Hello all, for some bizarre reason this playback has gotten cut off at the end. The only thing I said after was that if anyone had trouble with #4 on the homework from last time, it's because it was a genuinely subtle question, meant only as a "bonus" of sorts.
Hey Grant ! I gotta just say..such amazing videos !! I couldn't stop jumping with excitement to see it all linked when the unit circle started popping up...and physics coming up in the end...wow just amazin'!!!!!
3Blue1Brown thanks for making me realise the actual connection between sinusoidal functions and the exponential function; here's my understanding: for f(x)=exp(x) f'' = f for g(x)=Asin(x) + Bcos(x) g" = - g = i² g this makes me get , at least some vague, idea why they appear as they do in Euler's formula
Especially considering interest rates represent the time value of money and they shouldn't be fixed at all, becuase a real interest rate is only the average interest rate as set by each and every actor considering interest in their actions towards ends.
I can't overstate just how lovingly humble you are. In spite of being a virtual god in mathmatics, you have so much understanding for very simple "misconceptions" and "mistakes"
so 2d rotation can technically be done with only 1 dimension(real numbers) but can be done "nicely" with 2 dimensions(complex numbers). 3d rotation can technically be done with only 3 dimensions but can be done "nicely" with 4 dimensions(quaternions). are you saying 4d rotation need 6 dimensions to be done "technically" or "nicely"? also what kind of rotation systems would octonions describe "technically", and "nicely" (sorry for using laymen terms!) edit: im reading every reply in this chain, so thanks in advance
@@kjetil1845 4-d rotations can be done with 6 technically. There are 6 generators for the Lie group. A standard representation is as pairs of unit quaternions, which is 8 real numbers, but 2 are somewhat redundant (signs matter, so not quite)
@@kjetil1845 In particular, the exact number of dimensions *technically* required is the number of possible planes you can rotate in. A helpful way to count that number is by looking at how many planes you can make out of the directions you have: in two dimensions, you have two directions, out of which you can make one plane; in three dimensions with three directions you can make three planes (1|2, 2|3, 1|3 - think if how many independent sides a cube has, if that helps); with four directions that bumps up to six (1|2, 1|3, 1|4, 2|3, 2|4, 3|4); and so on. You can work out what these numbers are in arbitrary dimensions, if you like. To go from the *technical* method to the *nice* method, you just want to add one dimension: this is the *scalar,* or ‘usual’ part of the number, which corresponds to the real part for two-dimensional complex numbers. The idea is now that to rotate in any given plane, you want to multiply by the *imaginary number,* say ι, associated with that plane (or, for an arbitrary angle θ, by cos(θ) + ι·sin(θ)). Having a scalar on top of your imaginary numbers means that you can rotate by an arbitrary angle, since when multiplying by cos(θ) + ι·sin(θ)), the first part is a scalar, whereas the second is imaginary. So you want both! That's why the *nice* system of rotation requires one more dimension than just the number of rotations that you want to be able to do. There’s actually a very elegant system that makes all of this very intuitive and, instead of introducing i by assuming it's the answer to a problem you couldn't solve, brings it about quite naturally by asking how you might want to rotate a shape. This system is called *geometric algebra,* and it quite naturally generalises to higher dimensions. Grant actually mentioned it in the last episode (I believe), and I'm very excited at the prospect of him doing a series on it! I've seen people argue that it should be taught instead of how vectors are generally taught today, and personally I agree: I think it's much simpler. That said, if you are a layman wanting to look it up, I will warn you not to get discouraged if you don't understand it: since it's still a relatively niche framework, there aren't many good explanations online teaching it at a fairly basic level. (Hence why I really really really hope this is relatively high on Grant's to-do list. Please!) Edit: octonions don't come up in my list of how many imaginary numbers you need for rotations in higher dimensions, which goes 1, 3, 6, *10,* 15, . . . . I'm not aware of them being used for anything else, but that doesn't mean they don't exist! Someone feel free to let me know if they do.
1) I love how you distinguish between things that logically follow from what's already known on the one hand and things that are arbitrarily adopted because they turn out to be useful. As a former teacher myself and observer of other formal and informal teachers, I find putting arbitrary decisions as obviously true is the biggest way smart teachers become less effective with smart students. 2) The sirens weren't audible at all, at least on my laptop.
39:38 You CAN make money in "i" interest compounded continuously. You just need to manage your money. After 0.8 years you withdraw your 71.73 real dollars and let the remaining 69.67 imaginary dollars ride for 1.5*pi years, leaving you with about 141.40 dollars total after that time
I wouldn’t mind taking out a continuously compounding imaginary interest rate loan. Take out the loan, get your $100, then wait for half a rotational period, then take out another 100 to return to a zero dollar balance
Absolutely awesome. Never have math concepts been so brilliantly translated. Its not my discipline but I come here for fun and to learn expecting to watch a single episode, and go to bed way into the wee hours pondering the beauty of the universe. The only truth in life is math. Bravo.
"You can see this as the venture capitalist approach". Thinking about that, it feels kinda like ANY business venture. Liquid capital is sunk into non-liquid assets like equipment, with real outflows from current liabilities, and non-current liabilities that might affect your non-liquid assets, in event of a bankruptcy, etc, but doesn't necessarily impact real revenue. This might actually be the graph of WeWork's CEO. Start out with some money, it turns into imaginary assets, which become real losses, with debt piled on to bail out the company, for the CEO to sell his share at a massive huge personal profit.
This is explanation is just pristine. "All it really means is after taking a bunch of steps that are perpendicular to your current position and you do it continuously such that those steps are infinitesimal, It's the same as walking around a circle".
I really wish we had you as our math teacher when me and my colleagues where young. It would have been and eye opening experience. Thank you for everything you do, this is truly inspirational.
These lectures are so useful in lockdown. As a student who took A level maths and further maths (going on to a degree in physics) I've spent a fair amount of time with these concepts (trig, complex no. etc), but its awesome to go back and think deeply about the fundamentals in a way we didn't during our course before moving on to the more complex content, pun intended. Really useful for keep my understanding up especially with not being able to take exams now (feels like I've lost a bit of a chance work everything into longer term memory). Never thought of complex numbers with SHM before, always gone a physical or differential equation route with it. Really interesting! Highlights yet again how much you can link topics from across subjects/curriculums. Thank you for your amazing content, and helping me stay sane ;)
You managed to sneakily introduce the idea of phase spaces, as well as solving a differential equation without calculus. That's education ninja, right there! Bravo!
I though that this was all nonsense too until I started dealing with AC circuitry and complex power in my electrical engineering courses. It's insane how often what initially seems like a completely artificial math hack is actually key to understanding nature.
