Math is beautiful

March 9, 2010 12:43 AM

It's been called the most beautiful theorem in all of mathematics.

Euler's number, e, is an irrational number originally discovered by Jacob Bernoulli, who was attempting to solve a problem involving compounded interest. π was discovered by the ancient Greeks trying to find the area of a circle. The imaginary unit i is the square root of -1, and was used to solve polynomial equations which had solutions that didn't exist on the real number line. 0 is the symbol for 'nothing', and was first used by Hindu mathematicians as a place holder in their decimal notation system. And 1 is, of course, the first number. 5 important constants, seemingly completely unrelated, which have a surprising relationship: e^πi+1=0 The proof is beautiful and if you take your time, surprisingly easy to understand 2, 3, 4, 5, 6, 7.

posted by empath (48 comments total)

(The Khan Academy series on calculus is pretty good in general, and it would probably help to do them in order before getting to the last links there, but I think its basically understandable even if you don't know exactly what he's doing at all times)
posted by empath at 12:45 AM on March 9, 2010

Once you have e^πi = -1, adding the + 1 seems kind of superfluous.

The basic 'trick' is the trigonometric identity e^xi = cos(x) + i * sin(x), which is called Euler's formula. Given that, it's obvious that e^πi would be an integer.

Also, e and pi aren't just irrational numbers like √2, but transendental numbers that cannot ever be created from rational numbers using algebra.

(so for example the solution to x² = 2 is √2, an irrational number. "x²" is the polynomial. But there's no finite polynomial that ever has e or π as a solution)
posted by delmoi at 1:02 AM on March 9, 2010 [6 favorites]

well, writing it as 3^πi+1=0 also has the fancy benefit of using all the basic operators (addition, multiplication and exponentiation) exactly once. I think it's prettier that way.
posted by empath at 1:04 AM on March 9, 2010 [1 favorite]

err -- i mean to type e instead of 3 there obviously
posted by empath at 1:04 AM on March 9, 2010

Also, e and pi aren't just irrational numbers like √2, but transendental numbers that cannot ever be created from rational numbers using algebra

Yeah, to me that's really the mindblowing part of it. Two numbers that to all appearances are completely random strings of digits, somehow combine to create a nice round number.
posted by empath at 1:07 AM on March 9, 2010

Not sure if it's been posted before but I found these series on Radio 4 excellent:

5 Numbers
Another 5 Numbers
A Further 5 Numbers
posted by slixtream at 1:41 AM on March 9, 2010 [6 favorites]

The thing is, though, they're not really random strings of digits, they are defined by their (infinite) polynomials. And their polynomials are really similar. this page does a great job of explaining it.

With e^x the polynomial is = 1 + x + x²/2! + x³/3! ... (basically xⁿ/n!) for each natural number n. (e would then just be e¹)

and then the equation for cos(x) = 1 - x²/2! + x⁴/4! ... and sin(x) is x - x³/3! + x⁵/5!

_{(so cosine is +/- x^y/y! for even numbers and sin is the same but with odd numbers)}

You can see right away that the formulas a very similar they both involve x raised to a power then divided by the factorial of that power. Except in the trigonometric functions, it alternates between adding and subtracting.

What makes it all fit together is is that iⁿ is either going to be 1, -1, i or -i. So when you take e^i*x all those components get multiplied by one of those four complex numbers. All the 'odd' powers of ix end up as imaginary numbers. Which are exactly the components of sin(x). And all the 'even' powers of ix are real numbers, which also make up the components of cosine.

So right there, the component pieces are in the right place for e^ix = cos(x) + i*sin(x). The last bit is the fact that the components of cosine are either multiplied by 1 or -1, which is exactly what we need. And the components of sin(x) are multiplied by either i or -i, which is again, exactly what we need.

