5 ms·

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 f

by exacube 11y ago

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)

mlitchard 11y ago
What is it you don't understand?
- jordigh 11y ago
  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.
  - madprof 11y ago
    But once they get all that they get stuck on what a Taylor series is, what the integrals of trigonometric functions are or what the squeeze rule is.
    - mauricioc 11y ago
      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.)
  - hasenj 11y ago
    I don't understand calculus and have no problem admitting to it. I hated calculus throughout highschool and college.
  - jameshart 11y ago
    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.
    - jordigh 11y ago
      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?)
- wehadfun 11y ago
  Lets start with those polynomials, then those integral coefficients.
  - dnautics 11y ago
    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.
umanwizard 11y ago
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 :)
- joe_the_user 11y ago
  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.
davvolun 11y ago
Note: Not a mathematician, by any means 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...
- jordigh 11y ago
  > 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.
  - Natsu 11y ago
    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 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?
    - chx 11y ago
      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?
hcarvalhoalves 11y ago
"Simple" in the sense it's minimal, not as in "easy to grasp".

by exacube 11y ago

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)

mlitchard 11y ago

What is it you don't understand?

jordigh 11y ago
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.
- madprof 11y ago
  But once they get all that they get stuck on what a Taylor series is, what the integrals of trigonometric functions are or what the squeeze rule is.
  - mauricioc 11y ago
    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.)
- hasenj 11y ago
  I don't understand calculus and have no problem admitting to it. I hated calculus throughout highschool and college.
- jameshart 11y ago
  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.
  - jordigh 11y ago
    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?)
wehadfun 11y ago
Lets start with those polynomials, then those integral coefficients.
- dnautics 11y ago
  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.

umanwizard 11y ago

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 :)

joe_the_user 11y ago
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.

davvolun 11y ago

Note: Not a mathematician, by any means 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...

jordigh 11y ago
> 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.
- Natsu 11y ago
  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 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?
  - chx 11y ago
    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?

hcarvalhoalves 11y ago

"Simple" in the sense it's minimal, not as in "easy to grasp".