We don't. There is only one unit circle. Your unit circle is actually the same circle as my unit circle. If that were not the case then mathematics would be different for the two of us. Which would not work out very well. But who said circles exist anyway? And what is that unit thing? Can you point out either of those things i the real world?
I love how he doesn't just ignore the minority of people that get the wrong answer even if they are very few. Instead he tries to understand why they got the wrong answer and what was could've been their thought process while answering and then he points out where the thought process went wrong and then gives the right idea to think about it. It's just lovely how great of a teacher he is. ❤
Grant, I struggled hard with trig in school. It discouraged me so badly that I had left it as something I wouldn't understand and so I never moved on to higher math. Your lectures in this video and the last, and following along with the test questions, not only made me realize how beautiful and interesting trigonometry is, but also rekindled a long-forgotten love for math and confidence in my ability to do it. Thank you, thank you, for making these videos.
learning is all about resources! so amazing how teaching math in a way that prioritizes actual understanding leads to actual understanding... great to hear you're enjoying math once again
About conventions i or j: In electric engineering the imaginary numbers are normally represented by “j”, instead of “i”. The reason is that the letter “i” is already used to represent current.
Also Python complex literals (which exist; the floating-point properties are as a separate real and imaginary part each one being a double) also use a suffixed j.
@@sober4769 It is a lot more common that an electrical engineer uses current instead of current density, in their calculations. It if does end up mattering, capital J would be current density, and lowercase j would be the imaginary unit. Current density is for the physics behind a lot of electrical components and their theory of operation, but it is rare that you deal with the continuum mechanics of electricity as an electrical engineer. Current itself infinitely more common, than current density, for electrical engineers.
@@carultch I've learned that u (or v), j, and i are used lower case when talking about alternating current and upper case when talking about direct current.
@@topilinkala1594 I can't say that I'm familiar with that convention, since I've always used capital I and V for electrical calcs. I had wondered why they needed to use j instead of i, if capital I would stand for current anyway, and thought maybe it is just to avoid confusion when talking about the equation aloud.
Regarding 1:04:56 : One my electrical engineering professors said that if mathematicians hadn't come up with complex numbers, electrical engineers would have. Dealing with electrical circuits that involve capacitors, inductors (and alternating currents) without complex numbers is very difficult, having to deal with differential equations and trig identities, but if you interpret inductors & capacitors like resistors, but with an imaginary resistance, you get an incredibly beautiful and simple way to work with them. In general, there is pretty much no area of electrical engineering that does not benefit greatly from using complex numbers. Especially everything involving AC.
@@cubing7276 electrical signals are sine waves or can always be expressed as sums of sine waves, see the videos on fourier transformations. So you can express the signal at any point in time with amplitude and phase angle, which is extremely convenient to do as a complex number. You can think of a hand/needle/pointer/*phasor* that's spinning around in circles as time goes on. When the resistance is 1, voltage and current always have the same value at any point in time. If it's 2, the voltage will be 2x the current. So a non-imaginary resistance simply scales this complex signal. When you have a capacitor or inductor, the peaks of voltage and current are no longer at the same time, they're out of phase by (ideally) 90 degrees. So we just multiply our phasor by i, and there we go. Of course, any real-world part like a wire has a resistance, capacitance and inductance, and we can use complex numbers to describe this. hope this helps, I'm not studying this in English, so there might be some errors in translating to the correct technical terms.
Thank you! I won't need it because I watched the stream, but this will help a lot of people. You might want to change 61:45 and under to 1:01:45 so the links actually work though, but that must've taken a long time.
I converted the bottom half to hours: 1:01:45 Q7: Prompt 1:03:00 Ask: Can we do without complex numbers? 1:05:10 Q7: Results 1:05:40 Q7: Solution 1:10:10 Q8: Prompt 1:10:50 Ask: sum/difference of angles 1:13:40 Q8: Results 1:14:50 Q8: Solution 1:16:10 Desmos Example 1:20:20 Bringing it all together 1:20:50 The cis shorthand explained 1:22:00 Q9: Prompt 1:23:45 Q9: Results (1:05:40 Closing Remarks)?
@@RodelIturalde That would only be true if there were an infinite number of length units, which would be incredibly inconvenient. Also, it isn't necessarily length units that define circles in general. Mohr's circle for instance, has units of psi or Pascals, and the radius represents the physical quantity of maximum shear stress within the plane.
I also liked last time when he actually said a pretty interresting fact: squaring it gives 4761, and cubing it gives 328509, together they contain all digits from 0 to 9 exactly once (apparently its the smallest number with this property). A bit base 10-y tho but still. Funny when he went "because it has that property, thats probably why you submitted this"
Working out cos(75) geometrically instead of plugging it into a calculator just singlehandedly allowed me to finally grasp quaternions These streams are incredible
This was an absolutely fantastic lesson! The explanations are so simple and elegant. Thank you very much for your effort. Can't wait for the next livestream!
I worked on the graphics engine for the space shuttle. And we used quaternions a lot. Mostly because we didn't have to worry about gimbal lock, they are much faster than matrix multiplication and make relative rotations more intuitive.
I actually like the name imaginary. It makes them sound whimsical and interesting. The moment I heard of them I wanted to know more. On the other hand I think complex is a poor naming choice. It makes them sound complicated or hard to understand, which they really aren't. I'd prefer "compound" or "combined" numbers in that regard.
I think people hear imaginary and then their next thought is "doesn't exist,no application, how can this help me, they don't matter so I don't care" or they get confused on either how a number can be imaginary, or that all numbers are "made up and thus imaginary", a lot of ways hearing imaginary number can go wrong.
All these teachers say terrible name for complex numbers or imaginary numbers. I also think it is interesting. I found it more interesting than real numbers or rational numbers!
I find it fascinating that the most intelligent people on the planet (doctors, lawyers, Grant, etc.) usually have terrible penmanship. Maybe it’s because they all have bigger things to worry about and think about…who knows? But Grant’s handwriting/printing looks like it was written by a 8 yr old.
Agreed. Whenever I make a mistake I immediately jump to the conclusion that I'm not good enough, that all my grades were luck and that I'm going to fail but seeing someone who is really good at maths make mistakes gives me more faith in myself
Grant, I've gotta say. What sets you apart from nearly every math teacher I've ever met is your presentation and humility. Despite the fact that you're unbelievably smart, you know exactly what kinds of logical questions that we who aren't as knowledgeable will ask. It not only makes us feel known and understood, but it significantly boosts your credibility and enhances your teaching. We're not simply taking things at face value because we know that the person teaching us has thought through things the same way we are and can address our concerns. Thank you.
Honestly, can we do this format even after the pandemic? Maybe just once a month or twice a month, I find out that your vids about these fundamentals of math have taught me a lot more than what I learnt in university for entire 4 years. (I'm studying Mechatronics engineering, and the more I learnt, the more I realize the power of mathematics, but it's kind of too late/too hard for me to build my base mathematical knowledge now.) Anw, really good content, take care and keep up the good works.
Imaginary numbers was a derogatory term that Descartes used that, unfortunately, has stuck hundreds of years later. Ironic that the father of analytic geometry thought that complex numbers were nonsense.
@@MrBorderlands123 Sticking with derogatory terms for the long haul happens often in maths and science. Big Bang cosmology was originally called that by detractors, who wanted to point out how absurd the idea is. Well, absurd or not, it appears to be factual. Or "climate change". They originally wanted to call that the "climate catastrophe", but some of the scientists involved with that felt that was too alarmist.
56:45 Good Lord, that's the defining property of exponential functions! Suddenly, I see the link between the two concepts. The click in my head was audible! Thank you, Grant, you're an awesome teacher.
