Proof that pi is rational - lol

[shmoe]

Crappy why? Based on what argument? That you said so?

I already explained that there were problems with trying to treat these 'dx' as 'infintessimal numbers'. They don't fit in nicely in the real number system that you're used to. If your course was trying to treat these dx terms as numbers (infintessimal or otherwise) without adressing these concerns, then it was crappy. Don't take my word for it though, go find a non-standard analysis book and see how they deal with this. However, something was apparently mentioned in your course (the bold part):

We did limits of riemann sums as n went to infinity, and thats how we started on integrals. dx was specifically explained to mean a infintessimal value although now that you mention I vaguely remember the guy mentioning something about them being a source of confusion in the math community or something like that.

and I suspect that you've misinterpreted your instructor and somehow have come to believe that 'dx' is some kind of number or value.

We had a textbook but I dont buy them. The one my roomate has is the same one I think used for like 3 calculus classes and its edwards and penny early transcendental calculus.

I don't have Edwards and Penny handy, but if memory serves they (as most other intro calc texts.) they waffled over the actual meaning. They defined the integral as the limit of Riemann sums, where the limit is taken as the norm of the partition heads to zero. The jump from the Riemann sum notation (which had delta x's) to the integral (which has the dx) when you consider the limit is usually described to suggest that this dx represents what you might consider an 'infintessimally small quantity' but that this is vague and they aren't going to go into detail to attatch precise meaning to this comparison or what this comparison would actually mean. The dx essentially means that in the Riemann sums that are the base for this definition of the integral, you are taking the "weight" of your intervals in the partition to just be their length (compare Riemann-Stieltjes integrals for example).

You were probably also warned about this when they introduced the notation dy/dx for the derivative. Again, this is meant to suggest the idea of a ratio of 'infintessimals', but it's really defined as a limit of a certain function (the f(x+h)-f(x)/h or equivalent thing) that is still taking on only real values. You would have been warned not to try to treat dy/dx as an actual fraction, even though it sometimes looks like that's what you're doing (I mentioned the chain rule previously as an example).

 

[shmoe]

"Saying a rational number is one that you can write as p/q is ambiguous and does not rule out infinite summations of rational numbers."

I think you're missing the spirit of the definition. I'd guess it can be shown that rational numbers may consist of some finite summation of other rational numbers. Like 1/4 = 1/8 + 1/8 or 1/4 = 1/16 + 1/16 + 1/16 + 1/16. If however, you have to involve an infinite summation... wouldn't that mean the number is irrational? Maybe I've missed something.

This is trivially true wes. A number is irrational if and only if it cannot be expressed as a finite sum of rationals. Any finite sum of rationals is rational and any rational on it's own is a (trivial) sum of rationals.

A point kriminal was making is a valid one. Just because pi can be expressed as an infinite sum of rationals does not mean it's irrational. The claim "pi cannot be expressed as a sum of a finite number of rationals" isn't really saying anything except "pi is irrational" and does not constitute a proof that pi is irrational. See Sarkus' post for a link to Nivens proof (or a variation of it) for one way to do this.

 

[Silas]

Thanks. I see why my statement is absolutely false. Pi is irrational since a perfect cirlce requires either the diameter or the circumferance to be irrational.

If you take a rational, finite diameter - i.e. 3.000000000n and multiply that by Pi, does that not then make the new circle circumferance become a rational number?

Strangely enough there is nothing conceptually impossible about an irrational number in real life.

Using the most accurate vernier scale and the finest pencil, draw a line exactly 1 inch long. Using the most accurate right angle, draw a line perpendicular to it at one end, exactly one inch long. Draw a straight line to complete the triangle. That line is precisely sqrt(2) in length. You've just created an irrational number.

Measure it with a standard ruler. You'll find the length of it is a little over 1.4 in. Take a photocopy and blow it up ten times, and measure the length. You'll see it's now a little over 14.1 in. Blow up only the section from the last rule marking to the end. It's a little over .4 in. Blow up the section from the last rule marking to the end. It's a little over 0.2in. Blow up the section from the last rule marking to the end. It's a little over 0.1in. Blow up the section from the last rule marking to the end. It's a little over 0.3in. This can go on for ever and ever!

