I feel really bad when I read stuff like this and don't understand, especially when it's titled "Simple"
(granted I didn't spend more than 5 minutes trying to following the proof)
Well, you have to spend as long as it takes to understand, and you have to understand calculus, and probably have at least a non-zero amount of practice following proofs from a university level or strong high school level math course.
It's "simple" relative to the rest of mathematics, not relative to daily life.
The point is -- don't feel bad; math doesn't come easily to anyone :)
The one other thing about this proof compared to the exposition of upper division college math is that it doesn't follow an easy linear path but is instead using the theories and tools in an ad-hoc manner. Getting an idea of theories and proof methods as a toolbox rather than just a linear progression of ideas is one of the milestones of "mathematical sophistication" - especially because it's important to not lose sight of the linear development of the field as well.
I'm not a fan of how the proof is explained, specifically why are we doing x or y.
I would prefer this --
Let pi be a rational number thus ( * by the definition of rational numbers) pi = a/b, the quotient of positive integers. We will show no such a and b can exist, therefore pi cannot be rational.
( * is not completely necessary, since the definition of rationals is so elementary)
<Next Paragraph, and so on>
I suspect a lot of mathematicians prefer the format given because it is more obtuse...
> I suspect a lot of mathematicians prefer the format given because it is more obtuse...
No, because as you say, defining what a rational number is seems pretty pointless here, as why would you be reading the proof of something whose definition you don't even know? And you also want an explanation of what a proof by contradiction is, which also seems to be way too elementary. Proof by contradiction is one of the most basic techniques.
Spivak's version of this proof explains it a bit more, but still requires work from the reader. Any proof does. Mathematics cannot be a spectator sport.
A lot of people struggle at math because they're missing things like that. With today's technology, there's no reason things can't be broken down to arbitrarily small steps, so that people can be filled in on the cracks in the foundations of their understanding of math.
The usual reason for not including everything is that it becomes ridiculously laborious. With mathematics being abstraction piled atop abstraction, this is quite reasonable. Just look at how much work it is to prove 2+2=4 -- http://us.metamath.org/mpegif/mmset.html#trivia
But, as you can see from the above, there's technology to simplify things. I wonder if someday we'll be able to break things down in a friendly way so that for any piece a student doesn't understand, they can get a proof in terms of things they do understand?
Erm, the linked metamath page asks what's the longest path one can take. Well, geez, it's long. Proving that 2+2=4 from the usual Peano axioms isn't that hard or long. We define addition by a+0=a,a+S(b)=S(a+b). Now what we mean by a positive integer number is just a shorthand for that number of successive S(). So 2+2=4 is just S(S(0)) + S(S(0)) = S(S(S(S(0)))). To prove this, let's apply the second part of the addition to S(S(0)) + S(S(0)) we get S(S(S(0)) + S(0)). Let's repeat this "move the S() from the second to the outside" and you will get S(S(S(S(0)) + 0)). Now we can use our first half of definition where +0 can be left out: S(S(S(S(0)))). Now, that wasn't that long, was it?
I'm not a mathematician, so please bear with me. I'm trying to understand your definition of addition. Surely that can't be enough? At the very least a+b=0 satisfies your formulas (should I use the word axiom there?).
What I'm trying to understand is how I am supposed to think in order to prove/disprove my statement above. Disproving something by contradiction is easy enough, if you can find a contradiction but what if you can't? How can you be sure that you've covered all variations?
This is something one struggles with as a developer too. There are cases where you think that a fundamental function does the right thing for all inputs but then you discover an edge case where that isn't true. The number of cases where this has been the case even in high profile libraries suggests that I'm not alone in this.
What is missing from chx's description is the definition of the natural numbers (and the S (successor) function). The standard definition is:
0 is a natural number
S(n) is a natural number, if and only if n is a natural number
x=y if and only if S(x)=S(y)
for all x, S(x)!=0
These axioms are sufficient to show that the natural numbers form a line.
At this point, we can use the definition of addition that chx provided:
a+0=a
a+S(b)=S(a+b)
It is true that, without the restrictions on S, than defining a+b=0 would satisfy these. However, because of the restrictions on S from our definition of the natural numbers, it is impossible for S(a+b)=0.