Multiplying complex numbers makes sense. Thinking complex numbers as vectors make sense. But combining both are making things confusing. As multiplying complex seem like transformation which seems like 2*2 matrix. Multiplication with vector. So, we are considering one complex number as matrix and one as vector. But if we see just at multiplication terms it seems like we are doing matrix and vector multiplication.
Same. Completed full undergrad engineering curriculum. Just now learning that adding rotations equates to multiplying complex numbers. What a joke. Grant, you're a treasure.
@@kdawg3484 True which indicates how much of a bubble the current type of business aka universities are. Full of useless pHDs that can't teach (probably because they never understood it deeply either if they had same type of teachers? who knows), but yet you manage to get a bachelor and you seem like a scientist to the average popualtion :D .
Lectures like these make me feel that all fields in math are connected very fundamentally, but since we learn then in different chapters, we just don't see it.
Thank you for this. Several of your videos use complex numbers to explore some concept or another, and there's always this assumption that the viewer is familiar with them (never stopped me from watching them anyway). Having never formally learned about them, I had this vague idea of what they were, but now I actually feel like I know enough that I can go back to those videos and get something new out of them. When I was trying to memorize the trig identities for my calc exam, I looked up methods to remember them, and the only thing I got was "rederive them using complex numbers" which was very unhelpful, since I had no idea that trigonometry had anything to do with complex numbers. It's cool to see how they actually connect in a really fundamental way, and where the identities come from, instead of the teacher going "here's an identity I just pulled out of my ass, now memorize it". I'm very excited to learn about how this all ties into exponential formulas and what Euler's identity is. Sorry for the long comment, I'm just really glad to finally actually learn what the hell is up with complex numbers after all these years of people talking about them without explaining them. You'd think they'd teach them (or at least mention them) earlier since they're so central to trigonometry and tie into a bunch of areas of math really elegantly. (for reference I just finished first year calculus)
Update: had a lecture on complex numbers in my mechanics I class (2nd year) because we needed to apply them and half the class hadn't learned about them yet. I'm noticing a pattern where my physics class applies a math concept before my math class teaches it, and it's very annoying (for example, vectors are used a lot in 1st year physics but not taught in math until 2nd year).
@@sarahp6512 that is indeed a common pattern in physics. Its kind of cool because physicists should have the intuition to use mathematics even if they don't understand the full rigour. Physics is also more rewarding when using math. I do agree that not teaching complex numbers by 2nd year uni is a big mistake from the math (?) department though.
Revisiting 3 years later, I am 100% convinced that in the whole world, math teachers in the school were completely incompetent. Most likely they never understood the math and didn't know how to teach the math subject. Majority of people went through hell. It took decades to appreciate math because of such beautiful math UA-cam channels and kind teacher/professor. 3b1b channel is a gem. It teaches us how to think, how reason. If math is not directly useful in your life, don't worry, at least you will learn how to think, logical reasoning. It's very useful skills to have in your daily life.
I've always been interested in complex numbers because of games, as in, 3D rotation relating to quaternion which themselves are the next step up from complex numbers and just thinking about it differently helps so much! I've really been enjoying these lectures you've been doing, I graduated from high-school 6 years ago now, and while I did try to go through university, it just didn't fit me, I ended up enjoying learning from math papers and just playing around to understand more than just listening to the lectures they gave, so I dropped after changing uni once, then having to validate my first year in 2 times, so really not for me. Now I still love math but programming school I found doesn't have lessons as the school is more of a "figure it out yourself" (School 42 in Paris) which I love, but it has it's draw backs when it comes to trying to find new was to solve problems, you too easily get stuck into what your brain is used to, so all those math videos are always a really good way to just think differently. I'm a bit slow so I often have to pause to do the math but I can pull through. I know this format is more meant because of the virus and lock downs and I can't be surprised if the logistic makes it a bit hard to do regularly when all of this calms down buuuuuuut I would really love even just one like those every month :D And while I'm here, big up to Ben for all the interactive tool! Been enjoying his videos for a few months now, quite something too!
Something you hear on Mathologer is "We've found that two completely different methods give us the same result. Do we just say 'Oh, that's nice'? Of course not, we try to find out why they are the same, what is the connection?"
ca. 22:10 I just figured it out. Each complex number is written as the sum of a horizontal vector and a vertical vector. (The real and imaginary components.) Each of these lines is just one of the unit vectors (1 or i or -1 or -i) multiplied by a constant, so it's easy to see how the fact that multiplying by i rotates these four numbers (by virtue of multiplying i by i being how i is defined) would extend to rotating the horizontal and verticle components of any complex number. If you rotate these components than you rotate the number (ad makes sense visually). (That felt like "figuring it out" to me because it explains how the symbolic math is related to the visual rotations and how it all stems from "i×i=-1".)
Rule #1 for presenters: Never criticize the AV people during the presentation. They KNOW when there is a problem and - you can be sure - they are FRANTICALLY trying to fix it. Chastising them publicly is just a jerk move. Much better to say to the audience: "Please have patience. My AV folks are really great, so there must be something really unexpected for this to happen."
@@kirenireves I think it pretty clear that he was just being funny and the AV people probably knew this. But thank you for the insight, I'll certainly keep that in mind for the future.
@@thasyashetty3797 He says it a few times, even encouraging the audience to give the AV folks a "stern word" over twitter. Then there is a tweet at 26:29 where someone compliments Ben Eater and says he doesn't "deserve the harassment" that Grant is throwing his way. (So it's not just me who noticed this.) Grant does *not* take that moment to say "I'm just joking...These guys are doing great." People who are presenting feel anxious when AV does not go smoothly and so they project it back on the AV folks so they are absolved of blame in the eye of the audience, but it just makes them look small. AV people get alot of abuse. Just try doing that job with an ungrateful set of presenters, and it'll drive you to drink. Thankless job.
After watching your series, its as if all the math I've learned up until now was crap!! Your way of geometrical interpretation of all problems isn't something which is taught or rather known by many teachers. Great thanks for making this awesome series. This has increased my love for math to a higher extent.
1:20:22 We are considering only the real part of doubling the angle, that is, cos(2Φ), but if we also take into account the imaginary part which is isin(2Φ) and compare it with the algebraic result, we get sin(2Φ)=2sinΦcosΦ ! Amazing how the familiar trig identities just pop out of complex numbers. Incredibly elegant!
I like to play around with 3d graphics and I've come across quaternions multiple times. They are actually used, math libraries designed for graphics or games always include quaternions as far as I know. That said quaternions are very often transformed into rotation matrices before heavy computations that involve rotations (like transforming millions of vertices in a modern 3d videogame) bacause GPUs are designed to be fast at matrix math. They can also be multiplied togheter before applying them to all of the vectors so you can do rotation, scale, non uniform scales, translations, projections(that involve the use of homogeneous coordinates in some cases) ecc. with a single matrix multiplication. Of course many of those things could be done with quaternions alone but I think that having hardware that can do matrix multiplication really fast is just more usefull than having hardware that can compute quaternions really fast.
59:52 It takes some heavy abstract algebra to prove, and I only mostly understand the proof, but algebraic closure of a field is unique, so anything that would require access to all solutions over polynomials in the reals must use complex numbers, or a construction that is isomorphic to them, or something more complicated that has a subfield isomorphic to it. So, the answer to this question is effectively no. Using vectors and matrices is an extension that allows skews in additon to rotation and scaling, but if you limit it to only matrices that are shape preserving, then you reduce exactly to the complex numbers.
Very reassuring how he says it's ok to not know things from time to time. As a 10th grader from a non english speaking country, the online math community can feel very confusing and intimidating. But I'm just here to learn, aren't I? No matter how long it takes.
