Proof that pi is rational - lol

[Sarkus]

You wouldn't have a circle until you drew an infinite number of points. And you would never finish doing that of course.

Just because something requires an infinite number of something does not mean that it can't be done....

There are an infinite number of (x,0) points between (0,0) and (1,0), agreed?
Yet if you drew a straight line from point (-1,0) through to (2,0) you would have surely drawn through all those infinite points, no?

Yet this does not take an infinite time to do.

You are viewing things as discrete - the "dx" - when there is also the continuous.

 

[kriminal99]

Just because something requires an infinite number of something does not mean that it can't be done....

There are an infinite number of (x,0) points between (0,0) and (1,0), agreed?
Yet if you drew a straight line from point (-1,0) through to (2,0) you would have surely drawn through all those infinite points, no?

Yet this does not take an infinite time to do.

You are viewing things as discrete - the "dx" - when there is also the continuous.

Well my point was this doesn't give any more solid a basis for a definition of pi as it is still the limit of an infinite effort. Now your comparing reality with our limited understanding of it (math) But anyways...

What you mean to say, is that there are an infinite number of infinitely small points from 0 to 1. However the tip of a pencil is not infinitely small. The force applied to the pencil by our muscles does not exist in infinitely small quantities.

 

[Sarkus]

However the tip of a pencil is not infinitely small. The force applied to the pencil by our muscles does not exist in infinitely small quantities.

So you're saying that if we did have a pencil with an infinitely small tip then we would never be able to draw a continuous line through (0,0) to (1,0)?
Ever heard of Xeno?

 

[kriminal99]

How the flip should I know? What does it even mean to have a pencil with an infinitely small tip? Its size would be constantly shrinking at the rate we can concieve of something smaller than what was already there. Remember infinite numbers and infinitely small numbers don't have a cardinality.. (set size)

Xeno - Yep. That paradox is bunked by what I said before: The force provided by our muscles doesn't exist in infinitely small quantities.

IE past a certain point you wouldn't be able to move small enough distances anymore. Also at some point, obviously, your ability to measure half the distance would become way too imprecise.

 

[HallsofIvy]

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.

Okay, I thought you were objecting to defining "rational number" as a number that CAN be written as a fraction. You were only objecting to defining it as a number that IS written as a fraction or requiring that the numerator an denominator be known. That's fair.

However,

Like, you supposedly could with pi, except you don't know what it is yet.

is not true. There exist several different proofs that pi CANNOT be written as a fraction. There are not a simple as the proof that sqrt(2) cannot be be written as a fraction but they should be clear to an undergraduate math major.

 

[shmoe]

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.

Did he bother to define what an 'infintessimal value' was? Can you explain "always+shrinking"? What do you mean by 'cardinality' here?

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.

There's really no difference in what I said. Knowing pi as an infinite sum of rationals on it's own does not imply that it somehow cannot be written as a ratio of integers, we may have just not found a way to do so. In other words, just because you haven't found a way of doing something doesn't mean that something is impossible to do.

Like, supposedly, you could do it with 10^1000/ 2, but youd probably die first.

10^1000/2 is already in a form that shows it's rational since 10^1000 is an integer (I should point out that something like 10^3 is no less an integer than 1000, in case that's somehow not obvious). Or do you want all the digits written out? This really won't take very long, but will be an eyesore.

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.

I was agreeing with this. Remember when I said "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."?

 

[AndersHermansson]

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.

I don't know if i understand where you are getting at? That it's impossible to draw a perfect circle by hand or by machine in the real world or that it there exist no perfect circle mathematically?

Well given that you accept real numbers you get 2-space (x,y) (pairs of numbers in two real sets) in which you define distance between two points as d=sqrt(x^2 + y^2). Then you can easily define a circle as the set of points equidistant relative some fixed point C. You can call this circle perfect if you'd like. If you then plot this set on paper you will see something that you would intuitively call a circle. It is not necessary to plot the circle or do any calculations for it to exist mathematically.

Pi also follows from this, and we don't even have to calculate the value of it in decimal form or take any limits.

 

[chroot]

Why can't dx be an infinitessimal? That's easy to show.

You're probably accustomed to the Riemann sum being expressed as:

Sum( f(x) Delta x )

And you learned that as Delta x tends to zero, the Riemann sum becomes the integral:

Integral( f(x) dx )

The problem with this is the "Delta x tends to zero" part. The limit of Delta x as Delta x approaches zero is... zero. Not "infinitessimal." Zero. It's a plain, simple limit. If this were really true, the integral would disappear.

The reason you were taught this way is because your teacher was lazy. The reason you accepted it (like the mental sheep you are) is because you did not yet have a satisfactory understanding of limits.

Personally, I can't believe you're so quick to call me names when you can't even be bothered to do a little investigation yourself.

- Warren

 

[kriminal99]

Why can't dx be an infinitessimal? That's easy to show.

You're probably accustomed to the Riemann sum being expressed as:

Sum( f(x) Delta x )

And you learned that as Delta x tends to zero, the Riemann sum becomes the integral:

Integral( f(x) dx )

The problem with this is the "Delta x tends to zero" part. The limit of Delta x as Delta x approaches zero is... zero. Not "infinitessimal." Zero. It's a plain, simple limit. If this were really true, the integral would disappear.