Basically you can think of i as causing a kind of loop as it goes through it's powers, which filters out the powers of x that are present in e^x into the right spot for the cos(x) and sign(x)

(I don't know if that was covered in the video or not)
posted by delmoi at 1:44 AM on March 9, 2010 [6 favorites]

You think that's amazing? e^{π √163} = 262537412640768744 exactly. Doesn't even need an i in it.
there may be a slight but surprisingly small deviation from fact in the above statement
posted by edd at 1:46 AM on March 9, 2010

"Surprisingly easy to understand" being a relative term, I'm hoping.
posted by l2p at 1:50 AM on March 9, 2010 [1 favorite]

Just go with the videos, they're pretty tough going in the beginning when he's talking about derivatives (especially if you don't know what they are, obviously) but if you just assume he knows what he's talking about in the first 2 or 3 videos, it starts to click -- if you really want to understand the proof completely, he's got a complete series on calculus you can get to from here.
posted by empath at 1:54 AM on March 9, 2010

Our cheapness changed the world. Indians are so dedicated to being so cheap for so long, that Indian people actually created the number zero. You know how much dedication that took? That means, back in the day some Indian guy was looking at the numeric system ...

*Indian accent* "1 2 3 4 5 6 7 8 9 ... Hmmm ... None of those are the amounts I want to pay."

Then his friend came along and drew a circle.

"Whats that?"

"Nothing."

"Whats inside of it?"

"Nothing."

"What's its value?"

"Nothing."

*Sniff* It's beautiful (shedding a tear). We shall call it jeero. Take it and go."

~Russell Peters
posted by bwg at 2:03 AM on March 9, 2010 [4 favorites]

This is great stuff, thanks. Although I have to say (just because) that, as wondrous as Euler's theorem was, nothing in it can hold a candle to the beauty of the later books of Apollonius of Perga's treatise on conic sections.

I've been reading (or trying to get through, for a third time) a fantastic book by Jacob Klein called Greek Mathematical Thought and the Origin of Algebra. It's dense stuff – Klein was a student of Heidegger's, and this was (I believe) his graduate thesis, so it takes some digging – but it's highly interesting. Chiefly it deals with the fact that modern mathematics has come to see the algebraic model of mathematics as preeminently reflective of reality, whereas the Greeks would clearly have seen this as a fundamentally flawed perception of the universe; it's clear even from Euclid, who is more conducive to algebra than most Greek mathematicians, that the road of geometrical equivalence with algebra was a long and fraught one.

I think that's a fair enough thing to say. I remember hearing a student of Klein's once opine that Richard Dedekind's ideas about number theory (he's the fellow who came up with the number line) "suck all of the meaning out of numbers." Interestingly, I think Heidegger would probably have agreed; there's a lot more to "one" and "many" than Dedekind leaves in it. And Dedekind is, after all, just the last in a long line of mathematicians who moved further and further away from believing that numbers and geometric figures have any inherent significance or real meaning beyond a kind of simple equivalence. Modern mathematics flattens all of this and ignores what math is ostensibly about – quantities, equivalences, numbers, amounts. I think ancient and modern mathematics are probably concurrent, and don't disagree on major factual points, but they do differ in this respect: modern math starts from assuming that numbers underly geometric figures, in much the same way that modern science assumes that numbers underly physical reality – whereas ancient mathematics started from the opposite assumption, insisting that commensurability be proven somehow.

Anyhow, that's just random guff, an interesting indication that modern mathematics may be somewhat misled or at least held back by its form. Euler certainly follows Descartes' wholly modern number theory on this point. Sorry to thread-jack somewhat; just thought it was interesting.
posted by koeselitz at 2:11 AM on March 9, 2010 [10 favorites]

Another way to think about it that's much simpler, but with the downside that you need to understand vector spaces and I just thought it up, so it could be totally wrong:

--

Did you know that 0! = 1? (that means x⁰/0! = 1).

Because of that, if you think of an infinite vector space where the bases are xⁿ/n! for each n, then (i think) e^x then would be the vector

1,1,1,1,1,1,1....

cos(x) would be the vector

1,0,-1,0,1,0,-1...

and sin(x) is the vector

0,1,0,-1,0,1,0,-1...

so if you multiply the sin(x) vector by the scalar i you get

0,i,0,-i,0,i,0,-i...

and so of course the linear combination of cos(x) + i*sin(x) =

1,i,1,-i,1,i,1,-i...