In some sense, proofs as presented in a textbook should not hold your hand. The point of learning about proofs is being able to create proofs yourself, even if they are trivial variations on proofs you have seen.
Computer analogy: would it be a good idea if someone showing an implementation of a double-linked list explained every little detail about pointers and arrays etc.?
However, there are many things we can do syntax-wise to facilitate the understanding of proofs. I remember we experimented with using indentation (on paper) in proofs when I did functional analysis, and that was quite useful for keeping track of "variable scope" etc.
While it's true that you won't survive long with a lot of 'cracks' in your foundation of understanding, I think there's quite a bit of worth in helping people identify and explore the holes in their own understanding and making this as painless as possible.
I understand that we can't trod new ground if nobody is willing to put up with the difficult slogs, but there's no reason to make things difficult for the sake of difficulty. There are plenty of challenging exercises one can do simply by learning to apply a newfound understanding without having to leap from perch to perch because nobody would explain how to bridge a certain gap.
I think that things are trending in this direction already, actually.
It is immediately obvious that he is attempting proof by contradiction, because the first line supposes the opposite of the title of the paper. As in the proof of the irrationality of the square root of 2, he sets up his assumption that pi is rational by defining it as a/b.
In a proof, we don't have to know why something is done, just that it logically follows from our assumptions and is used as a step in the proof. Explaining how the author came up with the proof should be left to an educational textbook or an "Author's Notes" if the proof is several pages long and would help readers be guided along.
I'd prefer something similar to your version, like: Let's suppose that pi is rational, pi = a/b ...
I usually like to understand what is the intended meaning of the symbols and how they will be used later, in spite it's not necessary to have a correct proof. I was (slightly) surprised when I found out that a/b "was" pi.
But, each one has a preferred writing style, ...
But in this case, I suspect that the proof is optimized for space. An alternative title could be "A proof that pi is irrational in one page" so some details and constructions are skipped to put everything in one page.
Using pi as a name is only a very weak cue for "proof by contradiction". The title of the paper gives enough hint to detect it, but I would still use "suppose pi is rational", because it is the dead giveaway for proof by contradiction.
And as to 'obtuse': how many C programmers comment their 'main' function to help people unfamiliar with the language learn that it is the entry point of the program and receives its command line arguments as input? At some time, you have to assume your audience knows some lingo.
My main objection to this proof is that I do not find it beautiful; it feels like one shouldn't need to do integrals over goniometric functions to prove it.
Because it uses such fairly heavy weaponry, it isn't immediately clear to me that the logic isn't circular.
It would have been nice for the author to explain the motivation for this construction. Note that the polynomials are the construction for the Taylor series of exp(ix) around the point (a/b), and remember Euler's magic formula.
Usually people who don't understand stuff like this are unable to even begin to explain what they don't understand.
I don't really understand people like this. I have tried to guess many times what the problem is. I think they may be too embarrassed to admit they don't know what an integral or factorial sign means, or perhaps something like why f is a polynomial, or what a polynomial is.
I'm not the OP, but your attitude here is unnecessarily dismissive. Not everybody has the calculus skills to understand this, let alone an awareness of mid-twentieth century calculus notation conventions.
For my part, I do know what polynomials are, what a Taylor series is, and so on, and in theory I can trace through the steps here and agree that yes, one follows from the other. Yet I find this proof unsatisfying because it doesn't demonstrate clearly, to me, which particular properties of pi it is using that bring about the contradiction. When the proof makes use of pi, it doesn't explain why the statements it is making are specifically true for pi, and not for some other number.
Take this same proof, substitute the number 3 for every occurrence of pi. Now pinpoint for me the place in the proof where it is clear and obvious that the proof makes an invalid claim about the number three (but where for pi, it was clearly and obviously correct). If you can't find it, then this proof structure equally serves as a convincing argument that three is irrational. That's quite unsatisfying - though of course, a proof doesn't have to be convincing, it just has to be right. Nevertheless, to qualify as a 'simple' proof, I think it does have to appeal to intuitions and concepts in a way that simply convinces you of its truth.
This proof is short, but it is not simple. It doesn't satisfy because it doesn't show me how the ratio of a circumference to a diameter has to be irrational - only how a number called pi which has particular relationships (not specified clearly in the proof) to the sin and cos functions, has to be irrational.