Video Timeline 0:00:00 Welcome 0:00:55 Q1: Prompt (Would you take a interest rate) 0:02:05 "e to the pi i for dummies" video shoutout 0:02:45 Q1: Results 0:03:30 Q2: Prompt (two banks, two rates) 0:04:55 Ask: Beauty of connections in math 0:06:00 Q2: Results 0:07:05 Desmos for Q2 0:09:10 Q3: Prompt (savings growth rate, 6% every 6mo) 0:10:35 Q3: Results 0:12:35 Desmos graph explored 0:14:45 Breaking down an interest rate 0:18:00 An interesting interest equation 0:19:20 Q4: Prompt (100*(1+0.12/n)^2 as n → ∞) 0:21:05 Ask: Quaternions 0:22:35 Q4: Results 0:24:50 Explaining Q4 0:26:40 Defining e 0:28:40 The definition of e from previous lectures 0:30:45 The imaginary interest rate 0:32:35 Graphing this relationship 0:33:50 The imaginary interest rate animation 0:37:55 Compounding continuously with i 0:40:45 The spring & Hooke's law 0:43:20 Q5: Prompt (Δx & Δv for a spring) 0:44:50 Ask: Rotation in for multiple dimensions 0:47:45 Q5: Results 0:49:50 Rewriting the spring's position 0:55:00 Bringing it all together 0:59:00 Ask: Hints on last lecture's homework 1:03:25 Closing Remarks Liquid Drinks at: 20:4536:1044:1544:25 Edits: Spelling & formatting and updated some timestamps that drifted
These timestamps assumes the video is trimmed at or near 5:15. If watching the video before it is trimmed, add 5 minutes and 15 seconds to the above timestamps.
22:19 Quaternoins could be obtained from complex numbers by adding a "j" satisfying j z = \bar{z} j for any z from the "old" copy of the field of complex numbers, and imposing associativity and bilinearity of the multiplication. Here \bar{z} stands for the complex conjugate of z. A similar operation can get you from quaternions no (not quite associative) Cayley's *octonions*.
I think the difference is that quaternions are more on the "let there be some i j k such that [...]" because it's useful, whereas i is defined as the root of a polynomial and all its other properties are logical conclusions from that original identity. Don't get me wrong quaternions are awesome and useful, but 2d complex numbers are a lot more natural.
Just a little fun fact I wanted to share . 17:00 the equation Grant wrote is (M(1+r(delta t))) is also true for expansion of rod when it is heated... just replace r by temperature coefficient of that rod and M by the original length of that rod for small changes in temperature (
I love these lockdown math videos so much! They feel easier to follow than your shorter vids. I know part of the reason is because this is meant to be a more newbie friendly series, as opposed to the very knowledge dense shorter vids. But I think it's also because the short videos are too informationally dense for me to the point I get anxious I don't understand everything, and end up rewinding them a bunch of times and giving up somewhere in the middle. I guess my brain has a high inertia when it comes to science LOL. Hope to see more math lessons in this format in the future!! I think you're a great teacher.
I love this, thanks for this great content. I was happy that my first instinct was right, but after 3 years of electrical engineering this was expected. Although I noticed that I learned the e to the power of X expression like you showed it here. Funny enough my first thought when you said you would use a physics example was the force a moving, charged particle experiences in a magnetic field, which is orthogonal to the velocity and the magnetic field.
I haven’t watched the video yet and I already know this is gonna be some fun stock market-y stuff. I love when my interest oscillates from gaining money to owing it
I may be wrong but I think this question has a paradox in it. When you pick the 5th most common number, and if most people picks it too, it becomes the most common number, not the fifth.
@@Zeynep16th It's not a paradox. More than 80% of people will necessarily get that question wrong every time, but there's nothing wrong with that. It's sort of like a lottery.
@@EebstertheGreat Well je didn't say anything about right or wrong... He only pointed out that the original comment said "most people" while this by definition cannot be the 5th most popular choice
@@jilleswassink7762 It might be interesting to ask a question like that in which you can submit multiple answers, like maybe 5. Then it would actually be possible for a majority of people to guess the correct answer as one of their guesses, though it would be very unlikely.
Sorry, was the answer mentioned in this video? I skipped some parts when I first watched, but having rewatched those moments I can’t find the 5th most common number part.
Thank you for this lecture! You blew my mind with this new perspective and this is coming from a person who has already done a quantum mechanics module at open university.
What you do is take out a loan first to get your money to be negative, then wait for half the rotational period (where the money turns positive), then extract all the money (bringing the balance to $0).
(1+0.12/n)^n, if you replace 0.12 with 1 and when n blows up it approaches 'e'. In a sense, the exact opposite of option (E). I'm so glad that I choose option (E). This was a beautiful episode.
14:30 nobody should let the FED watch this video. I tkink it might introduce some shaky ideas. Grant is Great matematician, smart guy and the best UA-camr EVER!!!
You end up with 100 Real $ and $100 Imaginary $. Aha! Now I know how bitcoins work! Your explanation of e using interest rates reminded me of an old calculus text book called Calculus Made Easy by Thompson. As usual , great presentation.
Fantastic lecture, I have one comment. At time 17.05 it is stated that M(T) is the amount of money we have on the account at time T. I think this is misleading. By definition of M, M must always be the amount of money on the account at time T. Implying that M is a function of time is wrong. If r is given, then T is known, and the question is how the frequency of accumulation, n, affects our savings.
Animated ones take way long. Not even because of the visuals, but once you write something exactly, you hold a higher standard for the writing and story progression. That said, it turns out live lectures also take a lot of work to prepare! Especially given some of the work going into Itempool going on behind the scenes.
These lectures mixed with all the bells and whistles are very enjoyable. I was kinda smiling the whole time. In the future you should do live lectures whenever you feel like it and we're here waiting!! Thanks for your dedication.
@@SomeRandomGtaDude-zl3us It's hard to visalise things in higher dimension... However, if you have a diagonizable matrix then you are reduced to several 1-dimensional situations.
This is so marvellously informative and entertaining and just pleasant to follow. Thank you so much for putting out content like this. I never thought mathematics could be so interesting.
I didnt have so much fun doing "homework"in a long time. Thank you for this great series and giving a great save heaven from alot of more "boring homeworl" :)
Imaginary interest rates sounds good better than negative interest rates after some period but again risk value is high. Quiet insightful to look into it. Also about Quaternions, so how degrees of freedom get described if it is not by complex numbers in higher dimensions?
Dear Mr Brown, this is the first time i have lmao at one of your videos and at just 30 secs in for that matter. what a wonderful way to introduce the concept you are about to present. You are a master, sir.
As an engineer who learnt SHM starting from how displacement is represented as a sine wave and built upwards from there to include complex numbers, this video was a very interesting change in perspective.
One important thing with those interest rates at the beginning is usually when you calc this stuff in basic economics, the problem also states that the interest is capitalized every some time, meaning it goes back into the sum that then is used for calculating further interest. At least that's how it is where I live.
My intuition says that a complex interest rate would periodically increase and decrease in value.... wait does that mean that the REAL stock market has a complex interest rate as you have the boom bust cycle?
The stock market has a tendency to go up, though, so it's not as simple as randomly going up or down. Booms and busts also don't happen regularly, we've just had one of the longest bull runs in history, so you can't really describe it with an 'interest rate' unless you consider the average of many years of growth.
