Euler's Identity (and other maths) explained in 5 minutes

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:

  e ^ iπ = (1, π) = cos π + i sin π = -1

EDIT: clarification.

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 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.

Dear all,

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.

Cheers,

Oliver.

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.

Basically stand up comedy for mathematicians. :)

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.

Good job sir! Wonder if he does tuition? :)

Very impressive, and very good explanations! I just wish I could remember everything ;)

As someone not familiar at all with the material presented, I enjoyed it very much.

I wish I could talk that fast.

I liked the bit where I didn't understand anything.