Complex number fundamentals | Lockdown math ep. 3

today we are going to talk about one of my absolute all-time favorite pieces of math it's incredibly fundamental to engineering to mathematics itself to quantum mechanics but it's something that it has a terrible terrible name we call them complex numbers and worse than that the things that bring about complex numbers we call imaginary numbers and before we get into any of it what I want to do is start with kind of a poll just to pull the audience on seeing what you guys to consider to be well real what do you consider to exist when it comes to numbers so we've already been doing a couple polls in the warm-up animations but as a a serious poll of so it's one that's actually going to help me see where you're coming from before we begin the lesson here I just want to ask you a very simple question okay so let's go ahead and pull it up here pull it on up and oh for whatever reason it seems like we're having trouble pulling this one up hmm okay there we go took a little delay there among the values - square root of 2 square root of negative 1 and infinity which would you personally consider to really exist whatever really existing means to you so in theory if you guys go to three b-1b co / live you should be able to answer this and then the statistics based on your answers are going to start populating the screen we won't know what those answers refer to at the moment all we know is that you know there's someone who has a lead another one that's pretty close behind it and as your answers come in and as the server is kind of digest them we'll start to see some of the stats here if you go to that page by the way 3 B 1 B 2 slash live which redirects to item pool com what you're going to find is at the very top you can ask a question on Twitter and all that's going to do is basically open up a tweet that's going to have the hashtag in it locked down math and that's the way that we're gonna be doing questions to this instead of a live chat anytime you have a question or a comment that you want to you know insert into the lesson and let that be part of the discussion it's going to be pulled up here and it looks like we already have one this is from Yash Dave who asks if you could rename the complex numbers and the so-called imaginary numbers - something more intuitive - a name that conveyed the fact that they have numerous applications in the real world what would you name them I couldn't be happier that you asked - so I have one friend who seems very passionate about the fact that we should call them sneaky numbers personally I'm very fond of trying to connote spinning and rotation and this is one of the things we'll talk about in today's lesson is the fact that what we call complex numbers what we call imaginary numbers some of the main uses that they have come from very elegant descriptions of how to rotate stuff and I hope you'll kind of see what I mean as we as we proceed with that so on our poll asking you guys which numbers you consider to really exist which is of course a subjective question there's no right or wrong answer here I am genuinely curious actually how this breaks down because it's not at all a strong consensus in one direction it seems like we've got three top contenders and then three that are following pretty evenly behind that so let's go ahead and take a look what do you consider to really exist when it comes to numbers now I can imagine which ones might be the top two but I'm very curious about the fact that there's three all kind of coinciding with each other there and it looks like I I'm getting a little bit of a delay before the reveal so there's kind of this nice dramatic pause I'll tell you for me personally I feel like it's very silly to answer anything that's either it's not all of them are none of them I can maybe understand if someone wants to treat infinity as something different because it's ill-defined there's lots of different things that that might mean but insofar as numbers exist at all if you consider what a number is to be a real thing then it would oh man I I can't believe that we're stalling out on this one we had fixed it by livestream - but I guess there's going to be an oscillation between when it works and when it doesn't but for me personally basically anytime that you have a numerical construct that's helpful in the real world you know I'd consider that real I would say that if it's something that's actually useful in an application then it is as real as words are right you're never going to run into an abstract word like happiness out there but it has a kind of reality in our mind and things like the square root of two which you can't express as a fraction or things like the square root of negative one that don't show up among real normal numbers you know even if they might seem a little bit different oh this is such a shame I'm genuinely curious to see what your answers are but it's not showing up for me which I suppose means we'll have to move on with a lesson but this will presumably begin working by the end and we can maybe pull things up again so let me go ahead and take that away what I'd like to do for you today basically is show you the sense in which imaginary numbers are useful the complex numbers are useful and from there maybe try to imbue them with a little more reality I won't assume that you know what they are yet it's meant to be a basic primer but let's just dive right in okay the end by the way the very end here I want to talk about two different trigonometric functions and this is kind of a thing that we're going to build to two identities from trigonometry and I understand that maybe oh these complicated identities from trigonometry is not going to be the best way to lure some people into the understanding oh yeah complex numbers they're really useful you're really going to love them but I do think it's interesting that you can have a fact that has nothing to do with complex numbers of the square root of negative 1 it's just trigonometry it's everything we were talking about last time and you can have facts that are pretty hard to remember I remember when I was in school and we learned these addition formulas that if you want to know the cosine of the sum of two different angles you know it's this kind of long thing in terms of cosines and sines of the original two angles there's this minus sign that would always trip people up if you do the same for the sine it looks similar but there's a plus sign and instead of having Coast coast you have cosine it's something that's very error-prone if you're just trying to memorize it as it is however if you come at it with complex numbers this is not only much less error-prone it has a very beautiful meaning and it just falls right out so even if you don't necessarily believe in the reality of the square root of negative one you at the very least have to admit that it's interesting that it can make other pieces of math useful that other pieces of math a little bit more understandable too and trigonometry is just the tip of the iceberg if you talk to anybody who's in engineering anybody who's going into serious math they'll tell you the complex numbers are as weel a part of their work in their life as real numbers are but the starting point looks very strange okay when you start introducing this the very first thing you do is to say assume that there's some number I so that I squared is equal to negative one and I think through a lot of students there's maybe one of two possible reactions that you can have here one is no there isn't anytime I square a number even if it's negative if I take negative five for example and I square it well the negative times a negative is a positive so I get 25 any number that you squares if it's positive well that just stays positive so it seems like no matter what when I'm squaring numbers I always get a positive number I'm never going to get anything negative so this does not exist no such number however if a mathematician comes and says oh no it exists we've defined it so that that's the case i think the other reaction someone can have is hang on a second you can do that when you have a problem that you can't solve you can just say oh I've defined things so that we now magically have a solution okay next time I'm having trouble with my homework and I don't know what the answer to X is I will be like let X be the value define to be the answer to this question so if you're uncomfortable with this you're definitely not alone in fact Rene Descartes coined the term imaginary for these numbers as a derogatory it was meant to make fun of the fact that obviously there is no such answer and it shouldn't be taken as serious math and then we stuck with that as a convention and we still call them imaginary numbers which is genuinely absurd but that's not the only weird assumption that we make the second weird thing that you do when you start talking about complex numbers is to say there's not just such a number I but we're going to give it a home instead of the real number line which you know all of these numbers we know when we square them you can't get a negative what we do is say I lives in a different dimension I lives perpendicularly there's one above and then there's one below negative I and you can have negative 2i you scale it however you want essentially it's proposing that numbers be two-dimensional and that I has a very specific home one unit perpendicular perpendicularly above the real number line and okay if we want to extend our number system I get it maybe it's useful to put some kind of number up there but why I right why not say infinity is the number that sits one unit above zero or one divided by zero or any other problem that you couldn't solve before and you make up an answer to why should that live there what on earth does the idea of a point one unit above the real number line in a separate dimension have to do with squaring two negative one so I hope to answer this for you at the very beginning let's just talk about how if you're adding numbers that are two-dimensional like this the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be familiar with vectors so let's say hypothetically I have a number oh I don't know let me draw one here that's going to be 4 plus I ok and then I'm going to take a second number and it's helpful to draw them as vectors kind of an arrow from the number zero and this one is going to end up at negative 2 plus 2i so what I'm saying is you take the real number negative 2 and then you move in that perpendicular direction into the extension of our number system which again kind of asking the students to take a lot on faith here that you're ok to do that that you're allowed to just pretend that the numbers extend in this direction if you take that on faith and you follow hopefully the fact that it becomes useful helps to justify why we're doing any of this so my question for you is simply what happens when we add these two numbers now assuming that our question system has not broken down I should be able to do this as a proper Pole and let me go ahead I guess we can first check the previous Pole ok things seem to be working so we