It's the same thing with recurrent decimals, but with one difference - you can easily fold a piece of paper into three, and if you measure it, you'll see that the measurement is just over 0.3, and the length of the excess is 1/3 of the distance from 0.3 to 0.4. BUT if you fold a piece of paper and define the folds as 0.1, 0.2 and 1.0, you have a base-3 ruler that does not have this excess. The difference with irrationals is that there is no number base that allows you to eliminate the excess over the last measure point - ever.

 

[wesmorris]

Trivial is all I was shooting for. Thanks, shmoe. I just wanted to throw up the way it seems to make sense to me in case it happened to resonate. Proofs aren't really my thing. I'm a hack of a mathematician at best.

 

[kriminal99]

I suggest you look at the proofs of PI's irrationality provided by: Lambert 1761; Legendre 1794; Hermite 1873; Nagell 1951; Niven 1956; Struik 1969; Königsberger 1990; Schröder 1993; Stevens 1999; Borwein and Bailey 2003, pp. 139-140.

The one on THIS page is by Niven.
It starts with the assumption that pi=a/b (the requirement for rationality). And disproves its possibility.

PI does have a value - it is equal to the ratio of the circumference of a circle to its diameter.
The fact that we can not physically write the number in decimal form is irrelevant.
The same for "e", square root of 2 etc. They all have values by definition.



And thus I'm sure the link above will happily correct your understanding of whether or not pi is rational or irrational.

No. The only reason you can't is because it is IRRATIONAL. The number PI has finished being formed. It has been defined. It needs nothing more.
The fact you can't write it out in decimal, or Hex or binary is irrelevant. The number exists.

Furthermore, you can not "constructively" write a fraction with A and B both as a 10^(10^100) digit integer. But this would still qualify as a rational number.
Your definition is flawed.

Why it isn't meaningful to point to the fact that other people have supposedly proved something in an argument:

1) If you don't provide an argument yourself than there is no reason to believe that you understand the basis for your belief in it, therefore your opinion of it is worthless.

2) There are potentially an infinite number of supposed proofs and disproofs of everything, some of which are obviously going to be flawed. It makes no sense to spend your time reading flawed arguments, when if you are "wrong" about something (ie there is a better argument) you will encounter a consequence of this and can then find the solution.

3) Specifically in the case of mathematics, this type of reasoning is self defeating. The Godel incompleteness theorem supposedly prooves that math cannot be complete and consistent. Therefore it is possible that someone proove something with certain beliefs or assumptions that are relevant to a certain type of mathematics that does not even apply in another area. In fact, no specific grouping of mathematics into consistent belief sets has been done so there is no telling where contradictions may come up. This argument (like so many other "formal proofs" in mathematics) is something that would be immediately evident to anyone with a capacity for philosophical thought or has already been presented in philosophy.

It also makes doubly untrustworthy the concept of a proof by negation. Not only now do you have to consider that there is a non - trivial assumption that the mathematician is not consiously recognizing (which pretty much rules out proof by negation as a proof to begin with), the fact that math itself is inconsistent is always a possible candidate for this unless a proof for consistency in the specific area you are considering has been given. Which most likely has not been done for anything considering the infinite.

The only problem with your definition of PI is that there is no such thing as a circle. What we call a circle is really something like the limit of a convex polygon as the number of sides goes to infinity.... Then you have to define circumference... This nonmathematical fact (that there is no circle) is the reason why pi is an infinite summation of rationals. Pi does not define what a circle is as you seemed to indicate you believe in your other post. Nothing defines pi and nothing defines a circle, they are just goals to strive for that you can never reach.

The only thing we have for PI is an algorithm for calculating more decimals of it. The algorithm can be correlated with the physical action of adding more sides to a convex polygon used to approximate pi, and then cementing a decimal place when we see that the change from adding more sides will no longer effect that tens place. PI as in whats at the end of this sequence (which never ends) is not a number. The decimal places of pi that we use do not represent a circle. They represent polygons with large numbers of sides.