This is first time I have ever commented on a UA-cam video, as I feel guilty that even after such an amazing lecture if I can't appreciate your effort to help the students to get better understanding on complex numbers. You videos are really amazing and wonderful sir. My thoughts about math has changed drastically because of your videos👏👏👏👏❤️❤️❤️
(at 19:12) Reacting to the level of viewer participation: "This is genuinely delightful!" And u just know he MEANS that... and THAT is why I love this channel!
I like Gauss's solution to renaming imaginary numbers. He suggested the term "lateral numbers", while real numbers would be called direct (+) numbers and inverse (-) numbers. Not entirely on-board with calling the negatives inverse numbers, because inverse more often means reciprocal, but I think lateral numbers gives a much better understanding about what they are.
@@Sushantgupta12 the rotational attribute is not really of i, if you really think about it, but of the operation. The problem is that we confuse symbols with operations and with the attributes of the numbers. Maybe we shouldn't talk about scaling, complex product, vector product, and all that as if they were the same operation (and often same symbol) with different kinds of numbers, but rather different operations with just groups regular numbers (where each number in a group might be orthogonal in some way to the others).
@@Sushantgupta12 Discussing tiny little facts with such a great community really brings up new radical thinking and analogy to simple things. That's the joy of it.
Infinity has to be real, the universe exists in space, we do not know if the universe is infinite or not, but we for sure know that space has no borders, no shape, no limit, it just is in all directions, and according to our current understanding, it's expanding and just creating more and more space.
Why is he so pretty?? Like he’s amazing at maths, and more than that great at making it feel intuitive, but then on top of all that he’s properly gorgeous. Some people have all the luck in the world
@16:18 Ha! You spilled your vodka on the paper! (can't be tea or coffee, they would stain the paper, and the water won't dissolve the typographic ink!)
Really sorry if what I am saying appears to be silly or nonsense, but I think maybe, just maybe, this could be somehow useful. I am just curious to know whether there exists some thing(say x) which has the following property of |x| = -1 As we know, |x| means the distance of "x" from the origin. Having a negative distance doesn't make much sense (at least in general cases) because that's how we have defined "distance"(not including direction to be precise). Maybe this has some feeling of having the "distance" b/w any two points being shorter than the straight line joining the two points.. This, as it seems, if it were to make any sense, would have to be in some dimensional space beyond 3D. But if we talk of 4D or 5D..., distances are still positive(like def. In linear algebra). So such a number(x) could not be a quaternion or anything similar So maybe fractal or "negative dimensions" can come in to help (or maybe complex dimensions, sorry if that is silly and being panglosian). (There are, as far as I know, many definitions of "dimension", some of which allow negative dimensions.) Another thing one could do is "define" "negative distance" such that it coincides with the usual notion of distance(I have some ideas about this). This comes from one of Grant's videos namely "What does it feel like to invent math?" A similar approach could be re-defining what origin and space is(where numbers live). Maybe, such a thing already exists and I just don't know about that. Any help is deeply appreciated!! Even if it turns out to be nonsense, I will get to know of some more interestings aspects of math... -Pragalbh Kumar Awasthi
We really need to get this nailed down once and for all: Numbers do not exists, and in that sense, all numbers are imaginary. Grant said it himself, what numbers are, are human constructs, a linguistic shorthand for communicating the size of a set. In the real world, only the actual physical entities of these sets are what exists. The numbers themselves do not. To suggest otherwise is platonic gibberish, to quote AntiCitizenX
Even as someone for whom high school math was 15 years ago, I still learned a lot from seeing this perspective connecting complex numbers to trig. Did you consider starting first with two dimensional numbers (without talking about imaginary numbers; eg “apples on the horizontal and oranges on the vertical”) and then show that i is the result of wanting to rotate in this plane. And only then that it turns out to be the sqrt(-1). And then wrap it back into the trig.
7:35 “when you have a problem that you can solve you can just say ‘Oh I’ve define things so that we now magically have a solution’ “. Then you become part of the history of mathematics...
Nope. You become a figure in textbooks or on calculators only when you show that these definitions have nice identities, interpretations, consistency, etc.
I can't believe how many times I've studied complex numbers, and I've never realized why i^2 NEEDS to be -1. I thought it was a lucky definition that somehow turned out fine... So... thank you. I can't possibly express how grateful I am to you, for this moment of true insight, where everything makes sense. Mathematics is a beautiful imaginary world, but the way you commit to education and teaching makes our real world beautiful. Thank you!!
Answer to Ahmad Osama 57:50 More than 80% of Electrical Engineering is based on Fourier Series and Transforms, without Fourier S/T todays communication systems , signal filters and many other countless things were impossible... And Fourier Transform is impossible without complex numbers...
It's also provable that any alternate system to the complex numbers that could do all these things must be mathematically identical, or have a subset that is mathematically identical.
He’s great, but he doesn’t need to amass a lot of private property to be happy. Sharing the love of math, I’m sure, is what makes life worth living for him.
I am a retired engineer (electrical and nuclear) and very much enjoy your lectures. You have an outstanding way of conveying understanding. Keep up the good work!
33:05 an addition to the i and j thing. In electrical engineering the i is reserved for a flowing current, hence the similar looking symbol j is used. (where i study in germany at least)
There is a notion of multiplying 2-dimensional vectors, it's basically what the complex numbers are about. You can avoid talking directly about "complex numbers" if all you want to do is manipulate geometric vectors and that is by defining multiplication on 2-d vectors. If you say the formula for that looks weird, then try it in polar coordinates.
This has been amazing as usual. I wonder why they don't teach this way at the University. I liked the lesson progression: if I started after minute 40 I wouldn't have been able to follow.
they totally teach this way at uni, it's a pretty standard lecture format it's just that at uni level mathematics you have to be a lot more rigorous and the concepts are a lot harder to grasp
@@hassanakhtar7874 far harder content, same style is what I'm getting at. High school is a lot more interactive than this style which is quite literally a lecture; just like in uni
5:40 I learned a neat mnemotechnic trick to learn sin(a +- b) and cos(a +- b) If you kinda know what the formula looks like but you can't remember where is + or - or cos(a)sin(b) or cos(a)cos(b) etc. Just remember Cosine is a racist and a liar -> It separates cos and sin. -> When you see a + it's a -. When you see a -, it's a + so for exemple : cos(a + b) = cos(a)cos(b) - sin(a)sin(b) I know some French people learned it this way as well.
I love how Beavis and Buttheads put 69 in the answers and Grant continues to come up with mathematical reasons why somebody would have thought of that as the answer
man the past few months i was in a bit of a slump when it came to anyhting including science and i was losing my interset in it becuase i started focusing more on school, i found this video in my recommendations and forced myself to click on it despite thinking "ah i don't wanna watch 1 hour of this" and maaan i never appreciated just how much a good teacher is needed to make you like something, i'm now interseting in math again, now everyone on twitter is going to be tired of me talking abtou math again
@@SreenikethanI , yeah that's correct, but what i'm trying to saying is that in this simple equation you can see how complex number can cancel each other, and this is the principle of the three-phase electric power.
The question about the non-real cube roots of 1 is always going to have a special place in my heart. I remember one time I was being difficult and pestering my parents, so my father gave me that question. He didn't tell me how many there were; he just told me to find them. I had only just learned about complex numbers, so I didn't know about the analog between multiplication and rotation, but that problem let me figure it out on my own. Also, I think we were hiking, so I didn't have any pencil or paper, and I had to visualize it in my head (but that's easier for me than algebra anyway). That was a really fun few hours. I didn't even realize until he told me years later that he was trying to get me to stop bothering him, but apparently giving me difficult math problems was a strategy he used to use. It certainly worked, and I got a few really fond memories out it.
