Proof that pi is rational - lol

There is a very important difference between saying "we don't know for SURE that pi can be written as a ratio of integers" and saying "we DO know for SURE that pi CANNOT be written as a ratio of integers" and the latter is true. That's why we know for SURE that pi is not a rational number.

 

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Why don't you try to sit down and precisely define what an infintessimal is then. Avoid vague terms. List the properties you expect they should have. Explain how they fit in with the reals. Explain any kind of operations you think you can do with them (addition? Multiplication? order? etc.).

You can use whatever terms you like, be prepared to explain what they are others aren't familiar with them. I'm still not sure on what you mean by cardinality. In mathematics it's (loosely) used to refer to the number of elements in a set, I would have expected this convention would be used in a philosophy of math course as well. Are you claiming infinity is somehow a set that keeps growing?

I'm not sure what you mean by relation here either. Your limit example is anything but illuminating. Do you think limit x^2 > lim x as x goes to infinity? What would this even mean? Do you have different sizes of infinities? Of infinitessimals? Are they well-ordered?

If you want to actually understand how mathematicians deal with the kinds of limits you saw in caclulus, forget what you think you learned in your calculus classes. Go get yourself an honest to goodness Real Analysis text (Rudin is nice) or even something like Spivak's calculus text. You seem to have missed out on the whole epsilon-delta business. You might want to try to understand this if you actually are interested in understanding mathematics (whether you think mathematicians are "on to something" or just "on crack" or not).

I'm disagreeing with you. Your own 'proof' that pi is rational is assuming that limits (and rationals) have certain properties which they don't (this is one of the problems of relying on your 'intuition' here). Specifically you are assuming that the rationals are a closed set of the reals, they are not (what do you think the real numbers are anyways?). You have to understand that a series is not "an infinite sum", it's the limit of the partial sums. No one is actually adding an infinite number of terms.

I am agreeing that the claim "pi is rational because it can't be written as a finite sum of rationals" is not a proof of anything. Furthur arguments are required to show pi is irrational, see the proof of Niven's someone linked to for an example of how to do this. Many calculus texts will have a proof based on the series for pi from arctan as well (can't recall if Edwards & Penny does or not).

Out of curiosity, have you actually read any of gödel's works?




On the side, there's no need to define pi geometrically. You can take it to be the smallest positive zero of cos(x/2) and go on to show it fits the usual geometric properties. cos(x) itself can be defined in terms of a power series, so we can avoid any geometry here as well.

 

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The set of points you just mentioned takes an infinite effort to recognize because its an infinite set of points...

The limit of both of those terms in the inequality you provided are infinity. Infinite set of numbers, infinity itself whatever... It keeps growing in the sense that you can always concieve of a bigger number than you have already.

Different sizes of infinity? No... Different growth rates maybe, although only in cases like say limit x > inf (x^2/x) would it matter. Limit of x or x^2 as x > infinity is infinity... (oh sorry been using > to mean arrow)

But if you had two different variables such that Limit x goes to inf, y goes to inf (y^2/x) Its a different story....

Although I don't expect you to believe me, my definition of infinity is the right one - the one mathematicians will adopt if they haven't already (they probably have) because its the only way that blatant contradictions won't come up. This is due to the fact that I know how to tie the attributes of my ideas to the source of inspiration for them in reality.. I understand plenty already, thank you.

It is just as likely that infintessimals have been removed from lower math classes that YOU know about because of the POTENTIAL for people to misunderstand them. Potential that would not exist if people had better reasoning skills.

I am not relying on intuition, rather deductive reasoning. I'll concede one thing: My proof shows that if pi actually had a value it would be rational rather than it HAS a value and is rational.

Power series are also infinite summations. The result has no numerical value.

Stop making false statements without arguments by the way... What makes you think I actually believe you know what you are talking about if you can't even provide arguments for your statements?