@@Gregster427 Yeah I was imagining more of a "x+cos(x)" kind of function, while a complex number roughly follows just "cos(x)" in the real part. I found this interest problem more easy to visualize by remembering the definition of a product of two complex numbers. (a+bi)(c+di) = (ac-bd) + (ad+bc)i In a way, the imaginary interest leeches away at your real interest (due to the "-bd" term), and continually increases itself such that it will basically always overtake the real interest even if you start it off small. If a bank gave you a fixed real interest rate but then let you chose an imaginary interest b of your choice, I think the answer to maximize your real earnings would be to chose b = 0.
Can we take a moment to appreciate how spectacular it is that i^x and e^ix trace the same path. The proofs about powers to i seem almost obvious after acknowledging that fact, which also holds true for all positive numbers except 1, but it's actually an extremely rare property and that i just happens to be one of complex values it matches up at confounds just how extraordinary the discovery of how rising to the i works.
This video was mindblowing. By far the best I have seen. Thinking about imaginary interest rates is not insane at all. I even read the comments and I agree with some of them of the “sane” way of thinking about this. Could be a way of thinking of boom and bust cycles, as the way the spring behaves? Number “i” is related to sines and cosines which are also related to cycles. The second interesting way to understand this is by projects, you even mention the balance, cash-in, out of a project was well drawn by something as simple as your real cash balance depending on the periods. Projects my have an initial investment, then it is used to buy some materiales and after that you receive profits. The third way is as included in financial markets or as a kind of “implicit force”. Benoit Mandelbrot wrote a book called “the misbehaviour of markets” and he used his fractals -sets drawn and build on complex numbers- to explain the misbehaviour. The final way is that all the ways could be related! Congratulations!
You must be a pretty effing good teacher because I instantly knew the answer and I never completed anything more complicated than calc in high school... Except watching some of your videos... And some other UA-cam math guys, but your visualizations on the complex number plane (particular the Reiman zeta function) made this a piece of cake.
Every time after I watch your video, I feel both dumber and smarter at the same time. In Chinese, there is a saying that roughly translates to "Outside every mountain, there is a taller mountain, beyond every person, there is someone smarter". I think you fulfill that requirement for me, and I am grateful. Please keep up the great work. If math is indeed the language of God, then you bring me one step closer to God.
20:50 The basis elements i, j, and k commute with the real quaternion 1, (e.g. 1*i=i), each other quaternion squares to negative 1 (i.e. i^2=-1, k^2=-1, j^2=-1), and i * j * k = -1.
This just explained e to the pi i in the best way possible to me. I didn’t get what e meant in that it moved the spiral closer into the circle, but this graphical explanation was great.
The identity exp(A+B)=exp(A)exp(B) might not hold for all matrices, *but* it holds for all matrices A,B satisfying AB=BA. You need this property to sort products of A and B into expressions of the form A^n B^m, so that the binomial theorem and, subsequently, the identity exp(A+B)=exp(A)exp(B) both hold true.
Just noticed that AB=BA is still not enough, because we even have to raise the matrix to higher powers, and so the matrix need not obey this for higher powers, eg. ABA=BAB need not be true. I guess we have to narrow in on diagonal matrices only
@@gayatrisavarkar8196 Nope, AB=BA is enough. We do not need ABA=BAB to be true, we only need two products of A's and B's to be equal, if they have the same number of A's and B's in them. And AB=BA guarantees that we can rearrange any product of A's and B's into A^m * B^n where m is the number of A's and n is the number of B's in the product. In fact, we can rearrange any product of A's and B's into any other product of A's and B's as long as they have the same number of A's and B's.
I think the answer to Bobs question on viewing the Quaternions as an extended number system should be yes, by virtue of the Cayley-Dickson construction and/or the Frobenius theorem.
well, at the end the question about the exponential identity for matrices is pretty easy. exp(A)*exp(B)=exp(A+B) if A and B are real (or even complex) numbers, the identity holds. in case of real and complex numbers addition and multiplication are also commutative. but for matrices only addition remains commutative, matrix multiplication breaks this symmetry. this non-commutativity is the reason that the above identity is not valid anymore. why? because the exponential of a matrix is also a matrix. so the left hand side of the identity above is a multiplication of two matrices meanwhile the right hand side contains an addition of two matrices (then the exponential of this sum). so one side of the equation is commutative, the other side of the equation is non-commutative. this is a clear sign that the equation does not hold in general. because exp(A+B)=exp(B+A), meanwhile exp(A)*exp(B)≠exp(B)*exp(A), so two terms should be equal and not equal at the same time. contradiction...
It is FALSE that A*B ≠ B*A, for any matrices A and B. For some matrices A and B, A*B = B*A, for others, it is not. So, maybe it _could_ be the case that exp(A) and exp(B) would always be matrices of those types in which it commutting gives the same result. It is not the case, but it could be.
Guilherme Zeus Moura dude try to comprehend what I wrote. I was talking about IDENTITIES. identities represent generality. so when I said A*B≠B*A it meant that they are not equal in general (i.e. the identity is not necessarily true). so yes, you are right, but I was talking about generality and identity. non-commutativity is similar: the term “non-commutative” doesn’t mean that in specific cases it cannot be commutative, but in general it is not commutative.
When it comes to matrices, if you have a single matrix as input, and if it is a square matrix, only then will this equation be valid. The matrix has to be a square matrix before you even consider two matrices as inputs
I'm so glad I watched your video on ODE before this (making intuitive parallels between the spring and the [then] pendulum made this possible to follow. (I never took differential calculus before haha). SO thank you twice :)
The physics portion of the lecture had me thinking. So, even if the spring system is not subjected to any friction (54:45), if we consider time as quantized (not continuous) the graph would still spiral inward (the opposite of 36:27), so we would be still losing amplitude over time.
The spiral at 36:00 looks like the golden ratio which got me wondering whether there is a way we can approximate it using imaginary interest rates. And what will be the step size? Will there be any sort of cool relation between the two?
I think a really excellent way to illustrate the answer to Deepak's question about 3 dimensional rotation is with video games and 3D modelling. When you rotate something in 2 dimensions, you rotate it along a single axis (let's call it Z). It's like putting a pencil through a piece of paper and spinning it, that pencil is your axis. But if you pull out into 3 dimensions, you can now also rotate along the X and Y axes. Watch some videos on 3D modelling and you can see all 3 axes in play as objects are rotated along them. And the people writing game engines and 3D modelling software have to do all of this math we see in this video in order for those applications to work.
Question: How is i^i 0.207.... or e^-pi/2. Considering that when we work with complex numbers e^ix is actually the Exp() function and not the constant e, then how will one calculate i^i because that requires e to be the Andrew Jackson number to give the value of 0.207 approximately
Yeah, he didn't go into what x^y means when x and y are arbitrary complex numbers. First you express x as e^z, then you have x^y = (e^z)^y = e^(yz). Since i = e^(i*pi/2) we have i^i = e^(i*pi/2*i) = e^(-pi/2). At this point you can calculate e^(-pi/2) as 1 + (-pi/2) + (-pi/2)^2/2 + (-pi/2)^3/6 +... like usual. The value of i^i is actually ambiguous. You use, say, i = e^(i*5pi/2) and get a different answer.