can take a little step back in the lesson so I'm just genuinely curious I want to know how you guys answered on this one it looks like there's a there's a back and forth between answers F and D so f is all of them saying that all of these should be considered real and interesting D is the one that says you should consider to square root of 2 and negative 1 but not infinity so there's a good contingent of you out there who would just reject infinity as being considered real but are very comfortable with the square root of negative 1 that's awesome and then after that it looks like C people who reject the square root of negative fascinating I actually would have thought that none of them would have come higher than that none of them is much lower at a okay so it looks like we've got a cohort of people who are comfortable with negative one a large cohort are uncomfortable with infinity that's a topic for another day don't worry about it and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative one might be real let's see if we can convince you of the difference of that so for our first much more mathematical question that's kind of a warm up I just want to ask you to add these two before I've taught you how to add them make a guess at how it might work and I hope that it feels pretty straightforward addition is actually the least interesting part of this but it is it's a good thing to know when you're learning about complex numbers it's definitely one of those operations that you are going to need to know unfortunately and you can tell by the fact that I'm stalling and what I'm saying here it looks like the question is still not loading completely correctly so I'm going to have a stern word with Cam and either behind the scenes who have otherwise built such a beautiful beautiful interface that's helpful for this kind of back and forth between you guys and me I'm gonna have a stern word with them behind the scenes but in the meantime let's go ahead and move forward with a lesson here so I guess I can pull it up on the just on the piece of paper and you can follow along at home see what the addition might be it turns out to be relatively straightforward if you're moving four units to the right and then one unit up and you want to add the idea of moving two units to the left and then two units up will you just do each of those one at a time I'll go ahead and pull out black here the real part is going to be those four to the right then minus two to the left okay straightforward enough and then the imaginary part is going to be this one unit up and then these two units up one plus two times I so is that one I plus two I and then when you work that out four minus two is two one plus two is three a nice simple introduction here addition doesn't really have anything complicated going on which is great that means that it's one fewer thing for us to worry about what is so complex about complex numbers after all well where everything becomes interesting is when you try to multiply these numbers together so with victors there's not really any notion of multiplying them to get two vectors back at least when we're in the 2d plane you have some notions like cross products and dot products that in three dimensions can get you something like it but the rules end up being very different from that in the number system you can't really do algebra you can't do things like assume that if two numbers multiply to make zero then one of them has to be zero but complex numbers are going to end up behaving much like the real numbers so rules from algebra can carry over but to understand what that rotation rule is I know I'm giving things away what that multiplication rule is I just want to ask you a simple question which is basically suppose I have the point mmm 3 2 okay we're not even going to think of it as a complex member per se if I just have some sort of coordinate grid and I go to the point with x-coordinate 3 and y coordinate 2 what is the 90 degree rotation of this if I rotate it 90 degrees and let's say counter clockwise counter down geez writing is difficult counter clockwise okay now what's lovely about this is we can basically just turn our paper to figure it out you say okay if it started at three two and then I rotate 90 degrees counterclockwise I can just read that off now as being negative 2 in the X direction and then 3 in the Y direction if I had rotated the whole plane like that so what we've done here is we've taken 3/2 and then we convert it to negative 2/3 something which may be in our original system you know it looks like this negative 2 and then 3 that's going to be the 90 degree rotation and what's nice here is that that rule is very simple and it applies to any pair that we might have if I took a pair of numbers a comma B okay and then I said where is that going to rotate to if I rotate it 90 degrees it's going to end up by swapping the coordinates B a and then making that first one negative that's a 90 degree rotation and a nice gut check here is to say what happens when we do that twice what if we do that same very mechanistic operation again twice and I'm gonna go and take this I swapped the two coordinates we get a negative B but then that first one becomes negative so that was another 90 degree rotation well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a B and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree worked no I've done that wrong that will be the same as a 180 degree rotation which should look like this ignore the other vector that I drew which is just taking both of the coordinates and making them negative negative a negative B okay so that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to now why am I asking you this well I'm being told that supposedly I'm allowed to ask you questions again so I'm gonna have you do your very first complex product oh look a lot of people did submit answers very good okay let's let's grade the complex addition actually let's let's see if it is and straightforward to process as I was hoping it was see how much explanation is demanded okay so it looks like a majority of you did get the correct answer which is 2 plus 3i very good very good 52 of you answered simply 2 which would have been the real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer to reject the reality of imaginary numbers so you just don't even acknowledge that vertical component some of you answer negative 2 3 which I guess is just making a vestra swapping up whether you're taking 4 minus 2 or 2 minus 4 so that's completely understandable we've got 2 plus 3 which is maybe just dropping off the I so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap too so that's all very good let's go ahead and try our very first product though like I said so here because I already talked through one of the questions we're going to go ahead and skip ahead of it we know how to rotate something like 3 comma 2 so I'm not even going to give you time to do that and properly grade it stall stall words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not gonna have a stern word with and you guys can go at them on Twitter too under the same place that we asked questions and just say hey Cameron eater can't you make the live questions work a little bit better for us ok I think we're finally there everybody ready aha wonderful very simple question I want you to take the number I and I want you to multiply it by 3 plus 2i and even though I haven't really talked about the rules for multiplication what I can say is pretend like it operates just like it does for normal numbers you've got things like the distributive property where you can distribute this throughout and then the defining feature of I is this idea that I squared is negative 1 that's the only special thing you need to know about that other than that just treat it like it's a normal number ok and then proceed forward with the product wonderful ok so it looks like we've got quite a few of you coming in to answer which is always lovely super exciting for me by the way just how many people are enthusiastic about coming and like getting back to the fundamentals of math in this lockdown and just you know we're gonna sit back for an hour and we're gonna learn about complex numbers and we're actually going to participant we're actually going to answer questions as you do it rather than sitting and passively watching this is genuinely delightful to me okay this is this isn't necessarily question I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not that would that would heavily inform where the lesson should go so that would be quite useful and I think that's probably enough time so I'm gonna go ahead and lock the answers into this one okay and it looks like the majority of you answered negative two plus 3i which is absolutely correct absolutely correct so there's two ways to think about this okay one of them is to walk forward with the algebra and just do it a little bit mechanistically okay so if we pull ourselves up our sheet if we take I times 3 plus 2 I 3 plus 2 I it just distributes I times 3 is going to be 3i I times 2i is going to be 2 times I squared by definition I squared is negative 1 which means that our final answer is going to look like negative 2 plus 3i ok and like I said it looks like a majority of you correctly did that product now it's one thing to just walk through it mechanistically it's another to step back and say what just happened geometrically right because what we just talked through was the fact that if you want to rotate numbers 90 degrees the rule is to swap the two coordinates and then multiply that first one by negative 2 well look it's what's happened here we've got 3 & 2 those coordinates have been swapped 2 is now the real part 3 is the imaginary part but that 2 got multiplied by a negative 1 because I has this defining feature of squaring to become negative 1 so that should give you some indication that okay multiplying by I has this action of rotating things by 90 degrees may that means that it's not a crazy thing to do to geometrically position I at a 90 degree angle with the real number line okay now to really see why that ends up being useful it turns out that if you have that if you have a number that behaves this way it gives you a computational mechanism for all of the other types of rotations that you might want to do that might not necessarily be ninety degrees and to show you why this works I'm going to go ahead and pull up an animation so let's say we have any number Z and in this case Z is going to be let's see where do I have it C is going to be a 2 plus I great and let's say I want to understand what is multiplying by Z due to every other possible complex number what we can go one by one the very first one that's relatively simple as if I ask what is Z times one where does it take the number one well Z times one is going to be Z so we're going to take the number one which I might draw with a little yellow arrow and we're going to rotate and stretch that arrow up to the point where Z is great a kind of trivial fact even though it's trivial I'm actually going to take a moment to write that down just so that we can oh no no that's for