Irrational values are ALSO at the end of an infinite effort and have no value. As I mentioned in an earlier post: If you use a 2 dimensional coordinate plane what does it even mean to talk about a length that is not perpendicular to either dimension? The diagonol line is really the limit of a stair case type thing as the size of the stairs goes to 0... What I mean again: Take the equation x=y, where x is apples and y is oranges. If you have 2 apples you have 2 oranges, (1,1) (.01,.01).. (.000001,.000001) etc but at no point do you actually have sqrt(2) "orpples".

 

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This is one of the funniest threads I've read in ages.

- Warren

 

kriminal99, you have some sort of gift for using infinity in your proof's don't you. How many times do I have to say that using infinity in your line of thinking is getting you nowhere and you don't see it. The orthodox method of proving that pi is rational doesn't involve infinity whatsoever and that is why it has been accepted and easy to understand. Plus, the word rational means that a number can be written as p/q. How can anyone prove that a number is rational (or not) without using the property that defines the word? Its like trying to prove red is red without using the word red.

Anyway, your "polygon" proof doesn't work either. This is like proving that the length of the hypotenuse of a right-angled triangle is the same as the sum of the lengths of the other two sides by first constructing the hypotenuse as "steps". You should know how this proof progresses: The length of all the tiny little steps obviously equal the sum of the other two sides but as the steps "go to infinity" all of a sudden their length doens't equal the lengths of the other two sides. (you could probably find an illustration of this phony proof somewhere on the internet).

The point is, this is a proof that seems perfectly logical EXCEPT at the defining moment where we expect to see that the length of the hypotenuse is the same length of the two sides, the proof involves INFINITY. It doesn't work here and it won't work in your proof.

Also, I'd like to see your definition of infinity.

PS. There are not enough people in the world with your attitude toward seeing further than everyone else has. I think it is an admirable quality (I know I don't have it.) But I think you are looking into the wrong thing. Proving pi is rational is too irrational. *pun intended*.

 

Yes I gathered that use of > (though pointing it out is appreciated). I was trying to figure out if you consider an inequality a 'relation'. If so, by your belief that a limit would 'preserve relations', you'd necessarily have limit of x^2 greater than limit of x. Or maybe you're using 'preserve' in an unknown manner here.

You really haven't been able to explain what you mean by infinity. Set? Number? Something else? How does it 'preserve relations'?

Care to explain this story?

What definition? From what you've explained here all I see is a bunch of handwaving and nothing at all precise on what you mean by infinity.

No it doesn't. You seriously don't understand what a mathematician means by a limit, that is what your proof shows. Can you tell me how the real numbers relate to the rationals? What property do they have that the rationals don't?

Again, you don't understand what a limit (or series, or both) is or you wouldn't say this nonsense. First of all a power series is something that converges to a function (if it converges that is), not a numerical value. You probably mean infinite series (or maybe a power series evaluated a certain point). And they do have a numerical value (again, if they they converge). An infinite series is defined as the limit of the sequence of partial sums. If you think it's impossible for such a creature to have a numerical value, than why the hell do you talk about limits like they make sense? Integrals should then have no numerical values, derivatives either.

What false statements? For that matter, would you believe my false statements if I provide (presumably also false) arguments? Everything I've said here can be found in any analysis text. It's pretty clear to me that you won't believe anything I say anyway, so I've suggested a couple sources for you to look at. I already feel like I'm wasting my time here. If you don't think that a textbook that contains mathematics accepted by mathematicians everywhere is worth your time to read, why should I believe you'd think my own words are worth your time? Even though you may not find universal acceptance proof of it's validity (that's fine), surely it says more about it's potential than some random guy on the net.

So tell me, by this 'stair case' view of a diagonal line, what is the length of the line from (0,0) to (1,1)? Or do you think length somehow is undefinable in the plane? Or do you think there's no such line?

On second thought, I think this is bollocks. The only proofs in calc texts I can think of are essentially Nivens. In fact I don't think I've seen a proof based on the series (comparable to how you could do so for e). You're a dumbass shmoe.

 

Thats the whole point - the orthodox method of creating pi does rely on the concept of infinity just not in obvious ways. Noone has yet given a method of defining PI that isn't completely dependent on infinity ending.