(they're orthogonal)

And that's the same thing as the vector

i⁰, i¹, i², i³, i⁴

so the function e^x, sin(x) and cos(x) are all a part of the same 'space', if you think about it that way. They look different: e^x goes off to infinity and sin and cos just wiggle around between 1 and -1. But that's because e^x is all addition, while cos(x) and sin(x) have addition and subtraction.
posted by delmoi at 2:14 AM on March 9, 2010 [8 favorites]

modern science assumes that numbers underly physical reality

It assumes no such thing, except insofar as equations match with experimental results.
posted by empath at 2:18 AM on March 9, 2010 [1 favorite]

By the way, I mean to mention the fact that you're actually incorrect, empath (no offense) when you say that π was "discovered by the Greeks." For the Greeks, π was a completely different thing – principally, it was not a number, and an ancient Greek mathematician would have laughed at anyone who tried to claim that π was a number at all. For, she or he would say, first of all, how can a length be simply a number? Second, how can it be a number when it is not commensurable with any known related length? To the ancient Greeks, π was a length – an incommensurable length which had a relation to other lengths, but a length nonetheless. And that wasn't just because applying numbers to lengths didn't occur to them – they had very real reasons for believing that it was a mistake.
posted by koeselitz at 2:21 AM on March 9, 2010 [1 favorite]

me: “... modern science assumes that numbers underly physical reality...”

empath: “It assumes no such thing, except insofar as equations match with experimental results.”

That was a pretty sloppy way to say it, I admit. I should have said something like: 'modern mathematical physics assumes that numbers as algebra conceives of them underly physical reality.' Personally I think it's hard to deny that numbers underly physical reality in general – one and many are all around us – but that's not what I meant, really. And this isn't an attack – more an observation about how all of this stuff works now, in our minds. It's a language that's become so intuitive to us that we tend to forget that it's a language at all.

For example, things like sine, cosine, tangent – all these numbers and equations which define geometric images in our minds – we take it for granted that thinking in those symbolic terms is the sum and substance of mathematics. For the Greeks, mathematics was fundamentally different, and when Euclid tried in book 4 of his Elements to get to the point where he could let symbols stand in for lengths, even he did so very tentatively, and there was some controversy about whether it was the right thing to do. I only like the idea of questioning the language, and where it came from – from Vieta, from Descartes, and from a few people who translated the Arabic version of geometry in the 1400s and 1500s.
posted by koeselitz at 2:40 AM on March 9, 2010 [1 favorite]

I'm having a shit-ton of fun with these links, by the way, empath. Thanks for this.
posted by koeselitz at 2:41 AM on March 9, 2010

transcendental: numbers that cannot ever be created from rational numbers using algebra

delmoi, after 3 years studying mathematics, and several years in engineering, you have expressed in plain english and made me understand the concept of a transcendental number in a simple way, for the first time.
posted by molecicco at 2:55 AM on March 9, 2010 [3 favorites]

Once you have eπi = -1, adding the + 1 seems kind of superfluous.

No, then you wouldn't have the five most important constants in mathematics.
posted by ryanrs at 3:17 AM on March 9, 2010 [2 favorites]

an ancient Greek mathematician would have laughed at anyone who tried to claim that π was a number at all.

I think you might want to check your facts.
posted by empath at 3:18 AM on March 9, 2010

3 10/71 and 3 1/7 are not numbers.
posted by koeselitz at 3:42 AM on March 9, 2010

*waits for khan to get to quarternions and clifford algebras*

(oh and you forgot the khaaaaaaaaaaaaaaaaan! tag ;)
posted by kliuless at 3:45 AM on March 9, 2010

3 10/71 and 3 1/7 are not numbers.

what
posted by empath at 3:58 AM on March 9, 2010

an ancient Greek mathematician would have laughed at anyone who tried to claim that π was a number at all.

I think you might want to check your facts.

Somewhere, I have an mp3 of an ancient Greek laughing, but, back then they were not very happy about irrational numbers at all, let alone transcendental ones. The idea that 2 lengths could be incommensurable , that is, that 2 line segments couldn't always have some measuring unit which could be laid end to end a whole number of times in both segments, was very disturbing to all the ancient Greeks I know, with the exception of Pythagoras, who was something of a mystic and thus more open to weirdities.
posted by Obscure Reference at 4:54 AM on March 9, 2010 [2 favorites]

Metafilter: Gleefully doing the homework they dodged in school.
posted by Bathtub Bobsled at 5:05 AM on March 9, 2010

as wondrous as Euler's theorem was, nothing in it can hold a candle to the beauty of the later books of Apollonius of Perga's treatise on conic sections.