@@aapeli9662 I get that, what I am asking is, if u see the proof for i^i which are complex numbers, the final answer comes out to be e^-pi/2, then they substitute e as 2.7...... But according to Grant's last lecture exp() is only equivalent to e^x in case of real numbers, which these are not, hence why should we take e as 2.7... in this proof
@@aapeli9662 no I meant in the proof of i^i we take ln i.e log with base e, which itself shouldn't be possible because the constant e has nothing to do with the complex numbers, I think just Google i^i proof once, and u'll see the process where they use the constant e from the start of the proof.
Does "interest rate of i" mean multiplying by i every time or by (1+i)? Because when I hear "interest rate of 10%", it means multiplying by (1+10%). If it's (1+i), definitely worth it since the magnitude doubles every time.
Depends if the bank penalizes you when the real dollars go negative. (And the consequences of accumulating or "owing" negative imaginary dollars in the account.)
I agree that the wording is ambiguous. But "interest rate of i" is probably intended to refer to the coefficient of i, viz. 10%. Notice that (1 + i) is equivalent to (1 + 100% i). Nitpick: (assuming L2), the magnitude of (1 + i) is sqrt(2) = 1.414. But yes, (1 + 100% i) is much larger than (1 + 10% i).
This example seems to support the legitimacy of the term *"imaginary numbers",* the historic term that you and others have described as misleading, because they are just as "real" as *real* numbers. Even though it seems silly, a bank really *could* offer an annual interest rate of i, or some fractional multiplier of i, compounded *annually.* But, the account could only be liquidated and closed when the account balance was entirely *real* (negative or positive) and thus offset by depositing or withdrawing a corresponding amount of *real* money. At all other times, the account's "imaginary" balance would be mathematically valid as a ledger balance, even transferable to another account, but not representable or usable in the "real" world. In this example, isn't *"imaginary"* a reasonable and informative adjective for the account balance?
Thing is that constraint wouldn't even really be all that unusual in the financial world there are after all instruments with restricted withdrawal terms like bonds after all.
And 50 years from now, cash is abandoned, and all money is complex, even to buy groceries, and handling your finance is even harder than it is for many people today.
Your point with the matrices and quaternions lacking multiplicative commutivity is a good one but I would argue that it just means that we need to be more specific with the properties we say functions have and when they apply
I'd love a bank with continuously calculated imaginary interest rates for loans. I'd be happy to take a loan of an imaginary fortune and wait for about 1.571 years And if loans don't come with imaginary interest rates, only deposits, then I'll gladly open a bank account on which I deposit my imaginary fortune (a man, can dream, right?) and wait for some 4.712 years before I withdraw my total investment
Hello all, for some bizarre reason this playback has gotten cut off at the end. The only thing I said after was that if anyone had trouble with #4 on the homework from last time, it's because it was a genuinely subtle question, meant only as a "bonus" of sorts.
Hey Grant ! I gotta just say..such amazing videos !! I couldn't stop jumping with excitement to see it all linked when the unit circle started popping up...and physics coming up in the end...wow just amazin'!!!!!
And OMG the last graphs blew my mind!
3Blue1Brown thanks for making me realise the actual connection between sinusoidal functions and the exponential function; here's my understanding:
for f(x)=exp(x)
f'' = f
for g(x)=Asin(x) + Bcos(x)
g" = - g = i² g
this makes me get , at least some vague, idea why they appear as they do in Euler's formula
Could somebody explain how @ 6:00, the correct answer came out to be option D; and when was the solution explained in the video???
@@thetriankh 11:55
"So, tell me about your retirement portfolio..."
"Well, it's complex."
As long as it's not purely imaginary.
Would you be interested in buying some conjugate dollars?
They'll help you realize your gains.
> conjugate dollars
Great. Now whenever i read Conju-Gate, the next banking scandal comes to mind...
I'll be staying in a theoretical hotel with an infinite number of rooms.
To take full advantage of imaginary interest rates, you should clearly start by depositing negative money.
Borrow at the beginning, then borrow more after 3.14 years to pay back the amount initially borrowed!
I'm pretty sure most of society has that nailed or they are real close at this point.
Don't worry. There'll be negative interest rates before you know it. :)
kebman then just wait some more
What would be really nice is if they decided not to compound it continuously. If you knew the initial conditions then it would be a win win for you
Let me just invest -1000 dollars and take them out when they are +1000 and earn 2000 dollars...
*compound complex continuous interest
Overdraft.
The more you spend, the more you save.
So, basically invest in oil barrels whose prices went negative.
They give you $1000 and your shares, and after 4y your shares can be sold for $4000
This is called "buying a gun" in colloquial terms
It's better than current interest rates.
Especially considering interest rates represent the time value of money and they shouldn't be fixed at all, becuase a real interest rate is only the average interest rate as set by each and every actor considering interest in their actions towards ends.
Hahaha. Whatever happened to risk free assets?
@@jdtaylor81 risk free assets don't keep up with inflation generally.
@@defenastrator not when youre printing money at rates that would make 1920’s Germany go «these guys are craaazeyyy»
Turks would agree with you.
I can't overstate just how lovingly humble you are.
In spite of being a virtual god in mathmatics, you have so much understanding for very simple "misconceptions" and "mistakes"
Isn’t it the case that in 4D you need six “imaginary numbers“, not ten as Grant has stated?
Ah! Sorry, that was a misspeak, I jumped a dimension.
so 2d rotation can technically be done with only 1 dimension(real numbers) but can be done "nicely" with 2 dimensions(complex numbers). 3d rotation can technically be done with only 3 dimensions but can be done "nicely" with 4 dimensions(quaternions). are you saying 4d rotation need 6 dimensions to be done "technically" or "nicely"? also what kind of rotation systems would octonions describe "technically", and "nicely" (sorry for using laymen terms!)
edit: im reading every reply in this chain, so thanks in advance
@@kjetil1845 4-d rotations can be done with 6 technically. There are 6 generators for the Lie group. A standard representation is as pairs of unit quaternions, which is 8 real numbers, but 2 are somewhat redundant (signs matter, so not quite)
Indeed, the degrees of freedom should go by triangular numbers, so 1,3,6,10,etc
@@kjetil1845 In particular, the exact number of dimensions *technically* required is the number of possible planes you can rotate in. A helpful way to count that number is by looking at how many planes you can make out of the directions you have: in two dimensions, you have two directions, out of which you can make one plane; in three dimensions with three directions you can make three planes (1|2, 2|3, 1|3 - think if how many independent sides a cube has, if that helps); with four directions that bumps up to six (1|2, 1|3, 1|4, 2|3, 2|4, 3|4); and so on. You can work out what these numbers are in arbitrary dimensions, if you like.