I know that definition of a rational number ( THAT YOU GUYS WERE USING TO ELIMINATE AN INFINITE SUMMATION OF RATIONALS AS A RATIONAL ) is flawed. Thats my whole point. Your proof of this does not discriminate between the infinite summation of rationals and 10^10000000/2 or whatever else.

Strangely enough there is nothing conceptually impossible about an irrational number in real life.

Using the most accurate vernier scale and the finest pencil, draw a line exactly 1 inch long. Using the most accurate right angle, draw a line perpendicular to it at one end, exactly one inch long. Draw a straight line to complete the triangle. That line is precisely sqrt(2) in length. You've just created an irrational number.

Measure it with a standard ruler. You'll find the length of it is a little over 1.4 in. Take a photocopy and blow it up ten times, and measure the length. You'll see it's now a little over 14.1 in. Blow up only the section from the last rule marking to the end. It's a little over .4 in. Blow up the section from the last rule marking to the end. It's a little over 0.2in. Blow up the section from the last rule marking to the end. It's a little over 0.1in. Blow up the section from the last rule marking to the end. It's a little over 0.3in. This can go on for ever and ever!

It's the same thing with recurrent decimals, but with one difference - you can easily fold a piece of paper into three, and if you measure it, you'll see that the measurement is just over 0.3, and the length of the excess is 1/3 of the distance from 0.3 to 0.4. BUT if you fold a piece of paper and define the folds as 0.1, 0.2 and 1.0, you have a base-3 ruler that does not have this excess. The difference with irrationals is that there is no number base that allows you to eliminate the excess over the last measure point - ever.

Heheh nice try. Theres only one problem with this: You see you have created this thought experiment based on the belief that irrational numbers exist not on reality. In reality once you get small enough you are talking about molecules then atoms then god knows what, not lengths on a ruler or lengths of a piece of paper. So even if you were capable of infinite effort necessary to try and prove this in this fashion, you wouldn't be able to.

 

[wesmorris]

The definition is an arbitrary term annotating a phenomenon in math. I think James was correct.

Regardless, Pi would not fit the general form of a rational number since you can keep calculating it forever. A number expressed as A/B, would not take forever to calculate. (I'll leave this conversation to the mathematicians, pardon)

 

[kriminal99]

As sarkus just pointed out, if thats how you define rational number then neither would 10^100000/5 be a rational number.

Take any rational number. Now take another rational number. Now add them together. The sum will be rational right. Keep adding rational numbers and the sum will always be a rational number. We can conclude then that the sum of n rational numbers is always rational.

So 3 + 0.1 + 0.04 + 0.001 + 0.0005 is a sum of rational numbers and equals 3.1415 which is also a rational number.

But 3.1415 does not equal pi. Neither does 3.141592653589. Neither does any terminating sequence of decimal numbers. The only number which equals pi is the number 3.1415... (with an infinite number of decimals). If the decimal places terminate then it obviously isn't pi.

Now, can we write pi as a sum of rational numbers? Well, if pi had n decimal places (where n is not infinity obviously) then pi could be expressed as a sum of n rational numbers which would imply that pi is rational.

However, pi has an infinite number of decimal places and hence would require a sum of an inifnite number of rational numbers.

The problem arises when you sum to infinity. You say that the limit of 3 + 0.1 + 0.04 + ... as the sequence goes to infinity is pi, because this is just a sequence of rationals so pi must be rational. But the sequence consists of a infinite number of rational numbers. How on Earth can we even begin to think about adding an infinite number of things together. Any philosopher (and indeed a mathematician) should understand that strange things happen when you involve infinity, and actually infinity is generally avoided in mathematics because it doesn't make sense.

Try adding an infinite number of anything together. Say 1 + 1 + 1 + ... what does this equal? The sum has no limit and the sum is infinite. Is infinity a rational number? By your definition it is! becuase 1 is a rational number hence 1 + 1 + 1 + ... is rational. But now consider √2 + √2 + ... This is a sum of irrational numbers and the sum is infinity. Hence infinity is irrational. But haven't we just shown that infinity is rational?

This is all goobilygook if you are a mathematician and well versed in analysis, however I am trying to point out that once you involve infinity in your proof you can't rely on anything to make sense. This is why mathematicians have always avoided the use of infinity in their proofs or irrationality, simply because it doesn't make sense.