I knew it was coming from the stream on Tuesday and was scheduled well in advace for completely different reasons. Nevertheless, today is my birthday and I'm definitely going to enjoy listening to Grant explain complex numbers again. Thanks for the stream!
THANK YOU ! The best explanation so far on quaternion = 3D rotation ! Even you just very very breifly talk about it, extrapolating from 2D rotation with complex number to 3D rotation with quaternion make finaly sense with this video !!!! Now I want to understand Gimbal lock in a first place.
The question you answered around 59:46 There's *also* ways to "get rid of" a lot of matrices by using extensions of complex numbers instead. Just use Geometric Algebra or Clifford Algebra. (Same thing, different names) There are good reasons why one would stick to matrices anyway I think, but a lot of things that are rather clunky and awkward using matrices work out extremely beautifully with Geometric Algebra.
Hey Grant, thanks a lot for doing these live streams. You have a gift, really. It's amazing how well you're able to explain math to me and many others. And don't worry about your handwriting ^^
Really struggled with the cos(75°) question. I was still in the mindset of the previous lesson and used the half angle identity and the answer didn't match the choices. (1/2)sqrt(2-sqrt(3)) is what I got and turns out to be right but algebraically expressed differently to the answer given.
1:03:23 I have to disagree here. There are integrals that can only be solved to a closed form by complex-integrating along a semicircular contour. Also the characteristic function of a probability distribution is generally more reliable and useful than the moment generating function especially its identities about the additivity of random variables
I was asking myself what hell am I doing watching a lecture about complex number at 1:30am when I finished this video, the answer was "because it is fun". I didn't even notice the time passing. Thanks for all the work, everything on this channel is just brilliant
this is my all time favorite thing to watch on youtube. i genuinely get excited to see that these have been posted an i get to watch them. thank you so much for this:)
2010: watching youtube in math class 🥱
2020: watching maths on youtube 🤩
Did you just use math and maths in the same sentence?
If you weren't a native speaker i wouldn't blame you, i myself can't make my mind on which accent to fellow so at the end i use Ameritish
In 2010 you were an American math student. Now, you're a British maths student. 😜
@@zenbum2654 things can change a lot in 10 years 😄
So my life
It’s nice to live in an age where 148,000 people will sit and watch a 1.5 hour math lecture patiently
1,4Million now
Amazing
you mean 1,544,213 right?
Hey geniuses, if infinity isn't real, how can he have an infinite supply of unit circles?
Check mate
@@kebien6020 1-0
A nice quote by Prof. E. J. Farell: "There are many infinities, and the one you're most likely thinking of is the smallest one."
We don't. There is only one unit circle. Your unit circle is actually the same circle as my unit circle. If that were not the case then mathematics would be different for the two of us. Which would not work out very well.
But who said circles exist anyway? And what is that unit thing? Can you point out either of those things i the real world?
@@heater5979 I'm replying to your 1 comment. Does that count as a "unit" in the real world?
I love how he doesn't just ignore the minority of people that get the wrong answer even if they are very few. Instead he tries to understand why they got the wrong answer and what was could've been their thought process while answering and then he points out where the thought process went wrong and then gives the right idea to think about it. It's just lovely how great of a teacher he is. ❤
He did ignore the people that answered 69 lol
@@raquelsanchez4129 I noticed that too
Yes
@@raquelsanchez4129 🤣🤣 people are hilarous.. I was laughing the whole time. It's a way to refresh
"Three things are infinite: the universe, human stupidity, and Grant's supply of unit circles; and I'm not sure about the universe."
- Albert Einstein
Unit circle time
I laughed, very hard.
*-Randy
" Newton lied about inventing calculus " _Einstein_ , _2030_
*supply of
Grant, I struggled hard with trig in school. It discouraged me so badly that I had left it as something I wouldn't understand and so I never moved on to higher math. Your lectures in this video and the last, and following along with the test questions, not only made me realize how beautiful and interesting trigonometry is, but also rekindled a long-forgotten love for math and confidence in my ability to do it. Thank you, thank you, for making these videos.
learning is all about resources! so amazing how teaching math in a way that prioritizes actual understanding leads to actual understanding... great to hear you're enjoying math once again
51:00 its 2am but this has made me go get a paper and calculate cos(75). That's how powerful this math series is
I did the same at 2 am as well.
I'm watching at 2am as well, but I didn't bother getting a paper and just did it in my head
Nearly 2 too😁
I did the exact same at 5 am bros
12:43 and I am blown away by the fact that doing it by hand is way, way easier in terms of complex analysis than the trigonometric formula.
About conventions i or j:
In electric engineering the imaginary numbers are normally represented by “j”, instead of “i”. The reason is that the letter “i” is already used to represent current.
Also Python complex literals (which exist; the floating-point properties are as a separate real and imaginary part each one being a double) also use a suffixed j.
but,j is current density
@@sober4769 It is a lot more common that an electrical engineer uses current instead of current density, in their calculations. It if does end up mattering, capital J would be current density, and lowercase j would be the imaginary unit. Current density is for the physics behind a lot of electrical components and their theory of operation, but it is rare that you deal with the continuum mechanics of electricity as an electrical engineer. Current itself infinitely more common, than current density, for electrical engineers.
@@carultch I've learned that u (or v), j, and i are used lower case when talking about alternating current and upper case when talking about direct current.
@@topilinkala1594 I can't say that I'm familiar with that convention, since I've always used capital I and V for electrical calcs. I had wondered why they needed to use j instead of i, if capital I would stand for current anyway, and thought maybe it is just to avoid confusion when talking about the equation aloud.
Regarding 1:04:56 :
One my electrical engineering professors said that if mathematicians hadn't come up with complex numbers, electrical engineers would have.
Dealing with electrical circuits that involve capacitors, inductors (and alternating currents) without complex numbers is very difficult, having to deal with differential equations and trig identities, but if you interpret inductors & capacitors like resistors, but with an imaginary resistance, you get an incredibly beautiful and simple way to work with them.
In general, there is pretty much no area of electrical engineering that does not benefit greatly from using complex numbers. Especially everything involving AC.
Try signal processing or control theory without complex number
What does a resistor with imaginary resistance mean?
@@cubing7276 electrical signals are sine waves or can always be expressed as sums of sine waves, see the videos on fourier transformations.
So you can express the signal at any point in time with amplitude and phase angle, which is extremely convenient to do as a complex number. You can think of a hand/needle/pointer/*phasor* that's spinning around in circles as time goes on.
When the resistance is 1, voltage and current always have the same value at any point in time. If it's 2, the voltage will be 2x the current.
So a non-imaginary resistance simply scales this complex signal.
When you have a capacitor or inductor, the peaks of voltage and current are no longer at the same time, they're out of phase by (ideally) 90 degrees. So we just multiply our phasor by i, and there we go.
Of course, any real-world part like a wire has a resistance, capacitance and inductance, and we can use complex numbers to describe this.
hope this helps, I'm not studying this in English, so there might be some errors in translating to the correct technical terms.
Chy 75 it’s a zero ohm resistor 👍
Chy 75 either Inductive or capacitive reatance.
My mind is blown. Thank you for awakening me to what complex numbers are: vectors that have an awesome rotation definition for multiplication!
Video Timeline
0:00:30 W3: Results
0:01:00 W4: Prompt
0:02:00 Ask: What would you call 'imaginary numbers'?