To go from the *technical* method to the *nice* method, you just want to add one dimension: this is the *scalar,* or ‘usual’ part of the number, which corresponds to the real part for two-dimensional complex numbers. The idea is now that to rotate in any given plane, you want to multiply by the *imaginary number,* say ι, associated with that plane (or, for an arbitrary angle θ, by cos(θ) + ι·sin(θ)). Having a scalar on top of your imaginary numbers means that you can rotate by an arbitrary angle, since when multiplying by cos(θ) + ι·sin(θ)), the first part is a scalar, whereas the second is imaginary. So you want both! That's why the *nice* system of rotation requires one more dimension than just the number of rotations that you want to be able to do.
There’s actually a very elegant system that makes all of this very intuitive and, instead of introducing i by assuming it's the answer to a problem you couldn't solve, brings it about quite naturally by asking how you might want to rotate a shape. This system is called *geometric algebra,* and it quite naturally generalises to higher dimensions. Grant actually mentioned it in the last episode (I believe), and I'm very excited at the prospect of him doing a series on it! I've seen people argue that it should be taught instead of how vectors are generally taught today, and personally I agree: I think it's much simpler.
That said, if you are a layman wanting to look it up, I will warn you not to get discouraged if you don't understand it: since it's still a relatively niche framework, there aren't many good explanations online teaching it at a fairly basic level. (Hence why I really really really hope this is relatively high on Grant's to-do list. Please!)
Edit: octonions don't come up in my list of how many imaginary numbers you need for rotations in higher dimensions, which goes 1, 3, 6, *10,* 15, . . . . I'm not aware of them being used for anything else, but that doesn't mean they don't exist! Someone feel free to let me know if they do.
Bank: “Good news: there’s twice as much money,
Bad news: it’s all imaginary.”
Broke: there’s twice as much money
Woke: it’s all imaginary
Bespoke: all money is imaginary
jb76489 made me laugh
... And... It's gone!
@@jb76489 It's complex.
Meh. I'll wait for 4 terms for my positive real money
You know it‘s a good math lesson when halfway through, you‘re stoked that there‘s still 35min to go.
1) I love how you distinguish between things that logically follow from what's already known on the one hand and things that are arbitrarily adopted because they turn out to be useful. As a former teacher myself and observer of other formal and informal teachers, I find putting arbitrary decisions as obviously true is the biggest way smart teachers become less effective with smart students.
2) The sirens weren't audible at all, at least on my laptop.
can you elaborate on what you mean when you distinguish that which is arbitrarily adopted and that which logically follows
39:38 You CAN make money in "i" interest compounded continuously. You just need to manage your money. After 0.8 years you withdraw your 71.73 real dollars and let the remaining 69.67 imaginary dollars ride for 1.5*pi years, leaving you with about 141.40 dollars total after that time
This is by far the most entertaining explanation of the 'spiral' nature of the complex plane that I've heard. Reminds me of the end behavior of z^n.
I've got an investment in America that pays 0% interest
It makes no cents
Гой не достоин такого. Слава Яхве!
OMG THAT'S SO FUNNY!!!
Ahah, clever
That does sound like America
Well, wait until Trump and JPow introduce negative interest rates...
came for the square root of negative one. left with a primer on financial engineering. thank you 3Blue1Brown very cool
Just finished a course in electrical circuits using capacitors and inductors. This lesson was very applicable to that.
I wouldn’t mind taking out a continuously compounding imaginary interest rate loan. Take out the loan, get your $100, then wait for half a rotational period, then take out another 100 to return to a zero dollar balance
I would take out $200 after the half rotational period
If r= i, after a rotation you will have 16x the inicial amount because 1 + i = 2^1/2 < 45°.
I am a finance major (financial analyst intern) who also happens to love math. This is amazing.
Absolutely awesome. Never have math concepts been so brilliantly translated. Its not my discipline but I come here for fun and to learn expecting to watch a single episode, and go to bed way into the wee hours pondering the beauty of the universe. The only truth in life is math. Bravo.
no it is not! It is Krishna, the Supreme Personality of Godhead.
"You can see this as the venture capitalist approach".
Thinking about that, it feels kinda like ANY business venture. Liquid capital is sunk into non-liquid assets like equipment, with real outflows from current liabilities, and non-current liabilities that might affect your non-liquid assets, in event of a bankruptcy, etc, but doesn't necessarily impact real revenue.
This might actually be the graph of WeWork's CEO. Start out with some money, it turns into imaginary assets, which become real losses, with debt piled on to bail out the company, for the CEO to sell his share at a massive huge personal profit.
This is explanation is just pristine. "All it really means is after taking a bunch of steps that are perpendicular to your current position and you do it continuously such that those steps are infinitesimal, It's the same as walking around a circle".
I really wish we had you as our math teacher when me and my colleagues where young. It would have been and eye opening experience. Thank you for everything you do, this is truly inspirational.
I love how you completely explained the concept without even having to resort to calculus!
@3blue 1brown what does that mean?
These lectures are so useful in lockdown. As a student who took A level maths and further maths (going on to a degree in physics) I've spent a fair amount of time with these concepts (trig, complex no. etc), but its awesome to go back and think deeply about the fundamentals in a way we didn't during our course before moving on to the more complex content, pun intended. Really useful for keep my understanding up especially with not being able to take exams now (feels like I've lost a bit of a chance work everything into longer term memory).
Never thought of complex numbers with SHM before, always gone a physical or differential equation route with it. Really interesting! Highlights yet again how much you can link topics from across subjects/curriculums.
Thank you for your amazing content, and helping me stay sane ;)
Same position...same feeling
Your enthusiasm for mathematics is infectious!
One of the best UA-camrs ever, period!
I think the takeaway is that if your bank is offering you an imaginary interest rate, make sure you get an account with an overdraft.
You managed to sneakily introduce the idea of phase spaces, as well as solving a differential equation without calculus. That's education ninja, right there! Bravo!
I though that this was all nonsense too until I started dealing with AC circuitry and complex power in my electrical engineering courses. It's insane how often what initially seems like a completely artificial math hack is actually key to understanding nature.