So if you are going to try to prove something using infinity, no-one is going to listen (or understand you for that matter) UNLESS you actually have counted to infinity.

Hope this helps you understand.

Except Pi CAN be expressed as an infinite summation of rational numbers that has nothing to do with the decimals. This infinite summation of rationals is the definition of pi. There are several formulas for doing this, one given by leibniz was like 1 + 1/3 - 1/5 ... or something like that. Conceptually you can do it just by subtracting marginal lengths of diagonols of convex polygons as you increase the number of sides (if you only use even number of sides) If you want to use both even and odd number of sides then I think you would have to alternate the sign in the summation because the diagonols would get larger and smaller.

Hope this helps you understand.

 

[Sarkus]

The only problem with your definition of PI is that there is no such thing as a circle.

Okay, if there's no such thing as a circle, then what is the locus of X^2 + Y^2 = Z where constant Z > 0. ?

A "circle" is a geometric shape.
It can not be drawn perfectly but it is mathematically defined.

Pi is the ratio of the circumference of this circle to the diametre.

Except Pi CAN be expressed as an infinite summation of rational numbers that has nothing to do with the decimals.

I agree.
PI/4 = 1-1/3+1/5-1/7+1/9-1/11.... etc (Leibniz-Gregory-Madhava series).

There are plenty of other similar ones, by Euler et al...

1+1/4+1/9+...+ 1/n^2 +... = (PI^2) / 6 etc

Ultimately it is this requirement to go to infinity to calculate PI that makes it irrational (using the widely accepted definition).
If it could be done with just 10^1000000 terms then this would be rational - a pain in the proverbial to write down, but rational nonetheless.

 

[Wund]

Hah. Everyone knows that Pi equals 3.

Bloody Stupid Johnson

 

[AndersHermansson]

The only problem with your definition of PI is that there is no such thing as a circle. What we call a circle is really something like the limit of a convex polygon as the number of sides goes to infinity....

What a crock!
What is the curve of x^2+y^2=1 called? I think it's a circle.

 

[chroot]

"dx" is not an infinitessimal. It is only called such by introductory math professors who are too lazy (or stupid) to teach their classes correctly. In calling dx an infinitessimal, the professor is sacrificing your later understanding for his own momentary convenience. It makes reasonably good sense to think of dx as an infinitessimal when you're doing first semester calculus, since that helps to generalize the Riemann sum into the integral, but it's beyond sloppy: it's wrong. dx is not an infinitessimal, and infinitessimals don't exist in the real number system.

dx is a one-form -- a linear functional that maps vectors into the real numbers. This is the tip of an iceberg of interesting and beautiful mathematics: the study of differential geometry. The real beauty of this mathematics is currently obscured for you, kriminal99, by a couple of things:

1) You have very little education -- only a couple of basic calculus courses.
2) It seems the quality of this education was rather poor.
3) You seem to have delusions of grandeur, thinking that your basic calculus courses have taught you everything there is to know.

To reach the cutting edge of the field, you have at least ten years of formal classroom study ahead of you. You are, in a word, an infant.

- Warren

 

[superluminal]

chroot,

I'm a lowly engineer with a BS. You speak the truth. I have math envy. Our profs used to tell us that what we were studying wasn't even considered 'mathematics' by real mathemeticians.

 

[oxymoron]

Except Pi CAN be expressed as an infinite summation of rational numbers that has nothing to do with the decimals. This infinite summation of rationals is the definition of pi. There are several formulas for doing this, one given by leibniz was like 1 + 1/3 - 1/5 ... or something like that.

Ok, but you are still using the word "infinity". You can't rely on infinity to prove that pi is rational. Like I said in my post above, once you start involving infinity your summations and limits could pretty much do anything. The best thing you can do is say that 1 + 1/3 - 1/5 ... or whatever sum you want, is an APPROXIMATION to pi. It will NEVER equal pi because that requires an INFINITE NUMBER OF TERMS which is impossible to compute, define, comprehend, etc...

 

[chroot]

The advantage of math over the natural sciences, superluminal, is that learning it requires no resources beyond a book, a pencil, and some paper.