What are you talking about "using the property that defines the word" I did use that property I just didn't organize the proof formally. When you add any number of ratios of integers you get another ratio of integers. This is a property I used.

You have misunderstood the diagonol line argument. The step thing that I was referring to constantly crosses the diagonol line back and forth. The diagonol length is not the sum of the sides (oO) Im referring to the the infinite summation of infinitely small steps that cross the line back and forth. For that matter you could represent it with an arclength integral... The point is the irrationals such as sqrt(2) having a value is also dependent on infinity ending. The only relation I am claiming it has in relation to the two sides is that it is the difference of the sides as vectors...

The point of this was to point out that the "non rational" quality of sqrt(2) comes from the fact that we are trying to define a length that is not parallel to either axis of the coordinate plane we are trying to measure it on. That any length cannot be measured on a plane that is not parallel to one of the axes should be IMMEDIATELY EVIDENT because this length has to be translated using the limit of an infinite effort.

 

Sum of INFINITE rational nos. can be rational or irrational

See the examples below

1/3 = 0.3 + 0.03 +0.003 + 0.0003 + -----------------

Pi = 4 ( 1 - 1/3 + 1/5 - 1/7 + 1/9 ------------- )

L.H.S. is a rational no. or irational no.
R.H.S is a sum of infinite rational nos.

P.J.LAKHAPATE
plakhapate@rediffmail.com

 

Seems to me 99's argument boils down to labeling infinity a rational number.

 

Sum of INFINITE rational nos. can be rational or irrational

See the examples below

1/3 = 0.3 + 0.03 +0.003 + 0.0003 + -----------------

Pi = 4 ( 1 - 1/3 + 1/5 - 1/7 + 1/9 ------------- )

L.H.S. is a rational no. or irational no.
R.H.S is a sum of infinite rational nos.

P.J.LAKHAPATE
plakhapate@rediffmail.com

 

You're totally wrong about the staircase limit. The "limit of the staircase" is the base of the triangle plus the height of the triangle. To understand why, you have to consider that the total length of the horizontal portion of the staircase will always equal the length of the base and vice versa for the vertical portion of the staircase. Why is it not meaningful to talk about the length not perpendicular to either dimension? In the plane the distance between to points is defined as sqrt((x-a)^2+(y-b)^2). This definition both agrees with the special cases where both points lie perpendicular to either axis and with for example the hypotenuse of a triangle.

 

No it doesn't. It involves a FINITE series (that is, a series with n terms not and infinite number of terms. If you want to see the proof do a quick google search - I found several in just a few minutes.

I think you are making this more complicated than it needs to be. Lets not involve integrals, vectors, dimensions, etc... And what do you mean when you say "infinity ending"?

A piece of advice. If you want to start convincing people here on this forum (including me) then start writing your arguments using terms that we all understand and start organizing your proofs so that they are easy to read. Keep it simple and precise then maybe you'll have a chance.

"non rational" do you mean irrational? What does √2 have to with defining a length?

Have you done any work on vector spaces? The fact that √2 is irrational has nothing to do with its length being parallel to a plane? This is gibberish. By the way, how can a plane EVER be parallel to its coordinate axis?? You are not making any sense anymore!! What is "infinite effort"? Are you talking about force? Work? Why do we have to involve physics?

Please, from now on keep your arguements simple, clear and concise.

 

This genuinely made me laugh. Great line!

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You've just highlighted the very proof that you can not treat a sloped line as an infinite sum of ups and acrosses. Because the length of the staircase line is 2 regardless of the size of the steps, but the actual line length is really √2. This proves that a sloped line is not an infinite sum of horizontal and perpendicular steps.

Of course, my thought experiment about discovering the length of the hypotenuse was indeed just that. But the Universe's inability to go below planck lengths etc is a consequence of the physical laws of the Universe. That's not the fault of the square root of two. In another Universe which did allow infinite regress, you would be able to draw a perfect √2 line.