the best for later that is Randy don't you guys worry about him he'll be coming in in just a moment so I just want to write down three crucial facts that are getting an influence rotation three facts I'll call it three facts about multiplication the first two are going to look simple the third one is going to look innocuous but it ends up being extremely constraining so the first one is that if I have any number Z and I multiply it by one well I get the number Z back okay geometrically that means with that we stretch and rotate that vector one to get what we want the second one is that if I take Z times I from what we just talked about that is the 90-degree rotation what I might write is like rotate by 90 degrees the number Z okay so often our animation what does that look like let's see we've already multiplied by one we take I and then it's going to move to whatever the 90 degree rotation point for Z itself is okay two down infinitely many to go and we know what it does to one we know what it does to I let's see if we can understand what Z does to any other possible number well it turns out those two is really all we need to work with if we have the distributive property so the third fact that's going to look kind of innocuous is let's say I take this Z and I multiply it by C plus D times I where C and D are just any two numbers okay well this is going to distribute so Z times C I'm actually going to write that a little differently I'm going to write it as C times Z plus Z times D I which again I'm going to write in kind of a funny order and write that as D times Z I now the idea here is what we know where Z is we also know where Z times I is so if we're just scaling them up by some other constants that completely constrains where we need to go so let me go ahead and write this down with an example okay let's say that we go back here and I want to know what multiplying by Z does to anything I want to tell I want to convince you that it rotates all of the grid in a way that keeps these lines parallel it keeps them evenly spaced keeps them perpendicular to each other it applies this very constrained rule to the whole plane and really just think through any one particular point for this let's say that we have two times negative I okay so you move two units in the positive right direction and then negative one unit in the vertical direction well after the product where that's going to land has to be two times wherever zealand's plus negative one times wherever I lands okay and we see that right it's two times this yellow vector and it'll be negative one times the green vector so here even before you actually work out the product we could just read off the fact that Z here which looks like it's two plus I multiplied by two - I must land on the number five okay so I actually want you guys to work this out algebraically because I do think that it's it's edifying to see this work in practice right the fact that we have this geometric rule where we were just kind of stretching and rotating and this corresponds to an otherwise very mechanistic process of expanding out some so let's pull back our quiz and I'm going to it's a little bit silly because I've revealed the answer but what I actually want you guys to do here is take a look at the question which says what is 2 plus I times 2 minus I and if you have notes right now if you have a pencil and paper which I encourage you to always come to class with I want you to try working it out do the first inside/outside last distribution property just to see mechanistically what number ends up popping out from this and then we'll try to see how that squares with the geometric intuition so while you're doing that while you're working that out hopefully on pencil and paper it looks like we've got a question from the audience which is is I the same as I and J the vectors in physics great question actually that's that's a much more interesting question than you might think it is the short answer is no so in physics you often describe the rightward direction as I in the up and down direction as let's go get another one here let me finish answering the question first so in physics you often have this situation where instead of describing the XY plane you describe the right vector as I sometimes I hat and the vertical one as J hat so that's a separate thing that's how physicists will describe the right direction of direction for vectors they're not necessarily describing complex numbers totally separate thing what's more interesting actually is that the reason that convention came about I believe has to do with a number system that extends even beyond complex numbers it's called the quaternions so with the complex numbers we introduced something called I with the quaternions we introduced J and K these two other numbers both of which also have squares two negative one as it happens and the guy who invented them Hamilton he was trying to describe three-dimensional space very elegantly and using the purely imaginary directions I J and K became a nice way to do that and I think it's because of that and his attempts to push that forward that the conventions slipped into normal physics to just describe those as I and J of course that leads to serious confusion now because if we look at how physicist thinks of things you know I is in the right direction J is in the vertical direction but with complex numbers I is of course in the vertical direction so it just makes for a whole potential mess now what was our oh no it looks like they took away the comment oh man there was comment giving them some sass and then they got embarrassed so they pulled it away well it's on screen it's in there for everyone in the future to see now I think that's probably been enough time to work through our a little bit of algebra here so let's go ahead and grade it for everyone to see and it looks like the vast majority of you well I told you the answer so I'm gonna assume the vast majority of you are giving the correct answer Oh interesting okay so it looks like five five is correct and that's the most common answer but then the ones where missed were three and four and maybe we can try to understand why three and four would have been common misconceptions on this one 69 is close 0 i plus 5 is correct but I guess just formatted in a weird way so let's let's go ahead and work this one out by hand ok it's actually kind of a pleasant process to multiply complex numbers because it's so deterministic you know what you have to do you know where you start you know where you're going to end and it's really just kind of practicing through it so two times two that's the first thing that we get and then you take the inside parts I times two so that's going to be 2i and then the outside parts which is negative I times two so that's negative 2i and then the last part which is I times negative I so that'll be negative I squared the defining feature remember is that I squared is the same as negative one which means that the real component of our answer is going to be 2 times 2 or 4 now minus negative 1 which is plus 1 so that'll be 5 and then the imaginary parts cancel each other out that 2i cancels out with a negative 2i so to everyone on the quiz who answered 5 plus 0 I you know extra credit to you because that actually is a hundred percent correct you're even emphasizing that we're doing this in a complex context so there's no reason that you should have been graded incorrectly on that one however what's uh what's interesting here those of you who watch the very first lecture might notice the same with difference of squares and this is effectively a difference of squares type question and I'm going to guess that those of you who answered three or four it probably came down to if you answered three it would have been because you took this four and you subtracted one rather than subtracting negative one very common mistake no harm in that and then if you answered four it comes from forgetting the fact that I squared can be simplified into a real number so that you know that very much counts for the the real component of your final answer now what's interesting I think is that that a process of just walking through the algebra feels very different from this geometric idea that we had before of saying multiplying by Z has the action of rotating and stretching things and how it rotates and stretches things is in the way that's necessary to get the number one to sit on the number Z and in fact I have a have a good friend Ben Sparks who is a frequent number file contributor and as some of you might recognize and I asked him to whip up a little geogebra thing because he's just a geogebra whiz and he actually created basically the complex number equivalent of a slide rule that we can play with here so the way that this is gonna work is that I can take the number one and sort of drag it around the plane and as I do the rest of the grid moves in the way that it has to so that the origin stays fixed in place and then everything else kind of stays rigid with the exception of being able to stretch so the way that you might use this is to say let's say we want to know what Z times W is well I'm gonna stretch and rotate the number one so that it lands on Z and then I'm going to follow what happens to W which is now drawn as a a orange vector follow what happens to W and as it lands there we can just read off that it ends up at five now even though in practice computationally you would be working it out with the algebra like we just showed having in the back of your mind that this is how they operate helps draw the connection between complex numbers and geometry and also trigonometry which we'll get to in a moment and that turns out to be a much more helpful connection than you might expect so let me go ahead and ask you guys a second question here which is do to bring back our helpful little friendly quiz given what I just told you about how complex numbers act I want you to guess for me it's okay if you don't know it's okay if you just want to hazard a guess here what is the complex number Z so that multiplying by Z has the effect of rotating 30 degrees or pi/6 radians counterclockwise okay so you want to number Z so that when it's being applied as an action and you're saying what do you do to all the other possible complex numbers the corresponding action is to rotate by pi/6 radians or 30 30 degrees okay great we've got a lot of answers coming in strong consensus around one of them which is always a good sign let me show you how I might draw this out if I were doing things so I would take a look at a unit circle of some kind and luckily I have my infinite supply of unit circles sitting right here and I would say okay I know that the action that I want is going to be whatever takes the number 1 to something that's the 30-degree rotation of the number 1 it's it's up here whatever complex number Z this is is going to have the appropriate action and then from there it's all of the trigonometry that you and I were working through last time so let's look at our answers and let's go ahead and lock them into place now before we talk through the explanation and the correct answer in