I think there's a more intuitive explanation than the Taylor series expansions (whose correctness, after all, is not exactly obvious). It is not hard to show that the product of two complex numbers in polar form is:
(r1, θ1) ⋅ (r2, θ2) = (r1 ⋅ r2, θ1 + θ2)
That is, the radii multiply, but the angles add. If we consider only points on the unit circle, where the radii are all 1, clearly the product of any two such points is a third such point.
Now consider that
e ^ x = lim ((1 + x/N) ^ N)
N -> ∞
This equation defines real exponentiation in terms of integer exponentiation, which in turn is defined in terms of multiplication. To make it familiar, consider that
(1 + I/N) ^ NP
is the formula for compound interest, where I is the interest rate per period, P is the number of periods, and N is the number of compounding intervals per period. Let's say P = 1 so we can ignore P. If the interest rate is 12% annually and you're compounding monthly, then the balance after one year is higher by a factor of
(1 + .12/12) ^ 12 ≈ 1.1268249
But you could compound daily, hourly, by the minute, second, millisecond, ... As the compounding interval gets smaller, the result approaches
e ^ .12 ≈ 1.1274968
Okay, let's put these two things together:
e ^ ix = lim (1 + ix/N) ^ N [let "N -> ∞" be implicit]
= lim (r, arcsin x/Nr) ^ N
where r = √(1 + (ix/N) ^ 2)
As N -> ∞, r -> 1 and arcsin x/N -> x/N, so this reduces to
e ^ ix = lim (1, x/N) ^ N
= (1, lim (x/N) ⋅ N))
Remember, the radii multiply, but the angles add! This is the key step. From here it's easy:
e ^ ix = (1, x)
That's the general result; from there the special case is trivial:
You cannot just put these two together without making your proof cyclic. AFAIK, the second one, e ^ x = lim ((1 + x/N) ^ N), can be proven only for real x without using Euler's formula or almost proving it as an intermediate step. Thus substitution from #1 using complex numbers is not legal without proof.
Here's an analogy to consider:
1. Boyle's law: P1.V1 = P2.V2
2. Charles law: V1/T1 = V2/T2
Multiply LHS and RHS of the above two.
3. P1.(V1^2)/T1 = P2.(V2^2)/T2.
Yet, the gas equation is ...
4. P1.V1/T1 = P2.V2/T2
... and not #3.
While there is nothing wrong with the multiplication used for computing #3, this process adds the assumptions used for #1 and #2. Here's a more precise version then:
1'. Boyle's law: P1.V1 = P2.V2 when T1 = T2
2'. Charles law: V1/T1 = V2/T2 when P1 = P2
Multiply LHS and RHS of the above two.
3'. P1.(V1^2)/T1 = P2.(V2^2)/T2 when T1=T2 and P1=P2.
Derivation for #4 avoids those two assumptions.
In your proof of Euler's identity, the assumptions for your #1 and #2 cannot be considered to be compatible without proof.
I never used the word "proof". I said "intuitive explanation". Yes, it looks a bit like a proof because I wanted to break it down for people, but it contains hand-waving at more points than the one you identified.
It's a disappointing talk, simply because he wastes time reciting and explaining pi and e, then glosses over Euler's formula (http://en.m.wikipedia.org/wiki/Eulers_formula) which is the only real magic in this identity almost as if it's nothing, with hand waving Taylor series.
I agree -- I'm appreciative that he's trying to explain math to a wider audience, but realistically, reading through textbook definitions at 100wpm just confuses people.
(Shameless plug starting now).
I think we need to focus on the intuition behind _why_ things happen, not repeating the textbook definitions that confused us the first time.
Why are radians more natural than degrees? Because they are from the point of view of the mover, not the observer. Would you tell a runner to run 3 laps or run X degrees around the track? (http://betterexplained.com/articles/intuitive-guide-to-angle...)
What does e mean? It's the result of 100% continuous growth (http://betterexplained.com/articles/developing-your-intuitio...). The infinite Taylor series can be seen, intuitively, as your principal (1), the interest that principal earns (100% of 1 = 1), the interest your interest earns (1/2), the interest that 2nd level interest earns (1/6) and so on. e has simpler definitions and the Taylor series was only used to "explain" sine and cosine.
When you start combining these ideas (http://betterexplained.com/articles/intuitive-understanding-...) something neat arises. You can guess, intuitively, that if e^x means "100% interest for x years" then e^ix means ("100% rotated interest for x years"). What's rotated interest? It's change that pulls you sideways (90 degrees) and not forward (which compounds and makes you go further along the number line).
The neat thing about constant, 90-degree growth is that it moves you in a circle (imagine a stick on the ground with a firecracker mounted perpendicular). That's how gravity, etc. work too -- your velocity is always tangent to your position when orbiting.
The intuitive reading of "e^ipi = -1" (which is more clear... why the +1 = 0? Oh yes, we wanted to rearrange it so another constant appeared to make it more mystical) is "Start at 1 and intend to grow at 100% interest. Whoops, you are getting rotated interest (because of i). Go for pi seconds... this takes you 100% * pi units around the circle, because rotated interest does not compound (you don't spin faster and faster). Halfway around the circle is -1."
In general, sine/cosine give you vertical and horizontal position of an angle, so we can generalize even more, but this is confusing at first.
If you focus on the real meaning of the operations, not the DNA-splicing of the Taylor series, it's a lot easier to see what's happening. I didn't make sense of it until years after college because I kept looking at it from a formal mathematical viewpoint, instead of focusing on what clicked for me.