It is mental, like stuff you'd consider were people playing around with numbers for no apparent reason turns out to be a game changer for other people
Video Timeline
0:00:00 Welcome
0:00:55 Q1: Prompt (Would you take a interest rate)
0:02:05 "e to the pi i for dummies" video shoutout
0:02:45 Q1: Results
0:03:30 Q2: Prompt (two banks, two rates)
0:04:55 Ask: Beauty of connections in math
0:06:00 Q2: Results
0:07:05 Desmos for Q2
0:09:10 Q3: Prompt (savings growth rate, 6% every 6mo)
0:10:35 Q3: Results
0:12:35 Desmos graph explored
0:14:45 Breaking down an interest rate
0:18:00 An interesting interest equation
0:19:20 Q4: Prompt (100*(1+0.12/n)^2 as n → ∞)
0:21:05 Ask: Quaternions
0:22:35 Q4: Results
0:24:50 Explaining Q4
0:26:40 Defining e
0:28:40 The definition of e from previous lectures
0:30:45 The imaginary interest rate
0:32:35 Graphing this relationship
0:33:50 The imaginary interest rate animation
0:37:55 Compounding continuously with i
0:40:45 The spring & Hooke's law
0:43:20 Q5: Prompt (Δx & Δv for a spring)
0:44:50 Ask: Rotation in for multiple dimensions
0:47:45 Q5: Results
0:49:50 Rewriting the spring's position
0:55:00 Bringing it all together
0:59:00 Ask: Hints on last lecture's homework
1:03:25 Closing Remarks
Liquid Drinks at: 20:45 36:10 44:15 44:25
Edits: Spelling & formatting and updated some timestamps that drifted
These timestamps assumes the video is trimmed at or near 5:15. If watching the video before it is trimmed, add 5 minutes and 15 seconds to the above timestamps.
No way you wrote that in under one minute
My guess is you prepared this comment beforehand and pasted it here. That's a lot of dedication.👍
22:19 Quaternoins could be obtained from complex numbers by adding a "j" satisfying j z = \bar{z} j for any z from the "old" copy of the field of complex numbers, and imposing associativity and bilinearity of the multiplication. Here \bar{z} stands for the complex conjugate of z. A similar operation can get you from quaternions no (not quite associative) Cayley's *octonions*.
I think the difference is that quaternions are more on the "let there be some i j k such that [...]" because it's useful, whereas i is defined as the root of a polynomial and all its other properties are logical conclusions from that original identity. Don't get me wrong quaternions are awesome and useful, but 2d complex numbers are a lot more natural.
I had watched the e^pi video from Mathologer before, but this is what really made it click for me. Thank you!
Bank: "You owe us 100 imaginary dollars"
Me: "Okay, here you go!"
Bank: "Wait, what?"
Far better to ask for a loan of another 100 imaginary dollars, just billed to the same account.
I think I prefer these to the normal videos. I think having you there makes it a bit more engaging.
Too good to be true! I am talking about this lecture. Free. Insightful. Educational. Fun. What more can you ask for!
Just a little fun fact I wanted to share .
17:00 the equation Grant wrote is (M(1+r(delta t))) is also true for expansion of rod when it is heated... just replace r by temperature coefficient of that rod and M by the original length of that rod for small changes in temperature (
Them: Why would you ask such a silly question about Imaginary interest!?
Me: y₀?
veggiet2009 😂
Y sub zero
@@typo691 y not
@@Tuberex lol i know
B r u h
I love these lockdown math videos so much! They feel easier to follow than your shorter vids. I know part of the reason is because this is meant to be a more newbie friendly series, as opposed to the very knowledge dense shorter vids. But I think it's also because the short videos are too informationally dense for me to the point I get anxious I don't understand everything, and end up rewinding them a bunch of times and giving up somewhere in the middle. I guess my brain has a high inertia when it comes to science LOL. Hope to see more math lessons in this format in the future!! I think you're a great teacher.
I love this, thanks for this great content. I was happy that my first instinct was right, but after 3 years of electrical engineering this was expected. Although I noticed that I learned the e to the power of X expression like you showed it here. Funny enough my first thought when you said you would use a physics example was the force a moving, charged particle experiences in a magnetic field, which is orthogonal to the velocity and the magnetic field.
I haven’t watched the video yet and I already know this is gonna be some fun stock market-y stuff. I love when my interest oscillates from gaining money to owing it
Looks like "1" was the 5th most common number entered in that box, unfortunately most people got that question wrong!
I may be wrong but I think this question has a paradox in it. When you pick the 5th most common number, and if most people picks it too, it becomes the most common number, not the fifth.
@@Zeynep16th It's not a paradox. More than 80% of people will necessarily get that question wrong every time, but there's nothing wrong with that. It's sort of like a lottery.
@@EebstertheGreat
Well je didn't say anything about right or wrong... He only pointed out that the original comment said "most people" while this by definition cannot be the 5th most popular choice
@@jilleswassink7762 It might be interesting to ask a question like that in which you can submit multiple answers, like maybe 5. Then it would actually be possible for a majority of people to guess the correct answer as one of their guesses, though it would be very unlikely.
Sorry, was the answer mentioned in this video? I skipped some parts when I first watched, but having rewatched those moments I can’t find the 5th most common number part.
Thank you for this lecture! You blew my mind with this new perspective and this is coming from a person who has already done a quantum mechanics module at open university.
I think the reasoning for the "yes" answers for taking the imaginary interest rate is "it's not zero, which is what real banks offer."
What you do is take out a loan first to get your money to be negative, then wait for half the rotational period (where the money turns positive), then extract all the money (bringing the balance to $0).
(1+0.12/n)^n, if you replace 0.12 with 1 and when n blows up it approaches 'e'. In a sense, the exact opposite of option (E). I'm so glad that I choose option (E). This was a beautiful episode.
14:30 nobody should let the FED watch this video. I tkink it might introduce some shaky ideas.
Grant is Great matematician, smart guy and the best UA-camr EVER!!!
You're one of the best educators I've ever seen! Amazing series!
You end up with 100 Real $ and $100 Imaginary $. Aha! Now I know how bitcoins work!
Your explanation of e using interest rates reminded me of an old calculus text book called Calculus Made Easy by Thompson.
As usual , great presentation.
This is the explanation I have looked for for a year. Thank goodness
Fantastic lecture, I have one comment. At time 17.05 it is stated that M(T) is the amount of money we have on the account at time T. I think this is misleading. By definition of M, M must always be the amount of money on the account at time T. Implying that M is a function of time is wrong. If r is given, then T is known, and the question is how the frequency of accumulation, n, affects our savings.
Grant, you have made lockdown a complete pleasure with these live sessions! Almost don't want lockdown to end :)
32:52
"So you have what you had originally AND you can play monopoly" LMAO
Mathologer and you are the best math channels by far.
Do live lectures take more effort and preparations than a fully prepared animated video?
This is the fifth one he's done in the past few weeks and we usually get an animated video once a month so I would say no
Animated ones take way long. Not even because of the visuals, but once you write something exactly, you hold a higher standard for the writing and story progression.
That said, it turns out live lectures also take a lot of work to prepare! Especially given some of the work going into Itempool going on behind the scenes.
These lectures mixed with all the bells and whistles are very enjoyable. I was kinda smiling the whole time. In the future you should do live lectures whenever you feel like it and we're here waiting!! Thanks for your dedication.
@@SomeRandomGtaDude-zl3us Just evaluate the series defining exp at the given matrix...
@@SomeRandomGtaDude-zl3us It's hard to visalise things in higher dimension... However, if you have a diagonizable matrix then you are reduced to several 1-dimensional situations.
What a interesting start, more people know how a rotation by imaginary numbers works than a simple banking interest rates.
I've heard of octonions being a quaternion like representation for 4D rotations.