Later books? Are you thinking of Euclid?

As for Greeks not considering π a number because it was a length...I dunno. They did number theory via geometry and it seems doubtful to me they could have avoided noticing that the lengths were simply integers. It seems more likely to me that they only appeared to not think of numbers because the most powerful tool they had was geometry and they translated everything to that to do their work.
posted by DU at 5:15 AM on March 9, 2010

Oh, later books in the sense of the end of the set of books that comprise the treatise.
posted by DU at 5:16 AM on March 9, 2010

And Dedekind is, after all, just the last in a long line of mathematicians who moved further and further away from believing that numbers and geometric figures have any inherent significance or real meaning beyond a kind of simple equivalence. Modern mathematics flattens all of this and ignores what math is ostensibly about – quantities, equivalences, numbers, amounts.

I'd like to know more about what you mean by this, koeselitz.
posted by libcrypt at 5:22 AM on March 9, 2010

When math geeks whip their dicks out, we all win.
posted by seanmpuckett at 5:27 AM on March 9, 2010 [5 favorites]

modern mathematics may be somewhat misled or at least held back by its form.

Or, you know, maybe pre-modern mathematics was misled and held back by its form? Sure, you can do mathematics of a kind in which pi is not a number -- or for that matter in which 3 1/7 is not a number. In which infinities and probabilities are by definition not quantities, in which -1 has no square root.... but there are really good reasons that nowadays we operate in a different way, and as a result we understand things our ancestors were conceptually unequipped to grasp.
posted by escabeche at 5:59 AM on March 9, 2010 [4 favorites]

Euler's is the most beautiful? You've never seen the Pythagorean theorem in a bikini.
posted by dances_with_sneetches at 6:03 AM on March 9, 2010 [2 favorites]

But XKCD should not make a comic about the beauty of e. That would be wrong and make people angry.
posted by clvrmnky at 6:11 AM on March 9, 2010 [3 favorites]

Maybe, but this equation is also pretty beautiful.

Nice post. I cant wait to watch the proof videos.
posted by yeti at 6:31 AM on March 9, 2010

koeselitz: modern math starts from assuming that numbers underly geometric figures.

koeselitz: For example, things like sine, cosine, tangent – all these numbers and equations which define geometric images in our minds – we take it for granted that thinking in those symbolic terms is the sum and substance of mathematics.

Speaking as a modern mathematician, I don't think either of these statements reflect how I think, at least. It's possible I'm misunderstanding what you mean by "numbers underly geometric figures," but I can think of lots of interesting modern geometry that has nothing to do with numbers as such. And I don't think of the trig functions as either being explicitly geometric or symbolic, exactly, but rather multifaceted objects which can be described in either language. There's nothing explicit about the modern mathematical way of thinking that requires me to limit myself to one description or another.

I think modern mathematicians benefit from a flexibility of thought, which allows them to think in geometric, symbolic, numeric, or purely axiomatic ways depending on what the situation calls for or what produces the most beauty. You call this a "flattening," but I think it's just the opposite. As long as it follows the rules, it's all fair game.
posted by albrecht at 6:43 AM on March 9, 2010 [4 favorites]

e^{it} is just a matrix, well the matrix group of rotations about the origin in the real plane. euler's identity is just the unremarkable fact that if you rotate by \pi, "x" on the x-axis goes to "-x."

or, if that's unsatisfying, you can think of e^{it} as the "universal covering map" of S^1 (the circle.)
posted by ennui.bz at 7:33 AM on March 9, 2010 [1 favorite]

A long, long time ago,
Long before the Super Bowl and things like lemonade,
The Hellenic Republic was full of smarts,
And a question resting on the Grecian hearts was;
"What is the circumference of a circle?",
But they were set on rational numbers,
And it ranks among their biggest blunders,
They worked on it for years,
And confirmed one of their biggest fears,
I can't be certain if they cried when irrationality was realized,
But something deep within them died,
The day, they discovered, Pi.
    — Chan & Ferrier
posted by ob1quixote at 7:36 AM on March 9, 2010

Pretty numbers make bonobo's head hurt.
posted by bonobothegreat at 8:43 AM on March 9, 2010 [2 favorites]

But XKCD should not make a comic about the beauty of e. That would be wrong and make people angry.