The reason you were taught this way is because your teacher was lazy. The reason you accepted it (like the mental sheep you are) is because you did not yet have a satisfactory understanding of limits.

Personally, I can't believe you're so quick to call me names when you can't even be bothered to do a little investigation yourself.

- Warren

Except (good god) that an integral is an infinite summation of these infinitely small parts. How on earth did you come to the conclusion that if you had n of interval / n size blocks and took the limit of this (interval) * n/n as n went to infinity you would end up with 0? No matter how large the n on the bottom gets its canceled out by the n on the top.

You obviously have no clue what you are talking about. And it never ceases to amuse me when some idiot busts into a thread and immediately starts talking smack without any justification, but then bleeds from the heart the minute you have anything bad to say about them...

I don't know if i understand where you are getting at? That it's impossible to draw a perfect circle by hand or by machine in the real world or that it there exist no perfect circle mathematically?

Well given that you accept real numbers you get 2-space (x,y) (pairs of numbers in two real sets) in which you define distance between two points as d=sqrt(x^2 + y^2). Then you can easily define a circle as the set of points equidistant relative some fixed point C. You can call this circle perfect if you'd like. If you then plot this set on paper you will see something that you would intuitively call a circle. It is not necessary to plot the circle or do any calculations for it to exist mathematically.

Pi also follows from this, and we don't even have to calculate the value of it in decimal form or take any limits.

Not sure if its relevant with what you are talking about, but I also outlined a similar problem with the irrationals. But skipping that for now:

The thing you would draw from these points and intuitively call a circle only looks like a circle because it has too many sides to see clearly as a polygon with the human eye... Forget about plotting points.. It doesn't matter how many points you even concieved of you would never be finished constructing a circle even in your mind. You are basically thinking "hey it seems like I could do this forever... If I DID do this forever what would I end up with?" And the answer would be a circle. Unfortunately forever doesn't actually end.

My statements are mostly concerned with our mathematical representation of the world (and therefore any tools we would be using) My point is PI is not an actual number. It is the limit of an infinite effort, and since that effort never ends pi never exists. No matter how many decimal places you have for pi it could represent a polygon with a large number of sides. The circle is the same, the limit of an infinite effort and therefore never is actually constructed.

 

[chroot]

Since when has infinity / infinity been one?

- Warren

 

[chroot]

And since when has an infinite sum of zero-width rectangles had anything other than zero total width?

- Warren

 

[AndersHermansson]

He confuses taking the limit of sums as the lot of them tends to infinity with actually summing up an infinite amount of infinitely small terms. (I think I'm right in discriminating them)

 

[kenworth]

is it bad that i just take it on faith that if for a few thousand years no-one has proved that pi is rational an internet teen is not particularly likely to have done it?

 

[chroot]

Pi was proven to be irrational a long, long time ago.

- Warren

 

[kriminal99]

Since when has infinity / infinity been one?

- Warren

This isn't just any infinity of any "growth rate" divided by any infinity that we are concerned with.

Its the same n on the top and bottom that goes to infinity. Infinity has no size, but it preserves relations. If you have limit x > inf (x^2/x) this reduces to limit x > inf (x). Its the exact same thing.

is it bad that i just take it on faith that if for a few thousand years no-one has proved that pi is rational an internet teen is not particularly likely to have done it?

It is poor reasoning (and btw internet teen? oO) for several reasons. 1) Because if everyone reasoned this way, then of course noone would proove otherwise because noone would try... 2) The last line of the proof says "pi is rational or math is inconsistent"... but its already been proven that math cannot be consistent and complete. Not only that but no effort has been conducted to group math into belief sets that are consistent but can only deal with a certain subject. Therefore math itself recognizes that inconsistencies could pop up anywhere.

Did he bother to define what an 'infintessimal value' was? Can you explain "always+shrinking"? What do you mean by 'cardinality' here?

There's really no difference in what I said. Knowing pi as an infinite sum of rationals on it's own does not imply that it somehow cannot be written as a ratio of integers, we may have just not found a way to do so. In other words, just because you haven't found a way of doing something doesn't mean that something is impossible to do.

10^1000/2 is already in a form that shows it's rational since 10^1000 is an integer (I should point out that something like 10^3 is no less an integer than 1000, in case that's somehow not obvious). Or do you want all the digits written out? This really won't take very long, but will be an eyesore.

I was agreeing with this. Remember when I said "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."?

I don't really remember what he said in class. I didn't really need him to define an infintessimal though because I already knew or could figure it out when I came to the class (else I would have failed miserably without the textbook no doubt)

Infintessimal, example 1/infinity, infinity - subconsiously (how we construct the idea) defined as always growing, IE no matter how big a number you think of it its bigger than that.

Inspired by: recognition that objects in our view look similar which in turn inspires the thought that we can imagine another similar object grouped with these others, then by the fact that there is no limit to how many times we could add another object.