Steven Spencer Yes, correct. In general, the 2^(n - 1)-ions are numbers used for representing rotations in n-dimensional space.
This is so marvellously informative and entertaining and just pleasant to follow. Thank you so much for putting out content like this. I never thought mathematics could be so interesting.
I didnt have so much fun doing "homework"in a long time. Thank you for this great series and giving a great save heaven from alot of more "boring homeworl" :)
the only thing that makes me think during lock down days are these videos from grant!! thanks a lot
Imaginary interest rates sounds good better than negative interest rates after some period but again risk value is high. Quiet insightful to look into it. Also about Quaternions, so how degrees of freedom get described if it is not by complex numbers in higher dimensions?
Dear Mr Brown,
this is the first time i have lmao at one of your videos and at just 30 secs in for that matter.
what a wonderful way to introduce the concept you are about to present.
You are a master, sir.
@3blue 1brown huh?
I'd love to see a program that simulates saving with imaginary interest and lets you add or remove real and imaginary money in real time.
I don't think that would be too hard to code, might be a fun challenge :)
As an engineer who learnt SHM starting from how displacement is represented as a sine wave and built upwards from there to include complex numbers, this video was a very interesting change in perspective.
This would be more enjoyable if I wasn’t already dying from revision. Can’t wait to come back after exams
Joff good luck!!
One important thing with those interest rates at the beginning is usually when you calc this stuff in basic economics, the problem also states that the interest is capitalized every some time, meaning it goes back into the sum that then is used for calculating further interest. At least that's how it is where I live.
My intuition says that a complex interest rate would periodically increase and decrease in value.... wait does that mean that the REAL stock market has a complex interest rate as you have the boom bust cycle?
The stock market has a tendency to go up, though, so it's not as simple as randomly going up or down. Booms and busts also don't happen regularly, we've just had one of the longest bull runs in history, so you can't really describe it with an 'interest rate' unless you consider the average of many years of growth.
@@Gregster427 Yeah I was imagining more of a "x+cos(x)" kind of function, while a complex number roughly follows just "cos(x)" in the real part.
I found this interest problem more easy to visualize by remembering the definition of a product of two complex numbers. (a+bi)(c+di) = (ac-bd) + (ad+bc)i
In a way, the imaginary interest leeches away at your real interest (due to the "-bd" term), and continually increases itself such that it will basically always overtake the real interest even if you start it off small.
If a bank gave you a fixed real interest rate but then let you chose an imaginary interest b of your choice, I think the answer to maximize your real earnings would be to chose b = 0.
Can we take a moment to appreciate how spectacular it is that i^x and e^ix trace the same path. The proofs about powers to i seem almost obvious after acknowledging that fact, which also holds true for all positive numbers except 1, but it's actually an extremely rare property and that i just happens to be one of complex values it matches up at confounds just how extraordinary the discovery of how rising to the i works.
This is the embodiment of “but wait, there’s more!”
This video was mindblowing. By far the best I have seen. Thinking about imaginary interest rates is not insane at all. I even read the comments and I agree with some of them of the “sane” way of thinking about this. Could be a way of thinking of boom and bust cycles, as the way the spring behaves? Number “i” is related to sines and cosines which are also related to cycles. The second interesting way to understand this is by projects, you even mention the balance, cash-in, out of a project was well drawn by something as simple as your real cash balance depending on the periods. Projects my have an initial investment, then it is used to buy some materiales and after that you receive profits. The third way is as included in financial markets or as a kind of “implicit force”. Benoit Mandelbrot wrote a book called “the misbehaviour of markets” and he used his fractals -sets drawn and build on complex numbers- to explain the misbehaviour. The final way is that all the ways could be related! Congratulations!
I slept like a baby dreaming unit circles and money after this live lecture, its midnight 1:30 here.
You must be a pretty effing good teacher because I instantly knew the answer and I never completed anything more complicated than calc in high school... Except watching some of your videos...
And some other UA-cam math guys, but your visualizations on the complex number plane (particular the Reiman zeta function) made this a piece of cake.
Every time after I watch your video, I feel both dumber and smarter at the same time. In Chinese, there is a saying that roughly translates to "Outside every mountain, there is a taller mountain, beyond every person, there is someone smarter". I think you fulfill that requirement for me, and I am grateful. Please keep up the great work. If math is indeed the language of God, then you bring me one step closer to God.
Based take, comrade. Love it.
Exactly what I've been looking for. An imaginary interest rate for my imagination money.
Depends if another bank is offering an interest rate of 2i
20:50 The basis elements i, j, and k commute with the real quaternion 1, (e.g. 1*i=i), each other quaternion squares to negative 1 (i.e. i^2=-1, k^2=-1, j^2=-1), and i * j * k = -1.
I can't see his "3/4 blue 1/4 brown" eye that the channel is named after
Amazing video, and I did know where the lesson was going :D
Never felt so proud, love your channel, your lockdown videos really help me out
18:03 "INTERESTing EXPression" doudle pun haha XD
This just explained e to the pi i in the best way possible to me. I didn’t get what e meant in that it moved the spiral closer into the circle, but this graphical explanation was great.
Lesson begins at 5:14
The identity exp(A+B)=exp(A)exp(B) might not hold for all matrices, *but* it holds for all matrices A,B satisfying AB=BA. You need this property to sort products of A and B into expressions of the form A^n B^m, so that the binomial theorem and, subsequently, the identity exp(A+B)=exp(A)exp(B) both hold true.
They also have to be square matrices, in order to be able to raise them to powers.
@@gayatrisavarkar8196 True, but since the matrix exponential is only defined on square matrices, I didn't see any reason to mention it.
Just noticed that AB=BA is still not enough, because we even have to raise the matrix to higher powers, and so the matrix need not obey this for higher powers, eg. ABA=BAB need not be true. I guess we have to narrow in on diagonal matrices only
@@gayatrisavarkar8196 Nope, AB=BA is enough. We do not need ABA=BAB to be true, we only need two products of A's and B's to be equal, if they have the same number of A's and B's in them. And AB=BA guarantees that we can rearrange any product of A's and B's into A^m * B^n where m is the number of A's and n is the number of B's in the product. In fact, we can rearrange any product of A's and B's into any other product of A's and B's as long as they have the same number of A's and B's.
Borrow at imaginary rate, wait until I owe the negative of the principal, "pay off the loan".
I think the answer to Bobs question on viewing the Quaternions as an extended number system should be yes, by virtue of the Cayley-Dickson construction and/or the Frobenius theorem.
well, at the end the question about the exponential identity for matrices is pretty easy.
exp(A)*exp(B)=exp(A+B)
if A and B are real (or even complex) numbers, the identity holds. in case of real and complex numbers addition and multiplication are also commutative. but for matrices only addition remains commutative, matrix multiplication breaks this symmetry. this non-commutativity is the reason that the above identity is not valid anymore. why? because the exponential of a matrix is also a matrix. so the left hand side of the identity above is a multiplication of two matrices meanwhile the right hand side contains an addition of two matrices (then the exponential of this sum). so one side of the equation is commutative, the other side of the equation is non-commutative. this is a clear sign that the equation does not hold in general. because exp(A+B)=exp(B+A), meanwhile exp(A)*exp(B)≠exp(B)*exp(A), so two terms should be equal and not equal at the same time. contradiction...