this context is d which is cosine of pi/6 plus I sine of pi/6 congratulations to the majority of you who correctly answer that very interesting to me actually is that the second most common answer was a which is 1 plus PI sixths and I think that's that's an utterly reasonable guess actually and let's take a look at why so if we go back to our friendly unit circle the number we were going for is like I said sitting pi/6 radians around and from everything we talked about last time the X component of that is the cosine of PI sixths this is sort of how cosine is defined and then the Y component of that is going to be sine of PI sixths which means that that number Z that we're dealing with yeah he's going to have an X component of cosine of PI sixths and then an imaginary component what's in the Y direction of sine of PI sixths great now that second most common answer like I said that's very interesting um if we pull it up it was a which was 1 plus PI 6 over I let's take a look at where that sits with a I'll bring in a second cleaner unit circle here like I said I've got an infinite supply they were guessing that it's 1 and that instead of walking PI sixths around that you walked a straight up PI sixths okay which is actually quite close right especially for small angles if you're just walking straight up that's not too different from walking around the circle so if you were to try to use that number to have the action of rotating by 30 degrees it would have an action of rotating by something that's slightly less than 30 degrees and then they would stretch things out by something that's slightly more than a factor of 1 but it's actually close which is cool because usually the the misconceptions might be like very wrong can computationally but the idea that the the misconception here was like computationally quite close indicates that maybe the geometric reasoning was lining up with truth which i think is awesome so to give you an illustration of how this might actually be used okay we have a number that has this action of a 30 degree rotation I want to bring in of course a friendly PI creature for us to all think about Mandy so let's say that I have a PI creature and I'm I don't know animating a video and at some point for whatever reason I want that PI creature to rotate 30 degrees okay I just want them to rotate there's lots of ways you can do this one of the most common ones in computer graphics is to use matrices which might be a whole topic for another day but a tactic you could use is to use complex numbers and the way this would work is to think of our PI creature as living in the complex plane and then in any kind of computer graphics it's always defined with various points often these control points around it and each one of those control points I would give complex coordinates to so for example let's say the the lower right foot here or I guess Stage Left our right looks like it has coordinates about one negative three so that would be given the number one minus 3i and then from what you guys just told me in answering the question if we want to take that point and then figure out where it gets when we rotate 30 degrees about the origin what I should be multiplying it by is cosine of 30 degrees by sine of 30 degrees and if ever you wanted to be convinced that complex numbers are just as real as real numbers I think one factor to consider is that many programming languages have built into them complex numbers they are considered as real as real numbers so what I might do I can pull up a little Python terminal for us to play with and complex numbers if I write something like 3 negative 1 are considered a valid type in this case they use J I think basically because in programming parlance I is often used as an indexing variable so they just go with J instead that actually matches with electrical engineering convention to where they instead of using I for the imaginary number they use J because again I was kind of already used so 3 minus 1 J you can just read that as being the same as 3 minus 1 I and I can take this number and I can multiply it by other complex values I don't know like five for two if I multiply that it's going to apply the rules of complex multiplication which basically is just foiling it out you can think of this as being the real part is 3 times 4 or 12 minus negative 1 times 2 because remember that I squared gives us that negative so that's where the 14 comes from and then the imaginary part is going to be that negative 1 times 4 which is negative 4 plus 3 times 2 which indeed works out to 2 so that's kind of cool Python seems to be working all of this out for us and what that means remember is let's say that we wanted to rotate our pie creature and let's say I'm just going to focus on one point like his foot sitting here which is that one negative 3 so let's say I define the foot to be sitting at real part 1 imaginary part negative 3 the rotating number that I want Z is what you guys figured out for me we're going to need a little math for this so let's import math always a sign that you're about to have some fun so we're gonna call Z a complex number whose real part is the cosine well I can't write 30 degrees because this will all be in radians so I'll write it as PI sixths that's the same thing as 30 degrees and then the Y is going to be the sine of that sine of math PI over 6 great so if we look at its components the real part is around 0.866 and the imaginary part should be exactly 1/2 but you know often you get a little numerical error because computers they can't get all the real numbers perfectly there's just not enough information in the universe so they have to have a little bit of an error there some of you might recognize that that first part is the square root of 3 divided by 2 this is everything we were working out last lecture for computing things like the cosine of PI sixths now what this means for us is if we take Z times foot right or remember foot was that one negative 3z times foot should tell us where it goes when you just go through the mechanistic operation of complex multiplication this point should tell us where it lands and violently it's supposed to land at around 2.30 real-part and then negative 2.09 as the imaginary part and indeed when we play that out and we follow our little foot point here where does it land now all right it looks like the real part is a little above 2 and the imaginary part is a little less than negative 2 and any deviation there is probably because the foot wasn't exactly at that point 1 negative 3 like I said but you can see what I mean in principle you tell the computer to do this for every single point in a given image that you want and it will tell you where the outputs should be it'll tell you where the rotated versions should be so complex numbers are certainly as real as real numbers in that respect it looks like we've got a question from the audience so let's go ahead and pull that up seal programmer asks what is the biggest reason that people use vectors and matrices to represent rotation instead of complex numbers that is actually a very interesting question I don't know if there's a great reason actually because if all you're gonna do is rotate and then potentially scale things you could have everything defined in terms of complex numbers in 2d graphics in 3d graphics that actually wouldn't work complex numbers only describe rotations in two dimensions however if you want to use a number system to describe rotations in three dimensions instead of using matrices you could use these things called quaternions which are basically the complex numbers on steroids and in fact they expose you to fewer errors like there's errors that come about when you try to use matrices for 3d rotation this thing called gimbal lock talk to any robot assist or any computer graphics programmers they'll tell you that gimbal lock is a real pain and they're quaternion this sort of complex numbers on steroids helps to solve that problem there's a whole other contingent of people who feel passionate about something called geometric algebra that is a whole other topic potentially for another day but that's that's a really valid question you've got these two perspectives on one one possible operation but if we're just coming at it from the idea that complex numbers play a role in other parts of math and in other parts of engineering I think thinking of the rotation implications really helps us see that and this is particularly visceral for me when I think in terms of trig identities so if you remember at the very beginning I opened up by talking about one particular trig identity and without pulling it up again I don't want to go back and just read off of it I'm going to ask you one more question which this is going to be a tricky question this one will actually take some time to work through words that I want you to calculate for me the cosine of 75 degrees okay and let's go ahead and pull this up as a quiz question so let's pull up our quiz let's move on to the next question let's cross our fingers and hope the cam and either have not broken anything again and that's all it's asking what is the cosine of 75 degrees okay now let me get you started here on how you might think about it 75 degrees it's not gonna be based on any triangles that we might have seen maybe you've seen a 75 degree triangle and you happen to remember what its side lengths were but most people won't have however what you can do is note that 75 is the same as taking 45 degrees and then walking another 30 degrees around okay 45 plus 30 degrees and if I question is asking us for the cosine of 75 one way to think of it is if you first walk 45 degrees around and then you walk another 30 degrees around what is the x coordinate of that corresponding point okay so this is going to have some coordinates XY the cosine of 75 degrees is asking for the x coordinate of that now like I said I think this is this is not easy okay so it's gonna take some time to work out and while you guys are working this out I want you to actually pull out some pencil and paper and think it through I'm just gonna pull up a little pause and ponder music and give us I don't know I might give us a full 60 seconds or so to like noodle this one out and think it through and in the meantime if you want to ask any questions you know how to do so through Twitter it's with a hashtag lock down math and I might go ahead and take a couple questions while people try working this out by hand which again I heavily encourage you to do even if you're not watching this live even if you're watching this at a later date this is the time to say okay I'm gonna pull out my paper I'm going to try to think this through and and now I'll stop talking I'll let you just think certainly don't feel rushed by the way even if there are some answers rolling in right now that is by no means in the indication that you should have finished or anything like that so interestingly we do have a strong lead but it's not at all a strong second place so it seems like the the way that the assuming that the lead is correct the way that the error errors have happened is not in a consistent way which is