Bonus question: if someone claims e^i*pi + 1 = 0 is their favorite formula, ask them what i^i is. If they struggle, the probably only understand the mechanics and not the meaning of it. Even better, have them explain in words why i^i would be a real number.
I highly recommend Brian's visualization -- another way to see the identity is to imagine e^ix as a spiral that gets tighter and tighter as you compound more frequently (e represents infinite compounding).
This formula has so many cool interpretations, yet is often explained by resorting to Taylor series (which is like comparing the equations molecule-by-molecule instead of seeing the big picture).
I don't understand the point of the talk. The only way you could follow what he was saying was if you understood the material he was trying to "explain".
I think someone wanting to learn about about the relation between pi, e, i, sine and cosine wouldn't be helped by this talk.
Totally agreed here. I understood it only because I literally lived with Euler's identity as a signal processing engineer for many years.. and even I started to lose him near the end when he was going at 100mph through some of the most crucial steps in the derivation, not necessarily because I didn't understand it, but just because he was going through the tricky steps too quickly.
Entertaining, but hardly useful: the only people who would follow any of this are those who already understand it. Those who don't would be hopelessly lost.
But definitely a lot of entertainment value for those who already understand -- even if only somewhat -- what's being discussed.
I was going to try and attend this Ignite. I'm not convinced that this makes me regret not doing so.
That is: I don't think that this was presented at a speed where anyone not familiar with the mathematical principles will feel anything other than otherwhelmed by information overload, and people previously familiar might be amused by the presentation, but otherwise probably not that enlightened.
I used to speak that fast, and I still do when I forget to speak slow.
The problem is that most humans can hardly follow the speaker and think about it at the same time. I was actually forced to learn to speak slow in high school by my language teacher. My presentations in front of a class were too fast even for the teacher.
Speaking fast has its advantages, but it should only be used for simple presentations like this one. It certainly doesn't work for very scientific topics.
I am not sure of that. I don't have enough evidence to deny nor to accept it.
I remember once another classmate had a presentation and she spoke faster than normal and got criticized for it. However I did not think she was speaking fast until she was criticized. It seemed completely acceptable speed to me and I could very easily follow her line of thought.
This is the only example I remember and it's probably insufficient to accept your conjecture.
Somewhere in the video the neuroscientist talks about her cerebral cortex working differently than it does for most people, and indeed, she's able to understand fast speakers better.
Thank you for taking an interest in my talk! To give it some context, Ignite is a special format where you talk for exactly 5 minutes - you get 20 slides, each 15 seconds long, that advance automatically behind you. Just quantising what you have to say into exact 15 second lumps to fit the slides actually shapes what you say, and how, quite a lot. The video doesn't show the slide changes, so you don't quite get the feel of that.
I won't debate the "dumbing down" aspect - I'm sure it's equally valid for you to feel about what I did as it is for me to dislike general TV science output as being far too simplistic - we all have our levels.
However, I did just want to point out something that seems to have been missed, and that's that I'm not actually trying to prove Euler's identity in 5 minutes, or make the proof itself the point :) The talk had two main raisons d'etre:
1. To get people who had done A-level maths (that's around age 16-18 at school) many years ago to have their memories jogged, and push that long-forgotten bit of their brains back into service a bit.
2. To be, quite simply, unashamedly enthusiastic about maths, without being at all patronising or speaking down to the audience. There was a recent TV show on the BBC called something like "Beautiful Equations" which turned out to be an artist going round painting the damn things - they couldn't even explain E=mc^2 properly. This was a bit of a reaction against that.
Plus, of course, when you're talking to a room full of interested but not necessarily mathematical people, you have to be entertaining. Hence the pi recital (which also doubled as 15 seconds of getting people used to fast speaking so their ear was in for the rest of the talk), the pic of my daughter, etc. From the many comments I've received in person and online, I reckon the talk did a pretty good job at all that. Can't speak to everyone all the time, of course, but it's found a larger audience than I guessed it would have. That so many people can be interested in Maths With Equations In is a good sign.
I guess the beef some of you have is the Taylor series stuff: the reason I went that route is that a) I had 5 minutes (well, less, considering what else needed to be fitted in) and it's quick and easy to see visually how the equations match (far quicker than explaining about polar co-ordinates), and b) it's frankly the easiest way to explain it to the wider audience, and most likely to be understood/remembered by A-level students.
There's also the "missing the magic" complaint, but what was in there was seriously magical to the audience who were there: you really need a deeper understanding of maths to see the perhaps more beautiful explanations you've offered here. Plus just the result, the fact of itself, is the most magical thing for me, and that's what I wanted to get across most. Even leaving equations aside, the enthusiasm and humour in the talk seems to have inspired many to think about maths differently, and that's what I was after!
Anyway, thanks for your comments (they've made me think, certainly) and sharing all these links and posts, it's all good fun reading.
Now consider that
This equation defines real exponentiation in terms of integer exponentiation, which in turn is defined in terms of multiplication. To make it familiar, consider that is the formula for compound interest, where I is the interest rate per period, P is the number of periods, and N is the number of compounding intervals per period. Let's say P = 1 so we can ignore P. If the interest rate is 12% annually and you're compounding monthly, then the balance after one year is higher by a factor of But you could compound daily, hourly, by the minute, second, millisecond, ... As the compounding interval gets smaller, the result approaches Okay, let's put these two things together: As N -> ∞, r -> 1 and arcsin x/N -> x/N, so this reduces to Remember, the radii multiply, but the angles add! This is the key step. From here it's easy: That's the general result; from there the special case is trivial: EDIT: clarification.