0:06:40 Starting point & assumptions
0:10:25 W4: Results
0:11:25 Q1: Prompt
0:12:20 Q1: Process
0:14:05 Rotating Coordinates
0:16:40 Q1: Result
0:17:40 Q2
0:18:15 Q3: Prompt
0:19:40 Q3: Results
0:21:35 Rotation Animation
0:22:35 3 facts about Multiplication
0:25:40 Q4: Prompt
0:26:10 Ask: imaginary I vs physics i&j
0:28:15 Q4: Result
0:31:00 GeoGebra Demo
0:32:10 Q5: Prompt
0:33:30 Q5: Results
0:34:00 Q5: Solution
0:35:55 Rotating Images Example
0:37:10 Python Example
0:38:25 Python Image Rotation Example
0:40:35 Ask: Vectors & Matrices for rotation
0:42:40 Q6: Prompt
0:46:55 Q6: Results
0:47:25 Q6: Solution
0:52:20 Redefining Angle Addition
0:57:20 Q7: Prompt
0:57:55 Ask: Can we do without complex numbers?
1:00:10 Q7: Results
1:00:55 Q7: Solution
1:05:45 Q8: Prompt
1:06:30 Ask: sum/difference of angles
1:09:25 Q8: Results
1:10:25 Q8: Solution
1:12:00 Desmos Example
1:15:05 Bringing it all together
1:16:25 The cis shorthand explained
1:18:05 Q9: Prompt
1:19:35 Q9: Results
1:20:55 Closing Remarks
Edits: Changed timestamps to the hour format, moved them closer to event and updated them after video was trimmed.
Thank you! I won't need it because I watched the stream, but this will help a lot of people. You might want to change 61:45 and under to 1:01:45 so the links actually work though, but that must've taken a long time.
@@noahniederklein8081 Will do, It's all in a spreadsheet so its an easy fix. Going through now and double checking them.
I converted the bottom half to hours:
1:01:45 Q7: Prompt
1:03:00 Ask: Can we do without complex numbers?
1:05:10 Q7: Results
1:05:40 Q7: Solution
1:10:10 Q8: Prompt
1:10:50 Ask: sum/difference of angles
1:13:40 Q8: Results
1:14:50 Q8: Solution
1:16:10 Desmos Example
1:20:20 Bringing it all together
1:20:50 The cis shorthand explained
1:22:00 Q9: Prompt
1:23:45 Q9: Results
(1:05:40 Closing Remarks)?
Don't forget 31:44 : snarky remark.
Thanks.
"look, even python has complex numbers"
- opens i-python
dude, that was a joke ;)
Drink everytime Grant grabs a new unit circle.
My liver
Which pen he is using the black colored
Would you prefer if he deliberately had circles with a radius other than 1?
@@carultch aren't all circles unit circles.
Just with a different type of length unit.
@@RodelIturalde That would only be true if there were an infinite number of length units, which would be incredibly inconvenient. Also, it isn't necessarily length units that define circles in general. Mohr's circle for instance, has units of psi or Pascals, and the radius represents the physical quantity of maximum shear stress within the plane.
Every of this video reminds me the importance of a good teacher.
Thank you, for keeping my interest in math.
The part when he said 69 is close lmaoo I laughed so hard
Edit : 28:34
I also liked last time when he actually said a pretty interresting fact: squaring it gives 4761, and cubing it gives 328509, together they contain all digits from 0 to 9 exactly once (apparently its the smallest number with this property). A bit base 10-y tho but still. Funny when he went "because it has that property, thats probably why you submitted this"
Saame; 33:50
hehe 69 amirite xDDDDDD
I just like said close to whst???😕😕
Video was trimmed. This is now at 28:34
You are an amazing teacher, I am rediscovering complex numbers and the beauty that hides inside of them. Thank you very much.
I learned all this 43 years ago, but this is the first time I saw the animations. Fascinating educational tool!
look up his calculus series, it's fantastic
His calculus series saved my grade and life. Not exaggerating
Teach me please pretty please
Working out cos(75) geometrically instead of plugging it into a calculator just singlehandedly allowed me to finally grasp quaternions
These streams are incredible
This was an absolutely fantastic lesson! The explanations are so simple and elegant. Thank you very much for your effort. Can't wait for the next livestream!
русские вперед
@@shaldee6814 wha
Man, i'm almost in my 40's, and i just learned a new intuition behind a tool I know and use since 20 years. You're an awesome teacher.
I worked on the graphics engine for the space shuttle. And we used quaternions a lot. Mostly because we didn't have to worry about gimbal lock, they are much faster than matrix multiplication and make relative rotations more intuitive.
What did you study in University? I wish to pursue this kind of career, thank you
Same
Fax I totally understood it by the way
@@GamerTheTurtle Electronics or Computer science may be. :D Just guessing.
Jack Martinelli do you mean the space shuttle simulator program?
Grant Sanderson is an asset to humanity, thank you for explaining to me what my brain didn't accept it fo decades. Many many thanks
I actually like the name imaginary. It makes them sound whimsical and interesting. The moment I heard of them I wanted to know more. On the other hand I think complex is a poor naming choice. It makes them sound complicated or hard to understand, which they really aren't. I'd prefer "compound" or "combined" numbers in that regard.
I think people hear imaginary and then their next thought is "doesn't exist,no application, how can this help me, they don't matter so I don't care" or they get confused on either how a number can be imaginary, or that all numbers are "made up and thus imaginary", a lot of ways hearing imaginary number can go wrong.
Thats exactly how they are called in polish language. 'Liczby zespolone' - ' Combined Numbers'.
All these teachers say terrible name for complex numbers or imaginary numbers. I also think it is interesting. I found it more interesting than real numbers or rational numbers!
Composite numbers
I like to call them dancing numbers personally, you know with Fourier Series and all
I find it fascinating that the most intelligent people on the planet (doctors, lawyers, Grant, etc.) usually have terrible penmanship. Maybe it’s because they all have bigger things to worry about and think about…who knows? But Grant’s handwriting/printing looks like it was written by a 8 yr old.
Ngl it looks like my handwriting and that is NOT a compliment
The little mistakes make it better. Such things are very comforting to insecure students.
hehe
Agreed. Whenever I make a mistake I immediately jump to the conclusion that I'm not good enough, that all my grades were luck and that I'm going to fail but seeing someone who is really good at maths make mistakes gives me more faith in myself
Grant, I've gotta say. What sets you apart from nearly every math teacher I've ever met is your presentation and humility. Despite the fact that you're unbelievably smart, you know exactly what kinds of logical questions that we who aren't as knowledgeable will ask. It not only makes us feel known and understood, but it significantly boosts your credibility and enhances your teaching. We're not simply taking things at face value because we know that the person teaching us has thought through things the same way we are and can address our concerns. Thank you.
"Thank you for joining. Apologies for being mildly scattered throughou--" *video ends instantly*
Yeah.. " i " noticed that.." i " wants to know what happened ...
there's nothing important beyond that point. he just apologized for the sudden interruption that happened before that point in time.
@@nightmareshogun6517 i really likes what you did there.
Honestly, can we do this format even after the pandemic? Maybe just once a month or twice a month, I find out that your vids about these fundamentals of math have taught me a lot more than what I learnt in university for entire 4 years. (I'm studying Mechatronics engineering, and the more I learnt, the more I realize the power of mathematics, but it's kind of too late/too hard for me to build my base mathematical knowledge now.) Anw, really good content, take care and keep up the good works.
Imaginary numbers should be called "lateral." That name was actually proposed!
By Gauss.
Imaginary numbers was a derogatory term that Descartes used that, unfortunately, has stuck hundreds of years later. Ironic that the father of analytic geometry thought that complex numbers were nonsense.
Edward Hou that’s how they originally found a geometric interpretation for complex numbers though, rotations and scaling are a huge part of them
Yes u r right.