interesting and that'll be that'll be fun to dissect once we reveal what the answers are let's them let me check if there's any new questions that have rolled in no ah which suggests stew me that the two folks that I have behind the scenes who were supposed to feed me questions are probably desperately wrangling with something item pool related those of you who have already submitted an answer or if we think you already know the answer by the way I want you to take a moment to seriously guess what you think the second most common answer is going to be and where exactly the misconception could have taken place or to double check your work you very well might be one of those sitting in the the misconception territory which you know it's it's a lot to work out there's a lot of different failure points along the way all right so with that I'm gonna go ahead and lock in the answers and if you wanted to still work it through even if the answer isn't going to be recorded that's totally fine we're going to walk through it in a moment but I'm gonna lock them in just so that we can see what the correct answer turns out to be and the correct answer is D which it looks like a majority of you got so that's awesome the second most common was C which really only differs by a minus sign and I can I can actually guess why that's true already so that's that's interesting after that is B where there's a square root of 3 instead of a square root of 2 okay so we have we actually have a pretty good distribution on this I think that this shows that this is not at all an obvious question and can take take a little bit of noodling to figure out so if I saw this on a test okay let's say that someone threw like a high school math test at me right now and it said cosine of 75 you know I would know that in principle I'm supposed to be thinking of this as in terms of friendlier angles I know that 45 degrees is a friendly angle for trigonometry and I know the 30 degrees is what I don't have memorized though are the addition formulas I learned them once slipped out of my head definitely don't have them memorized but I do know how to read arrive them on the fly basically the idea is that we take the complex number associated with a 45 degree rotation which is going to be cosine of 45 degrees plus I times the sine of 45 degrees I take that complex number and I'm going to rotate the complex number corresponding to a 30 degree rotation so I might even just pull up yet another unit circle here and say the one that sits 30 degrees away I want to multiply it by this number well what is that number its x-component is the cosine of that so the real component is the cosine of 30 degrees and then the imaginary component okay the vertical component here is the sine of 30 degrees which looks like it's you know the smaller leg of a 30-60-90 triangle which is going to be one half but we'll work out the numbers in just a moment sine of 30 degrees times i and by the magic of complex multiplication which I hope isn't magical based on what we walked through a little bit earlier the effect that this will have is to rotate by one and then rotate by the other so what are these values cosine of 45 degrees is the square root of 2 divided by 2 sine of 45 degrees is the square root of 2 divided by 2 and if you want to if you don't remember that and you want to work it out the way to think about it is that a 45 degree angle is going to create an isosceles right triangle so if we don't know what this side length is we at least know that it's the same as the other side length so if the hypotenuse is 1 we know that x squared plus x squared is supposed to equal 1 that's the Pythagorean theorem that tells us the 2x squared equals 1 so x squared is the square root of 1/2 and then that's actually the same as writing the square root of 2/2 you could also write it as 1 divided by the square root of 2 if you want but these are these are equivalent forms so if you forget you just draw the triangle it's no big deal cosine of 30 degrees well if we have our 30-60-90 triangle the long side of that from everything we talked through in the last lecture is square root of 3 over 2 square root of 3 over 2 and then the sine of 30 degrees is 1/2 and now we just carry out the complex product it's not exactly a fun prospect maybe but it's really just for operations so it's not going to be too bad the real part is going to be our cosine times our cosine okay so that's going to end up being our square root of 2 over 2 multiplied by square root of 3 over 2 okay I'm also going to package in when we multiply both imaginary parts okay I guess I was a little inconsistent I wrote I times sine and then sine times I but it's the same thing so I'm gonna include that square root of 2 over 2 multiplied by the 1/2 but because each of them has an I we also multiply it by I squared which by definition is negative 1 so we get this negative sign in here and that's going to be the real part okay which is actually all we need we just need the real part and that gives us the that gives us the x coordinate of the where they drop before here we go if we rotate 45 degrees then we rotated another 30 the X component of that the real part is going to be the final answer so we'll do the sine part in a moment too when we make it more general but just for the sake of answering the question this ends up being root 2 times root 3 which is root 6 divided by 2 times 2 so root 6 over 4 minus and that minus sign came about because of the I squared minus square root of 2 over 4 ok so all in all that ends up being root 6 minus root 2 divided by 4 and if we pull up the question I think it's very interesting that the second most common answer the the most common mistake you might say is only different because of that plus sign and if we look at the math where that came from as it happens is the idea that this was a minus sign which came about from the idea that there was an I sitting in front of each sign so the most common misconception came from maybe forgetting to apply the fact that I squared is equal to negative 1 at least if you were going about it using complex numbers which from the whole context of the lecture of course we want to so this actually corresponds to a much more general formula and I think it's edifying to write out that general formula for ourselves because anytime that I've forgotten what the sum of two angles in trigonometry should be it's totally ok because you can read arrive it relatively quickly if you're comfortable with complex multiplication and the more math that you do because it shows up all over the place you actually get very comfortable with complex multiplication even if you like me become very rusty on things like trigonometric identities but with that comfort you can very quickly read arrive what the angles should be or what the sum some identities should be sign of data writing and talking at the same time so remember the way to think about this is that this first number is what we get by finding a complex number that's basically just rotating alpha around the unit circle if you rotate an angle of alpha around that's what this first number is the second number tells us what you would get if you rotate an angle beta around so let's say there's some other angle which I might write down here the whole angle from the x-axis would be how should I write it I'll write it as beta that doesn't really connote that it's the fall angle just thing I'll just I'll just do it as a separate a separate friend over here beta so I think like when I when I see these terms cosine alpha plus I sine alpha I think in my head very visually of a point on the unit circle in the complex plane similarly I think of a point on the unit circle in the complex plane and I know that multiplying them together means that the action of one of them is rotating the other so the final result we know is going to be the point that sits alpha plus beta around alpha plus beta plus I times the sine of alpha plus beta wonderful but on the other hand if you go through the mechanistic foil operation you will get an alternate expression for the answer and by having an alternate expression you have an identity which i think is pretty cool it's a sign that when you have a non-obvious relationship like the algebra of complex multiplication and the geometry of complex multiplication you can leverage that non-obvious relationship to get yourself real identities okay so how does this work the real part is going to consist of multiplying those cosines so we have cosine alpha times cosine of beta and then it also consists of the two imaginary parts multiplied together because that I squared becomes real I times sine I'd say I won't write anything here it'll be that sine of alpha and then the sine of beta and because we have an I squared we can consider that a negative one okay so it's cosine alpha cosine beta minus it sort of accidentally started writing a plus so we're just going to live with that minus sine of alpha plus sine of theta sine of alpha times sine of beta okay obviously it's a lot of algebra but it is satisfying the idea that you don't need to memorize this and again when you do this regularly when you do complex multiplication regularly you can kind of see this pop out we're like oh yeah it's the the real put the two real parts - the two imaginary parts that as a pattern is something you start to recognize and that when you want the imaginary part of the product that you take the other two like in this case it's gonna be I times sine times cosine so the sine of alpha times the cosine of beta plus the cosine of alpha times the sine of theta the cosine of alpha times the sine of beta long complicated formula all of that times I this long complicated formula is giving you the angle on some identities and sometimes you can write this out in a much more compact form this this is not a commonly used notation and I'll explain why in a moment but sometimes cosine plus I sine of something comes about commonly enough it has this meaning on the unit circle that it's written as C is okay in this case C is of alpha so everything that we just wrote down could be written more compactly as CIS of alpha times CIS of beta is equal to C is of alpha plus beta okay this is communicating the very non-trivial fact that complex multiplication lets us you know lets us work out rotations in the way that we want to again that's not standard notation and I'll explain why there's a much more standard way to write these but it's weird so I don't want to just kind of throw it down there but before we do that I want to illustrate how you can do a little bit of magic when you have this proper intuition for complex numbers and and how they multiply so let me pull up our quiz again and let's move on to another question and this question asks which of the following values of Z satisfies Z squared equals I okay so there's two ways that you can think about this I'll talk about them both and I'm just going to give you a moment to kind of think this one through and while you work that out let me go ahead and take a question from the audience Achmed Osama asks what is an application that is impossible to achieve without