I didn't mean my attitude to be dismissive. It's just that truly a lot of people respond this way. "I don't understand it at all." And when you ask what don't they understand, they're unable to say it. Witness for example how exacube seems to have vanished and will probably never tell us what she or he did not understand.
I think what happens is that people are so overwhelmed with unfamiliar ideas when they encounter a proof like this that they just grind to a halt, curl up into a ball, and scream how much they hate it all and don't understand a bit of it. We have at least a couple of other people in this thread who have expressed their hatred of calculus. Starting from that it seems pretty hopeless to try to explain to them this proof.
Yet I find this proof unsatisfying because it doesn't demonstrate
clearly, to me, which particular properties of pi it is using that
bring about the contradiction.
Only one: that it's a root of sin(x). The proof actually works for any nonzero root of sin(x).
In fact, that's a great definition of pi: the least positive root of sin. It's a much easier definition to work with than ratio of circumference to diameter (how do you define cirumference? What is length? What is a curve?)
well sine has to come from lengths of curves, doesn't it? Because the geometric definition of sine is a function from angles to numbers, and to get pi in there you need to introduce radians as a way to measure angles - otherwise I could argue that the roots of sin are 0, 180, 360, etc. - and none of them are particularly irrational.
No. There's the analytic definition of sin that does not mention curves or lengths at all: the sine is the unique function that satisfies the following differential equation and initial value problem:
s''(x) + s(x) = 0
s(0) = 0
s'(0) = 1
It's a nifty way to define sine purely by its differential properties. Of course, it requires some work to show that differential equations have a solution and that this particular solution is sine (i.e. has the properties you want a sine to have), but once you do that work, it's pretty easy to prove things such as sin^2(x) + cos^2(x) = 1.
On the other hand, starting from the geometric definitions (and building the framework for that, such as arclength, which really requires calculus), it takes a longer route to get to the calculus of sine. Historically this was the route, but we have found shortcuts since then.
Dude I needed you when I was working through my Calc 2 stuff. I didn't ever see that definition as a route to use. Way nicer than memorizing the unit circle which eventually translated into "knowing" the answers to the derivatives and integrals.
Interesting - I was sort of wondering if you could get sine just from simple harmonic motion, and that's basically what those equations do - the first is equivalent to saying s''(x) = -s(x), so acceleration is opposite to and linearly proportional to (well, equal to in this simple case) displacement. That's a good point.
I suppose the complaint was to hint that mathematics that is older than 50 years is too old for modern tastes, but most of our modern notation was already established at the start of the 19th century. Even Euler uses almost completely modern notation, about 250 years ago.
Oh, I didn't even notice that. I encounter that notation frequently enough, but it wasn't the one I was taught in high school. Kind of hard to do a web search for its origins. I tried and failed.
I recognized it immediately but each math teacher I had in college used different notation for the definite integral. I really did think it was common. I need to study more math.
This is not directed at you specifically (I get your point), but here's an attempt at an explanation for those parts:
1) Verifying that f^(j) = 0 is 0 for all j doesn't require Taylor series (though, as 'dnautics pointed out, it does motivate the construction):
1.1) f^(j)(0) is zero for 0 <= j < n because every term of the polynomial f has degree at least n (and therefore you won't get a nonzero constant coefficient if you derive fewer than n times).
1.2) f^(j)(0) is zero for j >= n because once you derive n times you will get a factor of n! in each term, thus cancelling the only source of "non-integerness".
2) Point taken, you have to know how to differentiate a product and what the derivative of the sine and cosine functions is. With this in mind, checking the equation before equation (1) is routine, though. You then apply the fundamental theorem of calculus to get equation (1).
3) He is not applying the squeeze rule here, as it would not produce a contradiction. The squeeze rule would say that the limit of f(x) sin(x) (there's an implicit dependence on n here) is zero, which would say that F(pi) - F(0) is zero, which is not a contradiction.
The argument requires less machinery: For large enough n (not in the limit), f(x) sin(x) is strictly between zero and 1 (and is thus not an integer), because pi^n * a^n / n! is smaller than 1 if n is large enough. The simplest way of explaining this is that n! >= (n/2)^(n/2), as there are n/2 terms each larger than n/2 in the definition of n!. Thus the expression is at most ((pi a)^2/(n/2))^(n/2), and thus taking taking any n such that n/2 >= (pi a)^2 works for making the right side less than 1.