Too late.
posted by kmz at 9:31 AM on March 9, 2010

I have this open in a tab, and keep seeing "Math is beautiful" in the tab title. And now, running through my head, is:

Math is beautiful,
Math is fun
Math is best when it's
One and One
posted by Legomancer at 10:37 AM on March 9, 2010

I know that the first time I saw this equation it made zero sense to me, but I was proud a few years later when I saw the series definitions of trig functions and, knowing the Taylor series for e , I could see that it was true.
(or, what delmoi said)
posted by MtDewd at 10:51 AM on March 9, 2010

This thread? This is why I became a math major in college. And also why I subsequently dropped it.
posted by jabberjaw at 11:01 AM on March 9, 2010 [3 favorites]

Interesting thread and discussion. I have always remembered this equation even though I have long since forgotten most of the math I learned beyond basic algebra.

Since zero is a big part of the discussion, it bears mentioning that this mefite wrote the book on zero.
posted by TedW at 11:53 AM on March 9, 2010

Man, when I need some of this shit figured out, i'm just going to hire one of y'all to do it for me.

I'm going back to the thread about funny comics.
posted by Uther Bentrazor at 11:59 AM on March 9, 2010

I experienced a pretty strong visceral reaction against Dedekind cuts when I first read about them in Rudin's Mathematical Analysis, based mainly on the huge internal stucture they imputed to irrational numbers, which made them seem unwieldy and cumbersome without pointing to any new results about them whatsoever, and because it made rational numbers seem far too different from irrationals-- but wait! After we use use the poor simpleminded rationals to construct our cuts, they reappear in the new regime as cuts themselves and refuse to answer any questions about their previous more humble existence.

Questions such as:

What is the nature of the rational numbers that are now sitting inside the cuts? If they are the old rationals, they have no real existential status in the new definition of numbers and you don't really know what they are; if they are the new rationals, then you have given your new numbers an essentially circular definition, from which you can only escape, it seems to me, by seeing each rational inside the cut as an infinite descent toward the old rationals which can never be completed-- a particularly nasty combination of Zeno's paradox and an infinite set of Russian boxes.

I was inordinately (sorry) delighted to see an allusion some years later by J. H. Conway to somewhat similar objections:

Let us us see how those who were good at constructing numbers approached this in the past.

Dedekind (and before him the author---thought to be Eudoxus---of the fifth book of Euclid) constructed the real numbers from the rationals. His method was to divide the rationals into two sets L and R in such a way that no number of L was greater than any number of R, and use this "section" to define a new number {L|R} in the case that neither L nor R had an extremal point.

His method produces a logically sound collection of real numbers (if we ignore some objections on the grounds of ineffectivity, etc.), but has been criticized on several counts. Perhaps the most important is that the rationals are already supposed to have been constructed in some other way, and yet are "reconstructed" as certain real numbers. The distinction between the "old" and "new " rationals seems artificial but essential.

This is from the "0th Chapter"--- "All Numbers Great and Small" of Conway's On Numbers and Games.

I believe you can read the entire 0th chapter from my link, and I recommend it highly to anyone who wants to see what numbers are to a great mathematician who knows them inside out-- a mathematician who has, in fact, demonstrated the amazing strength to be able to turn the entire number system inside out, to the accompaniment of horrified gasps from the crowd of onlookers.
posted by jamjam at 3:29 PM on March 9, 2010 [1 favorite]

I, uh, have this tattoed on my wrist. Thanks for the awesome post!
posted by beepbeepboopboop at 5:25 PM on March 9, 2010

Euler fucking rocks. He has his own number, his own characteristic, his own path, and a bazillion theorems.

I fear that my life will never generate that much.
posted by twoleftfeet at 9:58 PM on March 9, 2010

Oh, and in my book (or God's great book of proofs) the Euler Characteristic beats that silly formula by a mile for beauty.
posted by twoleftfeet at 10:03 PM on March 9, 2010

All I ever learned about math came from this simple module.
posted by not_on_display at 7:53 PM on April 1, 2010 [1 favorite]