complex numbers convince me that they are needed they're fun to work with maybe but they made things easier but can we do without them I would say I believe yeah you can absolutely work without them because you can basically use rotation matrices in place of complex numbers so it's nice about complex numbers is that when you're adding things and when you're multiplying things you're dealing with the same type they're both complex numbers but if you wanted to describe this all with matrices and vectors and such when you're adding stuff you'd be adding vectors but then when you want to do rotation type actions and stretching type actions you'd be transferring to instead dealing with matrices so that's fine you just have to keep that jumbled and use them differently I'd be willing to bet pretty much anything you can do with complex numbers you could do there it'll become much less elegant and I guess one of the one of the main things that would be just really awkward is the fundamental theorem of algebra which we'll talk about in a moment but it basically says anytime you have a polynomial of degree n so the highest term you see is something like X to the N among the complex numbers it has n different solutions so that's very elegant it's very nice to work with it definitely makes the lives easier of anyone who's working with things where polynomials come about to include matrices and things like I don't want to like throw around terminology but when you're computing eigenvalues and such you find yourself with polynomials often of high degree and it's nice to have solutions to them I can conceive of constructing some kind of way to work with the numbers where you have like solutions to all of these where the solutions are expressed in terms of vectors and matrices and so maybe the strict answer to your question is you could do without them you definitely would not want to life for mathematicians and quantum physicists and really anyone who works with like linear algebra and polynomials would just become incredibly awkward so in in some very real sense they are a natural construct so much so that even if you could do without them conceivably you never would want to it would really make things quite the pain so coming back to our quiz here it looks like again we've got a lot of strong consensus which is good and let me go ahead and create this alright so the correct answer is a it's root 2 over 2 plus root 2 over 2 I it looks like the majority of you got that and ok so very interesting is that the second-most common answer was D that no solution exists without extending beyond the complex numbers because the question we're asking in this context is basically to take a square root of I so if you think of oh I is defined to be the square root of negative 1 I think a common question is to say you know how deep does this rabbit hole go okay if we had to make up an answer to the question square root of negative 1 do we have to make up another one for the square root of I and the square root of that and just kind of keep extending and extending and it's interesting and surprising that once you have I you have pretty much everything else that you want at least when it comes to solving anything that looks like a polynomial ok so the answer here there's two ways to think about it one of them is the brute force way and then one of them is the more elegant way so the brute force way looks like this you're reading through the multiple choices and you just kind of try them out and in this case when you try out the square root of 2 over 2 plus square root of 2 over 2 times I and you square it so we're going to multiply it by itself I'm going to actually do this just to illustrate that it's possible and that it's not super fun taking root 2 times root 2 for that first real part is going to give us 2 divided by 2 times 2 so 2 over 4 the real part will also include the products of the imaginary parts because of that I squared so we're going to have root 2 over 2 times root 2 over 2 which like we just saw well excuse me that was supposed to be 2 over 4 you can tell them a little all over the place today which you know just because sometimes complex numbers get me excited root 2 over 2 times root 2 over 2 is going to be another 2 over 4 but this time it's times I times I so we're going to have a negative in there yeah that's going to be the full real part and then the imaginary part is going to come from the cross terms something where there was only one I included so to over to I times root 2 over 2 it's the same product we're doing over and over it's always 2 over 4 and then we're gonna add 2 root 2 over 2 times root 2 over 2 I so we're gonna add the same thing and I'm not writing the eyes here because I'm gonna factor it out okay so the real part involves subtracting these the imaginary part involves adding them and indeed when we subtract them that real part is 0 and when we add them that imaginary part is 1 so we just get 0 plus I or I and okay it works but imagine that you looked at this question and it wasn't multiple choice right someone asks you to find a square root of I if the only option you have is this terrible brute forcing it seems like a very painful process right to try every number try squaring it if this is what you have to do but in an insane trial and error it would seem like magic if someone just pulled that answer out of a hat but if you're thinking about it geometrically it's actually a pretty straightforward question to ask because what you're saying is we want a number such that Z squared is sitting where I is right it's sitting one unit above well if we know the rotation corresponds to complex multiplication and that means what we want is a number that rotates by 45 degrees because if you have a number that rotates by 45 degrees then when you multiply by it twice that's the same as rotating 90 degrees so it gets you up to I all right well what number is that we could think of and in terms of sines and cosines if we want you know this is going to be cosine of 45 degrees this vertical part is going to be the sine of 45 degrees and it's root 2 over 2 for each of them this is something we were working out just by looking at the the isosceles right triangle each one of these is equal to root 2 over 2 so that's very fun simply by thinking of the geometry you could pull out of your hat the fact that the square root of I where I should say a square root of I is root 2 over 2 plus root 2 over 2 I another answer would have been what happens when you rotate not 45 degrees but if you were to rotate 40 45 plus another 180 degrees okay so 45 plus 180 actually gets us 225 degrees all the way around that's another number where if you multiply by it twice it'll have the effect of rotating you a grand total of 360 plus 90 degrees it'll kind of rotate you all the way around and then an extra 90 so in the same way that most positive numbers have two square roots you know square root of 25 is 5 but it's also negative 5 and in the same way that even negative numbers like negative one has two square roots not only is it I but negative I also satisfies this property that when you square it you get negative one even i itself has two square roots and in general everything except for zero has two square roots zero you might consider to have like a double root but again that's sort of a story for another time so that's pretty magical and now let me show you another bit of magic okay something where again you can come at a question that seems like it's going to involve just insane trial and error but if you're thinking about the geometry of it you're able to come to a pretty a pretty meaningful answer relatively quickly alright what's your question here the equation X cubed equals one has one real number solution x equals one makes sense you keep one you get one among the complex numbers there are two more solutions which of the following is one of them okay so this time it's actually asking us to cube some kind of complex number or to verify that when you cube it you're going to get one we'll give you some some time to sort of think about that and in the meantime let me take a sip of water and then take the question from the audience what do we have you showed that in order to find the cosine of a sum of two angles you can multiply two complex numbers does that mean that if I need the cosine of a subtraction of two angles I need to divide two complex numbers wonderful question yes yes it does mean that and dividing by a complex number is actually the same as multiplying by one in the same way that dividing by two is the same as multiplying by 1/2 you basically say 2 has the action of stretching the number line by 2 1/2 as a multiplier has the action of squishing everything down by a factor of 2 that's kind of the inverse action when it comes to complex numbers 2 divided by it you ask what's the inverse action so an example here might be let's say let's pull up something here let's say I wanted to rotate you know negative alpha that's what I want to do so I'm going to do like cosine of beta minus alpha or something like that that number could be considered the inverse the multiplicative inverse of what you get by rotating alpha the other way so up here we have I'm just going to write it as CIS 4 cosine plus I times the sine of alpha the inverse number over here is CIS of negative alpha and that number is 1 divided by this I could even write it out over here to say C is the number that's alpha the way a long and unit circle is equal to 1 divided by CIS of negative alpha and vice versa if I took one divided by CIS of alpha this is equal to C is of negative alpha now everyone who knows Euler's formula is kind of screaming at me that there's an easy way to see this in terms of Exponential's which is of course where we're getting with this but because it's weird this connection between complex numbers and Exponential's I don't want to just throw that down right away I do want to emphasize that this is this should all make sense before anything like Oilers formula enters the picture if you think of CIS as being what basically a function to ask where do you end up if you walk alpha units around the unit circle in the complex plane it's the Opera it's the number that you can use for for rotating things by alpha degrees so dividing by it will also get you rotating the other way but the overall answer there is the division is the same as multiplication so that's absolutely the right instinct you can derive for yourself the angle what would you call them the trigonometric difference formulas if you wanted to great so I think that's probably enough time if we punch back to our quiz and let's tone down the pause and ponder music here it seems like answers have sort of stopped rolling in so this is as good a time as any to grade things so the correct answer here of the four options that we were given ends up being drumroll please negative one half plus the root three over two times I now I just want you to imagine for a moment that if you didn't know any of the geometry of complex numbers someone asks you this question and if you were to able to spit it back decently quickly it makes you