(If you know the definition of the Euler constant e as a series, you'll see that (pi a)^n / n! appears in the expansion of e^(pi a). Since the series converges, this means that (pi a)^n / n! is less than 1 if n is large enough. But this requires more previous knowledge.)
Since the linked proof was published in a mathematical journal [Bull. Amer. Math. Soc. Volume 53, Number 6 (1947), 509], I think it's safe to assume that Niven was first.
Neat! I remember that presenting (from memory) a slightly longer proof that pi is irrational was the final question on my Analysis oral exam.
It is surprising how many people know that pi is not rational, yet how non obvious is the proof. Like the first one ever presented. This one is also a nice piece of cake, pulls this polynomial f(x) from a hat.
If the annotations are not working right when you expand them, particularly the second one, and you are using Safari, try reading in Chrome or Firefox. Those appear to work fine.
In Safari, it looks like when the height of the expanded annotation display region is calculated for dealing with scrolling, it is including the area under the sign up doohickey. It thinks you can see more than you can. This means the scrolling limits are set too small, and so you cannot scroll the bottom of the annotation up into view.
If your window size and text size make it so that the second annotation fits in the visible region, this won't matter.
> The first rigorous proof that π is irrational is from Johann Heinrich Lambert in 1761. He proved that if x≠0 is rational, then tan(x) must be irrational. Since tan(π/4)=1 is rational, then π must be irrational.
Of course, the proof for (tan(x) is irrational for rational x != 0) might be complicated, but at least it's easy to see why this proves that PI is irrational.
This one, on the other hand, I can't even follow in principle what it's trying to say.
That's of course because I don't understand integrals and differentials.
That just wraps up a mystery inside another. Why would tan(x) be irrational if x is rational? That seems like an even bigger mystery than why π is irrational.
It's worth noting that there's a number of annotations added that make the proof much easier to follow for someone whose calculus is not quite as... hum... fresh as the proof expects.
Every time I tried studying from a math textbook back in school I got the impression that examples and explanations were purposefully obscure to make it look more sophisticated. Then a friend comes along and explains it in 10 seconds and it makes perfect sense.
I started reading it, and then there was a nagging interstitial, begging for my email address so that I could be sent spam.
Closing that garbage, various buttons and unnecessary margin bits ("Click here to see more!" [bounce] [bounce] [bounce]...) kept jittering for my attention.
So, I closed the tab. I have no idea who is responsible for the design of fermatslibrary.com, but they should feel ashamed of what they've done. This is one of the most infuriatingly infantile designs I've ever seen. The old "punch the monkey!" ads have nothing on this.
I'm seeing the same thing: http://i.imgur.com/vYgP0Dj.png. It's bad taste in design, but I don't think they should "feel ashamed of what they've done".
For what it's worth, you can delete annoying parts of a page using the developer console in Firefox or Chrom(e|ium).
I don't know what to tell you, except that when I viewed the page, there was a reasonably well-formatted PDF in the center of the screen surrounded by unnecessary, fiddly, bouncing bullshit "click on me!" buttons.
There are sites that allow for annotations and manage to do so without bouncing, distracting "click on me!" buttons. Medium does a great job of this. I have no idea what the site's creator was thinking when they decided not to emulate the established, non-distracting examples instead of doing what they did.
(a-bx) is not zero, x is a variable. when x=pi, a-bx is zero, otherwise not. a-bx=0 is an equation whose solution is x=pi. The function f(x) is not identically zero, it is however zero when evaluated at x=pi: f(pi)=0.
And he does talk about f(pi) quite a bit. It's not like there are limits being discussed where we should consider something in the neighborhood of a 0/0 situation. He evaluates it at pi, and in doing so could just put a zero in there.
But nowhere does he proves that all derivatives of f(x) are not zero for x=0 and x=pi. f(x) clearly is zero there. So saying that F(pi) + F(0) is an integer proves nothing if both are equal to zero...?
You don't actually need F(pi) + F(0) to be non-zero. The proof shows that the integral is positive, but arbitrarily small. The means that F(pi)+F(0) must be positive, but arbitrarily small. If F(pi)+F(0) is a positive integer, it fails the arbitrarily small test. If F(pi)+F(0) is a non-positive integer (including 0), it fails the positive test.