@@MrBorderlands123 Sticking with derogatory terms for the long haul happens often in maths and science. Big Bang cosmology was originally called that by detractors, who wanted to point out how absurd the idea is. Well, absurd or not, it appears to be factual. Or "climate change". They originally wanted to call that the "climate catastrophe", but some of the scientists involved with that felt that was too alarmist.
56:45 Good Lord, that's the defining property of exponential functions! Suddenly, I see the link between the two concepts. The click in my head was audible! Thank you, Grant, you're an awesome teacher.
I'm already past my studies, I watch you only because I love math. It's really nice that you do what you do, keep it up, man! :)
Multiplying complex numbers makes sense. Thinking complex numbers as vectors make sense. But combining both are making things confusing.
As multiplying complex seem like transformation which seems like 2*2 matrix. Multiplication with vector. So, we are considering one complex number as matrix and one as vector. But if we see just at multiplication terms it seems like we are doing matrix and vector multiplication.
You're a wonderful human being and a great teacher. I send you all the love in this world.
I've been absolutely loving following through the lectures and would like to see more. Thanks :)
Damn 8 years in engineering and I didn't got imaginary numbers, 1 hour online lecture later my mind has opened
that too was a primer. nice
Same. Completed full undergrad engineering curriculum. Just now learning that adding rotations equates to multiplying complex numbers. What a joke. Grant, you're a treasure.
@@kdawg3484 True which indicates how much of a bubble the current type of business aka universities are. Full of useless pHDs that can't teach (probably because they never understood it deeply either if they had same type of teachers? who knows), but yet you manage to get a bachelor and you seem like a scientist to the average popualtion :D .
Lectures like these make me feel that all fields in math are connected very fundamentally, but since we learn then in different chapters, we just don't see it.
3b1b: get yourself some “frixxion” pens, they’re erasable.
Or a laminated unit circle
Or a whiteboard
Or imaginay paper
@@jonnyp1340 He's already using imaginary paper in some areas though.
I really want geometric algebra to be explained by 3b1b
Thank you for this. Several of your videos use complex numbers to explore some concept or another, and there's always this assumption that the viewer is familiar with them (never stopped me from watching them anyway). Having never formally learned about them, I had this vague idea of what they were, but now I actually feel like I know enough that I can go back to those videos and get something new out of them.
When I was trying to memorize the trig identities for my calc exam, I looked up methods to remember them, and the only thing I got was "rederive them using complex numbers" which was very unhelpful, since I had no idea that trigonometry had anything to do with complex numbers. It's cool to see how they actually connect in a really fundamental way, and where the identities come from, instead of the teacher going "here's an identity I just pulled out of my ass, now memorize it". I'm very excited to learn about how this all ties into exponential formulas and what Euler's identity is.
Sorry for the long comment, I'm just really glad to finally actually learn what the hell is up with complex numbers after all these years of people talking about them without explaining them. You'd think they'd teach them (or at least mention them) earlier since they're so central to trigonometry and tie into a bunch of areas of math really elegantly. (for reference I just finished first year calculus)
hearted
It's helping in AIME a little. Actually, a lot! Contest Math is different, but Pure Math has his own class.
Update: had a lecture on complex numbers in my mechanics I class (2nd year) because we needed to apply them and half the class hadn't learned about them yet. I'm noticing a pattern where my physics class applies a math concept before my math class teaches it, and it's very annoying (for example, vectors are used a lot in 1st year physics but not taught in math until 2nd year).
@@sarahp6512 that is indeed a common pattern in physics. Its kind of cool because physicists should have the intuition to use mathematics even if they don't understand the full rigour. Physics is also more rewarding when using math. I do agree that not teaching complex numbers by 2nd year uni is a big mistake from the math (?) department though.
Revisiting 3 years later, I am 100% convinced that in the whole world, math teachers in the school were completely incompetent. Most likely they never understood the math and didn't know how to teach the math subject. Majority of people went through hell. It took decades to appreciate math because of such beautiful math UA-cam channels and kind teacher/professor. 3b1b channel is a gem. It teaches us how to think, how reason. If math is not directly useful in your life, don't worry, at least you will learn how to think, logical reasoning. It's very useful skills to have in your daily life.
Quaternions are complex numbers on steroids - my favorite quote of this lesson
I described quaternions with that exact phrase to a friend last year. It's great to see Grant's mind and mine have something in common .
If that's not a common way to describe them, then mathematicians need to bulk up on colloquialisms
Can u tell me the timestamp where he talked about quaternoins.I watched the whole video already.Don't want to watch again
@@bhanusri3732 @40:55 or so
I'm bad at English. What are steroids?
Don't know what's more incredible; the way imaginary numbers fit so well on the two-dimensional number line, or Grant's teaching.
His infinite supply of unit circles
"Lets define x to be the answer of my question" - I love the applicability of these one.
I mean, there's hardly a problem in maths that doesn't use this.
I've always been interested in complex numbers because of games, as in, 3D rotation relating to quaternion which themselves are the next step up from complex numbers and just thinking about it differently helps so much!
I've really been enjoying these lectures you've been doing, I graduated from high-school 6 years ago now, and while I did try to go through university, it just didn't fit me, I ended up enjoying learning from math papers and just playing around to understand more than just listening to the lectures they gave, so I dropped after changing uni once, then having to validate my first year in 2 times, so really not for me.
Now I still love math but programming school I found doesn't have lessons as the school is more of a "figure it out yourself" (School 42 in Paris) which I love, but it has it's draw backs when it comes to trying to find new was to solve problems, you too easily get stuck into what your brain is used to, so all those math videos are always a really good way to just think differently.
I'm a bit slow so I often have to pause to do the math but I can pull through.
I know this format is more meant because of the virus and lock downs and I can't be surprised if the logistic makes it a bit hard to do regularly when all of this calms down buuuuuuut I would really love even just one like those every month :D
And while I'm here, big up to Ben for all the interactive tool! Been enjoying his videos for a few months now, quite something too!
I can just see Cam and Eder sitting in tiny chairs with their legs tied up in the corner of the room furiously working on their laptops
He was way too rude to them
Something you hear on Mathologer is "We've found that two completely different methods give us the same result. Do we just say 'Oh, that's nice'? Of course not, we try to find out why they are the same, what is the connection?"
33:06
Answer: 5
Grant: “69 is close”
To be fair it's closer than 1729 which is the other popular answer 🤦♂️
Phil Boswell that’s Ramanujan’s number
@@tortillajoe yes, and he mentioned it in the previous episode which would be why it's popping up here😜
Phil Boswell I’m aware
@@tortillajoe
Yeah right, I highly doubt you're a tortilla.
ca. 22:10
I just figured it out. Each complex number is written as the sum of a horizontal vector and a vertical vector. (The real and imaginary components.) Each of these lines is just one of the unit vectors (1 or i or -1 or -i) multiplied by a constant, so it's easy to see how the fact that multiplying by i rotates these four numbers (by virtue of multiplying i by i being how i is defined) would extend to rotating the horizontal and verticle components of any complex number. If you rotate these components than you rotate the number (ad makes sense visually).
(That felt like "figuring it out" to me because it explains how the symbolic math is related to the visual rotations and how it all stems from "i×i=-1".)
"I'm gonna have a stern word with them *behind the scenes"* - I think you just did.
Rule #1 for presenters: Never criticize the AV people during the presentation. They KNOW when there is a problem and - you can be sure - they are FRANTICALLY trying to fix it. Chastising them publicly is just a jerk move. Much better to say to the audience: "Please have patience. My AV folks are really great, so there must be something really unexpected for this to happen."
@@kirenireves I think it pretty clear that he was just being funny and the AV people probably knew this. But thank you for the insight, I'll certainly keep that in mind for the future.