look like a machine right because you would imagine that the only way to answer this is to look at each number and then cube it that you have to go through and do the whole first inside/outside last but twice because you're cubing it for each one of those and then deduce that answer B turns out to have this property that we want geometrically there's a very nice way to see it so let's go ahead and pull it up what we want is an action so that when you do it three times you end up back at the back at one basically I want if I were to multiply by this number three times I end up back at once let me clear for myself nice new piece of paper here so just remind ourselves what we're looking for which is X cubed is equal to one okay so one such action is the do-nothing action right if you multiply by one which does nothing ya do that three times you end up back at one that is our real solution the other ones would be to rotate a third of the way around the circle okay to rotate 120 degrees or you could rotate 120 degrees the which is equivalent to thinking of rotating 240 degrees around those are other actions where if you do them three times it's the same as doing nothing if I rotate 120 degrees then do that three times I cube it that's the same as doing nothing that's how you can read this equation which I think is very nice and elegant rather than just trying to plug in numbers and see what comes out so using the trig that we learned about last time this again comes from a 30-60-90 triangle you can see that the real part is negative one half and then the imaginary part is the square root of three over two so even before working it out I can know with confidence that if I were to go through the painstaking process of taking negative one half plus the square root of three over two and cubing that I'm going to get back one I don't have to go through that process to know that that's going to happen which is pretty magical this also answers for us a mystery that we had in the last lecture if you guys remember what we did is we pulled up decimals so let's go on back let's go on over to Chrome let's open up a desmos tab and let's get rid of whatever else was going on in here and what we played with was the idea that if you take a cosine graph and then if you square it the shapes look remarkably similar you have this bizarre relationship we're squaring cosine looks like squishing it and more specifically we were able to play around and say if you double the frequency of that cosine wave and then you manipulate it in some other ways we're going to shift it up and then we divide it by two this is all something we did last time we got an identity between them but this raised an interesting question what on earth does cosine of 2x have to do with cosine of x squared well now with complex numbers we can actually know the answer we can work this out we know that if you rotate around by theta degrees okay if we call that number Z then when we rotate around another theta degrees this new number is Z squared right and Z is what I was calling CIS of theta which we could expand out as saying cosine of theta plus I sine of theta and we can work out that square nearly algebraically we take cosine of theta plus I sine of theta the real part of that is going to come from two products of the real part so it's going to be cosine squared of theta but then also two parts of the imaginary and because of that I it becomes negative one so we'll end up with negative sine squared of theta and then the imaginary part is going to come from yeah I might as well just write this out in full cosine theta plus I sine I realize I'm kind of running against the edge which is why I was hesitant to do this but basically that cosine squared from comes from taking the cosine times the cosine the negative sine squared comes from taking I sine times negative I sine our remaining terms are going to be that cosine plus I sine so cosine theta times sine of theta and then there's going to be an I I'll just write it out times I and then we get exactly the same thing sine of theta times cosine of theta times I so sine of theta times cosine of theta times I which we could factor out this whole last part ends up just being two times the cosine of theta times the sine of theta all right all of that times I so what we've done here is basically said there's two ways to think about cosine theta plus I sine theta squared one of them is oh of course it doubles the angle and so that's going to be cosine of two theta on the other hand algebraically it's equal to deep breath cosine squared theta minus sine squared theta plus two cosine theta sine theta I now that might seem like a big pain but the big pain is showing us an identity it's showing us the double angle identity so what we found here just to summarize on the real part is that the cosine of two theta the real part of what happens when we rotate by that is equal to cosine squared of theta minus sine squared of theta now from another trig identity that we were working with last time sine squared of theta is actually the same as taking 1 minus cosine squared this was the alternate way of expressing the Pythagorean theorem so we can get everything written in terms of cosines and what we have is now 2 times cosine pardon the messiness of the handwriting here by the way 2 times cosine of theta all minus one that is evidently equal to the cosine of 2 theta and we could rearrange that further and see what we were playing around with in desmos that this cosine of 2x expression over here this cosine of 2x expression corresponds with squaring the cosine this was a reflection of the fact the cosine relates two complex numbers which i think is pretty cool yeah and is the very last thing that I want to mention I want to talk about why we didn't why nobody really uses the CIS notation and I was saying that CIS of alpha is not really standard notation when we're talking about rotating alpha around the unit circle as a reminder this was my shorthand for cosine plus I sine the reason is because what you're always going to see instead through electrical engineering through quantum mechanics through math through all sorts of things is instead the way that this is written is actually before I show you how it's written no no I'll just I'll just show you how it's written it's it's it's weird and we're going to talk about it at length the next time but you imagine that you're raising something to a power okay e to the I times alpha now this obviously makes a lot of people very uncomfortable first of all if you don't know about e what is e and then secondly what on earth does it mean to raise something to an imaginary power so that is a full lectures worth of material I do want to talk about it next time but just to to give an indication of why this might not be a totally crazy thing okay I want you to remember the core property that we wrote down earlier that when you rotate alpha around the unit circle and then when you rotate another beta around the unit circle this is the same thing as alpha plus beta which means our CIS function has this kind of interesting property that adding the inputs is the same as multiplying the outputs so with that I want to ask one final quiz question to try to relate this to other functions that you might be familiar with at this point it might be halfway obvious where we're going with this but is the very final question to cue you up for where we're going to go next time and maybe appreciate the relationship at play here and why it's not that crazy I want you to answer which of the following functions satisfies F of a plus B is equal to F of a times F of B okay I'm being told by the way that um when I advanced the questions too quickly it kind of reveals the answer of the next one which happened all throughout last lecture and you guys certainly told me about it again this is one of those things we're like this is what happens with live production we thought we had fixed that usually you know this is a very weird process for me because usually when I make videos you catch some kind of error so you edit out the error then you put out the video this is an experience for me is much more like alright world here it is raw here are the errors as they come I hope you enjoy watching them happen because there's no editing it does not get clipped out this is math presentation raw all right so as we do this it looks like pretty pretty strong consensus I'll give you a little bit more time to submit the answers all right and with that let's go ahead and take a look so the question is what function has this property we're adding the inputs looks like multiplying by the outputs so 2700 of you correctly answered the two to the X an exponential function behaves this way looks like the most common second answer was a that logarithms which makes sense there's an identity in logarithms that looks very similar it's basically the opposite so if we were to you know write this out I might say 2 to the a times 2 to the B is equal to 2 to the a plus B and when we're thinking in terms of repeated multiplication this makes a lot of sense because one of these is saying multiply in a copies of two then multiply in B copies of 2 and the result should be all a plus B copies of two the rule with logarithms is exactly the opposite and it's because logarithms are the inverse of multiplication on that front it looks like taking the log of a plus B ends up being the log of a sorry the log of a times B multiplied log of a plus log of B logarithms is another thing I think we're going to talk about in a later lecture of this series so if you don't know about them don't worry but to those of you who answered that totally understandable it's because it's a very similar property but it's just getting mixed up whether the outputs are being added or the inputs are now you'll notice is how strange this factor that comes about because 2 to the X is defined as repeated multiplication looks very similar to the rule that we found for complex numbers that sit on the unit circle that multiplying the outputs is the same as adding the inputs and that should be very suggestive that there might be some kind of unifying theory between them which of course there is it has everything to do with why we write things as e to the I theta which I actually think is kind of a bad convention the good thing about it is that it does make it pretty obvious when you take e to the I theta times e to the I beta sorry e to the I alpha that you're allowed to just add them it really highlights that fact which otherwise you kind of have to think through but maybe that's a good thing that you kind of have to think it through and you're not just blindly applying rules but I'm gonna talk all about that next time we're gonna dig deep into Oilers formula a way that you can try to understand what is this really saying like computationally it's not talking about repeated multiplication so what could you do if you are on a desert island with a bunch of paper to just work out by hand the fact that this is a reasonable expression all of that is to come next time so thank you for joining apologies for being mildly scattered throughout