- Warren

 

[superluminal]

Yeah, but I like to build things!

 

[wesmorris]

The advantage of math over the natural sciences, superluminal, is that learning it requires no resources beyond a book, a pencil, and some paper.

- Warren

(and a capable mind)

Nice to see you still check in from time to time Warren. We have recently briefly lamented your departure.

 

[kriminal99]

I already explained that there were problems with trying to treat these 'dx' as 'infintessimal numbers'. They don't fit in nicely in the real number system that you're used to. If your course was trying to treat these dx terms as numbers (infintessimal or otherwise) without adressing these concerns, then it was crappy. Don't take my word for it though, go find a non-standard analysis book and see how they deal with this. However, something was apparently mentioned in your course (the bold part):

and I suspect that you've misinterpreted your instructor and somehow have come to believe that 'dx' is some kind of number or value.

I don't have Edwards and Penny handy, but if memory serves they (as most other intro calc texts.) they waffled over the actual meaning. They defined the integral as the limit of Riemann sums, where the limit is taken as the norm of the partition heads to zero. The jump from the Riemann sum notation (which had delta x's) to the integral (which has the dx) when you consider the limit is usually described to suggest that this dx represents what you might consider an 'infintessimally small quantity' but that this is vague and they aren't going to go into detail to attatch precise meaning to this comparison or what this comparison would actually mean. The dx essentially means that in the Riemann sums that are the base for this definition of the integral, you are taking the "weight" of your intervals in the partition to just be their length (compare Riemann-Stieltjes integrals for example).

You were probably also warned about this when they introduced the notation dy/dx for the derivative. Again, this is meant to suggest the idea of a ratio of 'infintessimals', but it's really defined as a limit of a certain function (the f(x+h)-f(x)/h or equivalent thing) that is still taking on only real values. You would have been warned not to try to treat dy/dx as an actual fraction, even though it sometimes looks like that's what you're doing (I mentioned the chain rule previously as an example).

He did give some warnings regarding what things you can and cannot do with an infintessimal value. However, these things were immediately evident to me to begin with because an infintessimal, is based on the idea of infinity. The idea is basically composed of "always + shrinking", and has many of the same properties of infinity due to this. Like that it doesn't have a cardinality.

This is trivially true wes. A number is irrational if and only if it cannot be expressed as a finite sum of rationals. Any finite sum of rationals is rational and any rational on it's own is a (trivial) sum of rationals.

A point kriminal was making is a valid one. Just because pi can be expressed as an infinite sum of rationals does not mean it's irrational. The claim "pi cannot be expressed as a sum of a finite number of rationals" isn't really saying anything except "pi is irrational" and does not constitute a proof that pi is irrational. See Sarkus' post for a link to Nivens proof (or a variation of it) for one way to do this.

No the point that I was making is that rational is not universally defined as p/q with the addition that you have to be able to constructively write p and q. It is only universally defined as a number you >can< write as a ratio of integers. Like, supposedly, you could do it with 10^1000/ 2, but youd probably die first. Like, you supposedly could with pi, except you don't know what it is yet.

Simply adding to this definition that infinite summations of rationals "don't count" without justification and then pointing to that as "why pi is irrational" is a circular argument.

"dx" is not an infinitessimal. It is only called such by introductory math professors who are too lazy (or stupid) to teach their classes correctly. In calling dx an infinitessimal, the professor is sacrificing your later understanding for his own momentary convenience. It makes reasonably good sense to think of dx as an infinitessimal when you're doing first semester calculus, since that helps to generalize the Riemann sum into the integral, but it's beyond sloppy: it's wrong. dx is not an infinitessimal, and infinitessimals don't exist in the real number system.

dx is a one-form -- a linear functional that maps vectors into the real numbers. This is the tip of an iceberg of interesting and beautiful mathematics: the study of differential geometry. The real beauty of this mathematics is currently obscured for you, kriminal99, by a couple of things:

1) You have very little education -- only a couple of basic calculus courses.
2) It seems the quality of this education was rather poor.
3) You seem to have delusions of grandeur, thinking that your basic calculus courses have taught you everything there is to know.

