the analysis of AC circuits you can run other equations that will spin out the exact same solutions but the use of these complex numbers simplifies the process a lot and this is said to have accelerated the pace of innovation during the 20th century and using just what I showed in this video you can prove some very weird things like the cosine of I is about one point five four and I to the eye is about point 208 which I'll leave for you to try if you want to challenge yourself but all you need is Euler's formula to prove both of those there are more applications of course but due to time I'm not going to go into them again in college you can definitely expect to see a lot of this if you go into electrical engineering or even Computer Engineering as they share a lot of the same classes although other majors like aerospace engineers mechanical engineers by medical engineers and several more have to take a basic circuits class in which they'll learn a lot of the basics of this as well then most math majors as well as some physics majors and some electrical engineers will have to take a class on complex analysis in college which is all about complex numbers and the math behind them I took this class in college and loved it although we did not really go into any of the applications it was really more of a pure math class that does have applications which we did not go into in the course itself now before I end this video I think it's my responsibility to say don't let any of this scare you if you're going into any of these majors I was very hesitant to make the video as technical as I did but I really enjoy this stuff and one you guys can see exactly how it works as I said before I definitely am a math person but I remember learning this stuff in college and it taking a little bit of time to get used to just understand how these complex numbers apply to the things we were doing in our engineering classes but after a little bit of practice you definitely will get the hang of it and that's it for this video if you enjoyed be sure to LIKE and subscribe don't forget to follow me on Twitter and join the major Facebook group for updates on everything and I'll see you all in the next video

Channel: Zach Star

Views: 570,633

Keywords: majorprep, major prep, imaginary numbers, complex numbers, applications of imaginary numbers, what is an imaginary number, why do we need imaginary numbers, math, science, physics, engineering, circuits, signals, where are imaginary numbers used, where are complex numbers used, complex analysis, students, school, university, real world uses of imaginary numbers, imaginary number examples, euler, eulers formula, applied math, mathematics, the most beautiful equation

Id: _h49ilnTmW4

Length: 16min 48sec (1008 seconds)

Published: Tue Sep 11 2018