Necessary properties: It preserves relations but has no set size (this is what I meant by cardinality, its a term I learned in a philosophy of mathematics class sorry I don't like doing that) By relations I mean for example limit x > inf (x^2/x) = limit x > inf (x), reason being that however big of an x you concieve of you simultaneously have to recognize that x^2 is just that.

Are you agreeing or disagreeing with me? Are you saying the proof is wrong because we don't know for SURE that pi can be written as a ratio of integers and PI has been proven irrational? The problem with this is that sums of rationals are supposedly always rational by definition of things like addition and integer.

Unfortunately there is reason to believe that it is possible that something be simultaneously "proven" and "disproven" at the same time in mathematics. But more than that proofs by contradiction were illegitimate to begin with. Why?

A positive proof takes some assumptions and necessitates a cause. The only assumptions not listed must be shared by the reader or the argument makes no sense.

A proof by contradiction on the other hand depends on only the author's ability to consiously recognize all the assumptions he is making. Not only is it possible for him to miss an assumption, but this is pretty much EXACTLY what happens whenever someone is wrong about something. This is the source of people's disagreement on things, and people disagree ALOT. This kind of proof is not reliable.

 

[chroot]

Sure, the limit of (x^2/x) as x goes to infinity is well-defined -- the numerator grows faster than the denominator, so the limit is infinity.

You're concerned with n/n, however -- and the limit of this as n goes to infinity is not defined. Stop trying to change the subject.

- Warren

 

[kenworth]

It is poor reasoning (and btw internet teen? oO) for several reasons. 1) Because if everyone reasoned this way, then of course noone would proove otherwise because noone would try... 2) The last line of the proof says "pi is rational or math is inconsistent"... but its already been proven that math cannot be consistent and complete. Not only that but no effort has been conducted to group math into belief sets that are consistent but can only deal with a certain subject. Therefore math itself recognizes that inconsistencies could pop up anywhere.

i'm assuming you are a teenager because i think at some point you said you had just started an introductory calculus class.i also assume because pi is used in so so so many things that many of humanities finest minds would have spent at least some time pondering it.also the little man icon comes up if i post on here and i wanted to keep tabs on this thread cause im interested to see its outcome.

 

[chroot]

Because if everyone reasoned this way, then of course noone would proove otherwise because noone would try...

So some people should always try to be wrong, to add diversity?

2) The last line of the proof says "pi is rational or math is inconsistent"... but its already been proven that math cannot be consistent and complete.

No, it hasn't. It has been proven that, no matter how many axioms you have, there will exist true statements not derivable from those axioms. Godel's theorem says nothing about the consistency of those statements which are derivable from the axioms. Those had better be consistent, or you're just playing stupid games.

Not only that but no effort has been conducted to group math into belief sets that are consistent but can only deal with a certain subject. Therefore math itself recognizes that inconsistencies could pop up anywhere.

There are many such "belief sets." There are entire branches of math that spring forth from the choice of either of two axioms -- non-Euclidean geometry, set theory, etc.

The point of math is not to find what is "absolutely true." The point of math is to find all that is true given a specific set of axioms. If you change the axioms, you've got a whole new world of math to live in. Of course, some of those worlds are internally inconsistent, and therefore aren't very interesting.

- Warren

 

[kriminal99]

Sure, the limit of (x^2/x) as x goes to infinity is well-defined -- the numerator grows faster than the denominator, so the limit is infinity.

You're concerned with n/n, however -- and the limit of this as n goes to infinity is not defined. Stop trying to change the subject.

- Warren

Im not changing the subject, if you had any clue what you were talking about you would know this. The next logical step in your reasoning process should have been:

Limit x > inf (x/x) before even calculating this you can reduce it to:
Limit x > inf (1) = guess freaking what? 1 And using the reasoning you just mentioned concerning growth rates, they both grow at the same rate and perserve the relation x/x so you conceded this already. Infinity / infinity when the infinities are created by different means are the only kind that are difficult to define.

i'm assuming you are a teenager because i think at some point you said you had just started an introductory calculus class.i also assume because pi is used in so so so many things that many of humanities finest minds would have spent at least some time pondering it.also the little man icon comes up if i post on here and i wanted to keep tabs on this thread cause im interested to see its outcome.

I never said that... I said I have taken math courses up through linear algebra and multivariable calculus.. Some other joker said that about my taking intro calc classes... I think it was chroot. The problem with your reasoning in the second part is you are excluding yourself and anyone you know from potentially being one of "humanity's finest minds". Make no mistake that "humanity's finest minds" were still whiny smelly poopy hairless monkeys. This categorization is subjective and superficial no matter what the person accomplished.

 

[AndersHermansson]

My statements are mostly concerned with our mathematical representation of the world (and therefore any tools we would be using) My point is PI is not an actual number. It is the limit of an infinite effort, and since that effort never ends pi never exists. No matter how many decimal places you have for pi it could represent a polygon with a large number of sides. The circle is the same, the limit of an infinite effort and therefore never is actually constructed.

The thing is that you don't have to make an infinite effort to construct pi. Just define it as the ratio of the circumference to the diameter of the circle (the set of points equidistant to O, not your limit of polygons) and by the completeness property of the real numbers the number pi is bound to exist.