It is FALSE that A*B ≠ B*A, for any matrices A and B. For some matrices A and B, A*B = B*A, for others, it is not. So, maybe it _could_ be the case that exp(A) and exp(B) would always be matrices of those types in which it commutting gives the same result.
It is not the case, but it could be.
Guilherme Zeus Moura dude try to comprehend what I wrote. I was talking about IDENTITIES. identities represent generality. so when I said A*B≠B*A it meant that they are not equal in general (i.e. the identity is not necessarily true). so yes, you are right, but I was talking about generality and identity. non-commutativity is similar: the term “non-commutative” doesn’t mean that in specific cases it cannot be commutative, but in general it is not commutative.
When it comes to matrices, if you have a single matrix as input, and if it is a square matrix, only then will this equation be valid. The matrix has to be a square matrix before you even consider two matrices as inputs
I'm so glad I watched your video on ODE before this (making intuitive parallels between the spring and the [then] pendulum made this possible to follow. (I never took differential calculus before haha).
SO thank you twice :)
Unfortunately last time I tried to pay for anything with imaginary money I got arrested, therefore would strongly not recommend :(
Love how the answer bars right at the start grew almost identically.
39:39 By Grabthar's hammer, what a savings.
The physics portion of the lecture had me thinking. So, even if the spring system is not subjected to any friction (54:45), if we consider time as quantized (not continuous) the graph would still spiral inward (the opposite of 36:27), so we would be still losing amplitude over time.
The spiral at 36:00 looks like the golden ratio which got me wondering whether there is a way we can approximate it using imaginary interest rates. And what will be the step size? Will there be any sort of cool relation between the two?
I think a really excellent way to illustrate the answer to Deepak's question about 3 dimensional rotation is with video games and 3D modelling. When you rotate something in 2 dimensions, you rotate it along a single axis (let's call it Z). It's like putting a pencil through a piece of paper and spinning it, that pencil is your axis. But if you pull out into 3 dimensions, you can now also rotate along the X and Y axes. Watch some videos on 3D modelling and you can see all 3 axes in play as objects are rotated along them.
And the people writing game engines and 3D modelling software have to do all of this math we see in this video in order for those applications to work.
He is wearing the same shirt than in the episode 1, that mathematically means that he is going to be wearing the shirt of the episode 2 the next one
yes, correct calculation
You clearly haven't seen any of his videos 🤣
Maybe he is rotating his shirts with the fourth root of unity
*Thomas Bayes would like to know your location*
M O D U L O 4
Question: How is i^i 0.207.... or e^-pi/2.
Considering that when we work with complex numbers e^ix is actually the Exp() function and not the constant e, then how will one calculate i^i because that requires e to be the Andrew Jackson number to give the value of 0.207 approximately
um wow?
Yeah, he didn't go into what x^y means when x and y are arbitrary complex numbers. First you express x as e^z, then you have
x^y = (e^z)^y = e^(yz). Since i = e^(i*pi/2) we have i^i = e^(i*pi/2*i) = e^(-pi/2). At this point you can calculate e^(-pi/2) as
1 + (-pi/2) + (-pi/2)^2/2 + (-pi/2)^3/6 +... like usual.
The value of i^i is actually ambiguous. You use, say, i = e^(i*5pi/2) and get a different answer.
@@martinepstein9826 yes, I get that. But why do we take e as 2.7.... at all, as that is just exp(1) in the case of real numbers only right?
@@aapeli9662 I get that, what I am asking is, if u see the proof for i^i which are complex numbers, the final answer comes out to be e^-pi/2, then they substitute e as 2.7...... But according to Grant's last lecture exp() is only equivalent to e^x in case of real numbers, which these are not, hence why should we take e as 2.7... in this proof
@@aapeli9662 no I meant in the proof of i^i we take ln i.e log with base e, which itself shouldn't be possible because the constant e has nothing to do with the complex numbers, I think just Google i^i proof once, and u'll see the process where they use the constant e from the start of the proof.
I loved the comment about the VC money and how startup valuations are pretty much imaginary for the first couple of financing rounds:))
If your "i interest rate" was compounded once every 1.618... years, would you see a golden spiral on that graph?
A true math lover chooses to face a global pandemic by producing videos to help others learn.
Does "interest rate of i" mean multiplying by i every time or by (1+i)? Because when I hear "interest rate of 10%", it means multiplying by (1+10%). If it's (1+i), definitely worth it since the magnitude doubles every time.
Depends if the bank penalizes you when the real dollars go negative. (And the consequences of accumulating or "owing" negative imaginary dollars in the account.)
@@ratamacue0320 I mean if the bank penalizes you by taking money, that actually helps.
I agree that the wording is ambiguous. But "interest rate of i" is probably intended to refer to the coefficient of i, viz. 10%. Notice that (1 + i) is equivalent to (1 + 100% i).
Nitpick: (assuming L2), the magnitude of (1 + i) is sqrt(2) = 1.414. But yes, (1 + 100% i) is much larger than (1 + 10% i).
Yeah, the bank simply closes the account when the value drops to zero real dollars.
i love your videos. Nice to see a face ,on your voice and to see you in "live"…go on!
bank: haha imaginary money printer go brrrr
me: uhhh-- not worth it
I was just thinking and wondering about this out of the blue, and of course there's a 3b1b video about it! And I couldn't be luckier.
This example seems to support the legitimacy of the term *"imaginary numbers",* the historic term that you and others have described as misleading, because they are just as "real" as *real* numbers.
Even though it seems silly, a bank really *could* offer an annual interest rate of i, or some fractional multiplier of i, compounded *annually.*
But, the account could only be liquidated and closed when the account balance was entirely *real* (negative or positive) and thus offset by depositing or withdrawing a corresponding amount of *real* money.
At all other times, the account's "imaginary" balance would be mathematically valid as a ledger balance, even transferable to another account, but not representable or usable in the "real" world.
In this example, isn't *"imaginary"* a reasonable and informative adjective for the account balance?
Thing is that constraint wouldn't even really be all that unusual in the financial world there are after all instruments with restricted withdrawal terms like bonds after all.
And 50 years from now, cash is abandoned, and all money is complex, even to buy groceries, and handling your finance is even harder than it is for many people today.
Your point with the matrices and quaternions lacking multiplicative commutivity is a good one but I would argue that it just means that we need to be more specific with the properties we say functions have and when they apply
I'd love a bank with continuously calculated imaginary interest rates for loans.
I'd be happy to take a loan of an imaginary fortune and wait for about 1.571 years
And if loans don't come with imaginary interest rates, only deposits, then I'll gladly open a bank account on which I deposit my imaginary fortune (a man, can dream, right?) and wait for some 4.712 years before I withdraw my total investment