

Euler's Identity (and other maths) explained in 5 minutes  - jsavimbi
http://www.b3ta.com/links/Eulers_Identity_and_other_maths_explained_in_5_minutes

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ScottBurson
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.

~~~
alok-g
>> Okay, let's put these two things together:

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.

~~~
ScottBurson
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.

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rottencupcakes
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.

~~~
kalid
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...](http://betterexplained.com/articles/intuitive-guide-to-angles-
degrees-and-radians/))

What does e mean? It's the result of 100% continuous growth
([http://betterexplained.com/articles/developing-your-
intuitio...](http://betterexplained.com/articles/developing-your-intuition-
for-math/)). 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.

i can be seen as a rotation ([http://betterexplained.com/articles/a-visual-
intuitive-guide...](http://betterexplained.com/articles/a-visual-intuitive-
guide-to-imaginary-numbers/)). But more importantly, it's an "operation which
rotates". When you write 3 * i you're saying "take 3 and transform it in some
way -- rotate it".

When you start combining these ideas
([http://betterexplained.com/articles/intuitive-
understanding-...](http://betterexplained.com/articles/intuitive-
understanding-of-eulers-formula/)) 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.

~~~
ComputerGuru
Wish I could upvote this several times!

Guys, don't let the length and links scare you off, _excellent_ summary! And
nice website, too!

~~~
kalid
Thanks! It's my mission to share the explanations that actually worked for me
:).

------
mthomas
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.

~~~
jamesjyu
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.

Cool to watch, but not very useful, IMHO.

------
oliverhumpage
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.

------
hoag
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. :)

------
estel
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.

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

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

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benawabe896
As someone not familiar at all with the material presented, I enjoyed it very
much.

------
jimmyjim
I wish I could talk that fast.

~~~
pedrokost
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.

~~~
AlecSchueler
Do you think you find it easier than most people to make sense of other fast
speakers?

~~~
pedrokost
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.

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