@@thasyashetty3797 He says it a few times, even encouraging the audience to give the AV folks a "stern word" over twitter. Then there is a tweet at 26:29 where someone compliments Ben Eater and says he doesn't "deserve the harassment" that Grant is throwing his way. (So it's not just me who noticed this.) Grant does *not* take that moment to say "I'm just joking...These guys are doing great."
People who are presenting feel anxious when AV does not go smoothly and so they project it back on the AV folks so they are absolved of blame in the eye of the audience, but it just makes them look small. AV people get alot of abuse. Just try doing that job with an ungrateful set of presenters, and it'll drive you to drink. Thankless job.
@@kirenireves dude nobody really took I seriously. he was just joking, calm down.
@@ruhaanchopra8878 I'm really calm. I am just pointing out that criticizing AV people in front of the audience is not a good move.
After watching your series, its as if all the math I've learned up until now was crap!!
Your way of geometrical interpretation of all problems isn't something which is taught or rather known by many teachers. Great thanks for making this awesome series. This has increased my love for math to a higher extent.
1:20:22
We are considering only the real part of doubling the angle, that is, cos(2Φ), but if we also take into account the imaginary part which is isin(2Φ) and compare it with the algebraic result, we get sin(2Φ)=2sinΦcosΦ !
Amazing how the familiar trig identities just pop out of complex numbers. Incredibly elegant!
I like to play around with 3d graphics and I've come across quaternions multiple times. They are actually used, math libraries designed for graphics or games always include quaternions as far as I know. That said quaternions are very often transformed into rotation matrices before heavy computations that involve rotations (like transforming millions of vertices in a modern 3d videogame) bacause GPUs are designed to be fast at matrix math. They can also be multiplied togheter before applying them to all of the vectors so you can do rotation, scale, non uniform scales, translations, projections(that involve the use of homogeneous coordinates in some cases) ecc. with a single matrix multiplication. Of course many of those things could be done with quaternions alone but I think that having hardware that can do matrix multiplication really fast is just more usefull than having hardware that can compute quaternions really fast.
59:52 It takes some heavy abstract algebra to prove, and I only mostly understand the proof, but algebraic closure of a field is unique, so anything that would require access to all solutions over polynomials in the reals must use complex numbers, or a construction that is isomorphic to them, or something more complicated that has a subfield isomorphic to it.
So, the answer to this question is effectively no. Using vectors and matrices is an extension that allows skews in additon to rotation and scaling, but if you limit it to only matrices that are shape preserving, then you reduce exactly to the complex numbers.
I love how empathetic and willing to understand possible mistakes Grant is, made me feel not silly when failing :)
I'm fond of Welch Lab's label "Lateral Numbers" mentioned in the first few videos in their complex numbers series.
oh man! Knowing complex numbers would've been so helpful in high school.. so much better to derive the trig formulas than to memorize it.
19:30 "Writing is difficult" - Grant, 2020
edit: 19:08 after video trimmed
I don't get it. It seems like he is writing for the first time.😱
Very reassuring how he says it's ok to not know things from time to time. As a 10th grader from a non english speaking country, the online math community can feel very confusing and intimidating. But I'm just here to learn, aren't I? No matter how long it takes.
The "online math community" does not exist. You have to chose your teachers, your platform(s), tools, goals, lessons...
This is first time I have ever commented on a UA-cam video, as I feel guilty that even after such an amazing lecture if I can't appreciate your effort to help the students to get better understanding on complex numbers. You videos are really amazing and wonderful sir. My thoughts about math has changed drastically because of your videos👏👏👏👏❤️❤️❤️
(at 19:12) Reacting to the level of viewer participation: "This is genuinely delightful!"
And u just know he MEANS that... and THAT is why I love this channel!
YES! I specifically loved that phrase too hahahah
Grant, show your books for us from the background.
I like Gauss's solution to renaming imaginary numbers. He suggested the term "lateral numbers", while real numbers would be called direct (+) numbers and inverse (-) numbers. Not entirely on-board with calling the negatives inverse numbers, because inverse more often means reciprocal, but I think lateral numbers gives a much better understanding about what they are.
We can call:
Real numbers as "Horizontal numbers"
Imaginary numbers as "Vertical numbers"
Complex numbers as 'Circular numbers"
I like both of your nomenclature but i think Sushant wins here on the amount of info you can derive from the name.
@@Sushantgupta12 the rotational attribute is not really of i, if you really think about it, but of the operation.
The problem is that we confuse symbols with operations and with the attributes of the numbers. Maybe we shouldn't talk about scaling, complex product, vector product, and all that as if they were the same operation (and often same symbol) with different kinds of numbers, but rather different operations with just groups regular numbers (where each number in a group might be orthogonal in some way to the others).
@@Sushantgupta12 Discussing tiny little facts with such a great community really brings up new radical thinking and analogy to simple things. That's the joy of it.
I call complex numbers 2d numbers.
@@Sushantgupta12 Scalars and Rotators. This terminology is an absolute genius.
Infinity has to be real, the universe exists in space, we do not know if the universe is infinite or not,
but we for sure know that space has no borders, no shape, no limit, it just is in all directions, and according to our current understanding, it's expanding and just creating more and more space.
Why is he so pretty?? Like he’s amazing at maths, and more than that great at making it feel intuitive, but then on top of all that he’s properly gorgeous. Some people have all the luck in the world
He doesn't write well though ;)
NOT TO MENTION THAT HE HAS AN INFINITE SUPPLY OF UNIT CIRCLES
oh man. I wish I was that guy.
Well he is passionate about sth which he absolutely reasonably believes helps us when he shares it with us. This makes people gorgeous.
@@manupeter8050 your belief isn't very scientific, but then again it's very hard to argue against that! you might be onto something.
@@milanstevic8424 maths isn't science in the first place either.
@16:18 Ha! You spilled your vodka on the paper! (can't be tea or coffee, they would stain the paper, and the water won't dissolve the typographic ink!)
"(a,b) rotated 90° counterclockwise is (-b,a)" instant mindblow!!
This one is just too good a video! Grant, please keep these coming
Really sorry if what I am saying appears to be silly or nonsense, but I think maybe, just maybe, this could be somehow useful.
I am just curious to know whether there exists some thing(say x) which has the following property of |x| = -1
As we know, |x| means the distance of "x" from the origin.
Having a negative distance doesn't make much sense (at least in general cases) because that's how we have defined "distance"(not including direction to be precise).
Maybe this has some feeling of having the "distance" b/w any two points being shorter than the straight line joining the two points..
This, as it seems, if it were to make any sense, would have to be in some dimensional space beyond 3D. But if we talk of 4D or 5D..., distances are still positive(like def. In linear algebra).
So such a number(x) could not be a quaternion or anything similar
So maybe fractal or "negative dimensions" can come in to help (or maybe complex dimensions, sorry if that is silly and being panglosian). (There are, as far as I know, many definitions of "dimension", some of which allow negative dimensions.)
Another thing one could do is "define" "negative distance" such that it coincides with the usual notion of distance(I have some ideas about this). This comes from one of Grant's videos namely "What does it feel like to invent math?"
A similar approach could be re-defining what origin and space is(where numbers live).
Maybe, such a thing already exists and I just don't know about that.
Any help is deeply appreciated!! Even if it turns out to be nonsense, I will get to know of some more interestings aspects of math...
-Pragalbh Kumar Awasthi
Interesting idea!
We really need to get this nailed down once and for all: Numbers do not exists, and in that sense, all numbers are imaginary. Grant said it himself, what numbers are, are human constructs, a linguistic shorthand for communicating the size of a set. In the real world, only the actual physical entities of these sets are what exists. The numbers themselves do not. To suggest otherwise is platonic gibberish, to quote AntiCitizenX
Thanks grant....had a great time playing with complex numbers
As an engineer I have to say that when its coming to dynamics, you`re lost without the understanding of complex numbers and their usage.