Channel: 3Blue1Brown

Views: 987,384

Rating: 4.9210706 out of 5

Keywords:

Id: 5PcpBw5Hbwo

Length: 82min 11sec (4931 seconds)

Published: Fri Apr 24 2020

Reddit Comments

[u/commander_nice]

This is one of those topics I wish teachers would have gone more in depth with in high school. They introduce this new number i and don't explicitly say what you can do with it or what justifies considering a number whose square is -1. Add to that they don't explain any practical applications. What a way to kill any possibility that a student will learn to love math.

I would have appreciated at least a mention of a construction of the complex numbers as ordered pairs of real numbers. Define addition, multiplication, and a mapping from the reals to the complex numbers in a certain way and then you can prove all those properties of the complex numbers you implicitly use in high school. At least then I know exactly what i is.

I hope he goes in that direction but not counting on it. He only has an hour. He might explore some geometric properties instead, judging by the thumbnail.

[u/ITagEveryone]

I really wish Grant was releasing these videos back when I was in high school.

[u/botechga]

Wish i wasnt busy atm but can't wait to watch this later!

[u/NewtonMetre]

I would like to learn why imaginary numbers are, by definition, orthogonal to the real numbers. I mean, you force them to form pi/2 rad with the reals and all the rest, rotations, polar form and so on, are a consequence of that. Many things in mathematics are mutable, e.g. you may allow parallel lines to intersect, but this one is not. I'm probably misunderstanding something that it's obvious, but for me it's something intriguing.

[u/[deleted]]

Enjoyed watching

Waiting for 2nd part