To reach the cutting edge of the field, you have at least ten years of formal classroom study ahead of you. You are, in a word, an infant.

- Warren

I don't have to actually take the same class that you did to know that if I did I would be able to clearly explain to you how the two concepts are not mutually exclusive. It is clear to me that you are mentally the equivalent of a sheep as far as objective reasoning ability.

To begin with if you actually wholly understood what you were even talking about, you would have just shown how it is "wrong" to consider dx as an infintessimal. ("NUH UH! ITS CALLED A ONE-FORM!! NAH NAH!!"... lol) Except then you would KNOW the two ideas are not mutually exclusive. I'll give you a hint: It simply depends on what kind of properties you assign to an infintessimal. Im sorry you wasted 10 years of your life in hopes of being half as intelligent as I am. Perhaps you should try classes in a different department, like philosophy- read the post above regarding the limitations of the human brain and how mathematicians are fiercely dependent on the methods used to get around them. For that matter high level mathematicians need SOME capability to think in this manner. I wouldn't be surprised if you find yourself having some trouble... Perhaps that is the source of your bitterness?

 

[kriminal99]

Okay, if there's no such thing as a circle, then what is the locus of X^2 + Y^2 = Z where constant Z > 0. ?

A "circle" is a geometric shape.
It can not be drawn perfectly but it is mathematically defined.

Pi is the ratio of the circumference of this circle to the diametre.

I agree.
PI/4 = 1-1/3+1/5-1/7+1/9-1/11.... etc (Leibniz-Gregory-Madhava series).

There are plenty of other similar ones, by Euler et al...

1+1/4+1/9+...+ 1/n^2 +... = (PI^2) / 6 etc

Ultimately it is this requirement to go to infinity to calculate PI that makes it irrational (using the widely accepted definition).
If it could be done with just 10^1000000 terms then this would be rational - a pain in the proverbial to write down, but rational nonetheless.

This widely accepted definition - Are you trying to claim that it has simply been amended to exclude infinite summations of rationals, or are you trying to claim (erroneously) that it NATURALLY excludes them if all it says is that you must be able to write it as a ratio of 2 integers p and q.

In the first case, I would disagree that this is the universal definition of a rational, and it can not be accepted without argument. In the second case I would point out to you, once again, that saying that such a definition is ambiguous on such a matter.

As for the formula of the circle you have given. The "circle" defined by your equation falls prey to the problem of infinity no less than a real life circle or the definition of it I have given as the limit of a convex polygon as the number of sides goes to infinity.

Instead of increasing the number of sides of a convex polygon, you are ever increasing the vertexes of a complex polygon. Then you have defined the circle as the limit of that.

Now in regards to square roots (adressed towards the person with the paper folding technique)

Let me start by asking you the following question. If I take one apple and one orange, and put them together what do I have? I have one apple and one orange, not square root two "orpples"

Start with the two dimensional coordinate plane. Consider the simple equation x=y. What exactly do we mean when we say the slope of this equation is 1? We mean that whatever quantity we break down x into, we have that many y. If I had 1 apple, I would have 1 orange, If I had .01 apples, I would have .01 oranges. (liters of water and alcohol if you have a problem with apples and oranges usually being considered discretely)

That diagonol line is like the limit of a sort of a staircase deal as the stair size approaches 0.

 

[AndersHermansson]

kriminal99:

x^2+y^2=z is equivalent to the definition given in mathworld:

"A circle is the set of points in a plane that are equidistant from a given point O"

 

[kriminal99]

I don't think I have made my point clear enough. That is an algorithm for adding more vertices to a complex polygon. What I mean by this:

Say you have the equation x^2 + y^2 = 4. Then you take x = 2 and get y = 0. Then (0,2), then (-2,0), (0, -2). With those four points you have nothing more than a square. But you can keep doing it. THe more points you get, the more vertices a convex polygon drawn from point to point would have.

You wouldn't have a circle until you drew an infinite number of points. And you would never finish doing that of course. Now the diagonol that I mentioned before is set, but the circumference is constantly changing. Pi is now like the limit of the perimeter of this polygon as the number of vertices goes to infinity, divided by 2 times the sqrt of Z.