Even as someone for whom high school math was 15 years ago, I still learned a lot from seeing this perspective connecting complex numbers to trig. Did you consider starting first with two dimensional numbers (without talking about imaginary numbers; eg “apples on the horizontal and oranges on the vertical”) and then show that i is the result of wanting to rotate in this plane. And only then that it turns out to be the sqrt(-1). And then wrap it back into the trig.
“You will never find real happiness.”
-Grant Sanderson, 2020
7:35 “when you have a problem that you can solve you can just say ‘Oh I’ve define things so that we now magically have a solution’ “.
Then you become part of the history of mathematics...
Nope. You become a figure in textbooks or on calculators only when you show that these definitions have nice identities, interpretations, consistency, etc.
I can't believe how many times I've studied complex numbers, and I've never realized why i^2 NEEDS to be -1.
I thought it was a lucky definition that somehow turned out fine...
So... thank you. I can't possibly express how grateful I am to you, for this moment of true insight, where everything makes sense.
Mathematics is a beautiful imaginary world, but the way you commit to education and teaching makes our real world beautiful.
Thank you!!
Answer to Ahmad Osama 57:50
More than 80% of Electrical Engineering is based on Fourier Series and Transforms,
without Fourier S/T todays communication systems , signal filters and many other countless things were impossible...
And Fourier Transform is impossible without complex numbers...
Muhammad Qaisar Ali Nice you catch up on that!
@@drpkmath12345 Thank you
Muhammad Qaisar Ali Here to support! Lets communicate!!
@@drpkmath12345 sure...
My pleasure
It's also provable that any alternate system to the complex numbers that could do all these things must be mathematically identical, or have a subset that is mathematically identical.
Now that we know how to rotate a pi creature.
Just Imagine how painstaking 3b1b animations are.
Man how can you not love this guy, come on! I hope he becomes a billionaire. Grant, you're an inspitation, a person to look up to. Thank you so much!
He’s great, but he doesn’t need to amass a lot of private property to be happy. Sharing the love of math, I’m sure, is what makes life worth living for him.
Money not cool
I am a retired engineer (electrical and nuclear) and very much enjoy your lectures. You have an outstanding way of conveying understanding. Keep up the good work!
33:05 an addition to the i and j thing. In electrical engineering the i is reserved for a flowing current, hence the similar looking symbol j is used. (where i study in germany at least)
There is a notion of multiplying 2-dimensional vectors, it's basically what the complex numbers are about. You can avoid talking directly about "complex numbers" if all you want to do is manipulate geometric vectors and that is by defining multiplication on 2-d vectors. If you say the formula for that looks weird, then try it in polar coordinates.
This has been amazing as usual. I wonder why they don't teach this way at the University. I liked the lesson progression: if I started after minute 40 I wouldn't have been able to follow.
Of course they do
they totally teach this way at uni, it's a pretty standard lecture format it's just that at uni level mathematics you have to be a lot more rigorous and the concepts are a lot harder to grasp
Why would they teach you like this in uni? I think highschool more like it.
In india we do a lot harder questions to get into an IIT
@@hassanakhtar7874 far harder content, same style is what I'm getting at. High school is a lot more interactive than this style which is quite literally a lecture; just like in uni
5:40 I learned a neat mnemotechnic trick to learn sin(a +- b) and cos(a +- b)
If you kinda know what the formula looks like but you can't remember where is + or - or cos(a)sin(b) or cos(a)cos(b) etc. Just remember
Cosine is a racist and a liar
-> It separates cos and sin.
-> When you see a + it's a -. When you see a -, it's a +
so for exemple : cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
I know some French people learned it this way as well.
I've probably left a similar comment on almost all of Grant's videos, but 3b1b is, by a mile, the best channel on youtube.
Ben Eater has an awesome channel, too! If you haven't, check it out!
I love how Beavis and Buttheads put 69 in the answers and Grant continues to come up with mathematical reasons why somebody would have thought of that as the answer
If you have a soul you must be excited of the complex numbers.
Nice lesson again Grant.
man the past few months i was in a bit of a slump when it came to anyhting including science and i was losing my interset in it becuase i started focusing more on school, i found this video in my recommendations and forced myself to click on it despite thinking "ah i don't wanna watch 1 hour of this" and maaan i never appreciated just how much a good teacher is needed to make you like something, i'm now interseting in math again, now everyone on twitter is going to be tired of me talking abtou math again
x^3 = 1, the magic of the three-phase electric power in one equation
aren't the three 120° offset waves to be ... "added"? am i misunderstanding something?
@@SreenikethanI , yeah that's correct, but what i'm trying to saying is that in this simple equation you can see how complex number can cancel each other, and this is the principle of the three-phase electric power.
The question about the non-real cube roots of 1 is always going to have a special place in my heart. I remember one time I was being difficult and pestering my parents, so my father gave me that question. He didn't tell me how many there were; he just told me to find them. I had only just learned about complex numbers, so I didn't know about the analog between multiplication and rotation, but that problem let me figure it out on my own. Also, I think we were hiking, so I didn't have any pencil or paper, and I had to visualize it in my head (but that's easier for me than algebra anyway). That was a really fun few hours.
I didn't even realize until he told me years later that he was trying to get me to stop bothering him, but apparently giving me difficult math problems was a strategy he used to use. It certainly worked, and I got a few really fond memories out it.
I knew it was coming from the stream on Tuesday and was scheduled well in advace for completely different reasons. Nevertheless, today is my birthday and I'm definitely going to enjoy listening to Grant explain complex numbers again. Thanks for the stream!
Today is my birthday, too!
@@pronounjow HUZZAH!
You knew your birthday was coming due to the stream earlier in the week?
@@jasonlee3247 I knew this topic was coming due to tbe stream earlier this week.
Grant breaks down mathematics like potty training.
Calculus IS intuitive and he can show anyone.
THANK YOU ! The best explanation so far on quaternion = 3D rotation ! Even you just very very breifly talk about it, extrapolating from 2D rotation with complex number to 3D rotation with quaternion make finaly sense with this video !!!! Now I want to understand Gimbal lock in a first place.
Good news: Grant made another video on the relationship between quaternions and 3D rotations! It is amazing. Enjoy!
The question you answered around 59:46
There's *also* ways to "get rid of" a lot of matrices by using extensions of complex numbers instead.
Just use Geometric Algebra or Clifford Algebra. (Same thing, different names)
There are good reasons why one would stick to matrices anyway I think, but a lot of things that are rather clunky and awkward using matrices work out extremely beautifully with Geometric Algebra.
Hey Grant, thanks a lot for doing these live streams. You have a gift, really. It's amazing how well you're able to explain math to me and many others.
And don't worry about your handwriting ^^
Really struggled with the cos(75°) question. I was still in the mindset of the previous lesson and used the half angle identity and the answer didn't match the choices. (1/2)sqrt(2-sqrt(3)) is what I got and turns out to be right but algebraically expressed differently to the answer given.
1:03:23 I have to disagree here. There are integrals that can only be solved to a closed form by complex-integrating along a semicircular contour.
Also the characteristic function of a probability distribution is generally more reliable and useful than the moment generating function especially its identities about the additivity of random variables
I was asking myself what hell am I doing watching a lecture about complex number at 1:30am when I finished this video, the answer was "because it is fun". I didn't even notice the time passing. Thanks for all the work, everything on this channel is just brilliant
this is my all time favorite thing to watch on youtube. i genuinely get excited to see that these have been posted an i get to watch them. thank you so much for this:)
This guy is the CEO of maths