 

Actually you can show that sqrt(2) is irrational using the fundamental theorem of arithmetic. Silas posted a link to quite a few different proofs of sqrt(2)'s irrationality.

Or you could just define the length of a line with endpoints (a,b) and (c,d) to be sqrt((a-c)^2+(b-d)^2) and avoid relying on your ideas of what a limit is. Surely you've seen this before.

 

No I am not wrong. Thats if the staircase was UNDERNEATH the diagonol. I said the staircase was constantly CROSSING the diagonol. The distance formula is based on pathegorus's theroem. That is based on the concept of area which is something we defined as the length in one dimension times the length in the other. The point here is while the diagonol of a square is "sqrt(2)" length in relation to the side of the square, there is still NO evidence given that it is not simultaneously equal to a rational value that differs infintessimally from it.

If everyone arguing with this is trying to say the stairs are always rational sides and always end up just the sum of the differences in the respective dimensions between the two points that you start the stairs at (wherever that is), thats my whole point! However the fact that you have to keep making the stairs smaller and smaller doesn't allow you to get at that number through this method.

 

It doesn't matter if the stairs crosses the diagonal or not. By always travelling parallel to either the base or the side you will wind up having travelled the same distance. But at least now you've made it official. It doesn't seem like you know what you're talking about at all. You can't even spell diagonal and pythagoras. Now I'm almost sure that you're some guy playing a joke. I mean, "pathegorus", that's a bit over the top isn't it? =)

 

Yeah because I spend all day every day perfecting my spelling skills... maybe thats whats wrong with the rest of you... You spend so long trying to develop spelling skills to impress those that you allow to be concieved of as intelligent that you don't bother to develop your analyzation skills... Im not overobsessed with spelling because after all you knew what I meant didn't you? And I could care less about pythagoras seeing as I never met him and the world inspired his thoughts, rather than he inventing them.

As I said before, what you are pointing to is EXACTLY my point. Its the same thing as saying an infinite summation of rationals will be rational. Because fractions always add up into other fractions. However since it involves an infinite effort, you cant get the actual number that way. You can still define the diagonol as the limit of that staircase as the stairsize approaches 0.

Just what kind of thought process are you using here? That since that value should always equal a rational value, that can't be true because its supposed to be irrational? Thats circular reasoning when we are arguing regarding weather or not it is rational...

And according to some of these people (and I think the point of the definition of irrational) here its irrational BECAUSE you need that infinite effort to calculate it and no other reason. What I am trying to demonstrate is that it still obviously equals a rational value at the same time.

 

Finally, you understand part of what we are trying to say! This is good news. So if you apply the same logic to the infinite summation that represents pi, you can see that you cant actually get pi either! Infinite sums, infinite steps, infinite polygons, etc.. they are all merely APPROXIMATIONS.

Right, we've already said this. (refer to my example of circular reasoning when I was talking about infinity being rational AND irrational).

Ok, I've been waiting for this (an actual question). Your main question is that pi is obviously irrational (you know this for sure) but you also claim that pi is rational as well. Am I correct is assuming this? I hope so. Because now we have an actual claim, instead of random ramblings about god knows what.

The question of whether an irrational number, that can be approximated by an infinite summation of rational numbers, is also a rational number, is a fair question. I would now like to see you give a clear and concise proof of your claim. After which we will see if you have made any oversights, and give some feedback. Ok?

 

You mean that the staircase does not converge to the proper diagonal because it requires an infinite effort? I'm not sure what you mean but I'd say it's because you're taking the wrong limit.

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Which is true, given that the number has a decimal expansion that doesn't terminate AND doesn't repeat.

For your demonstration you need to show that either the decimal expansion terminates or it repeats (in which case you can write the number as p/q).

 

No the staircase does converge to the diagonol, and gives us the infinite summation which results in sqrt(2). Note that the start and endpoints of this staircase are not on the endpoints of this diagonol, but rather get closer and closer to the diagonol as the number of stairs increases. However they can be chosen as rational values. Infinitely increasing the number of stairs and shrinking the size of them is how we define sqrt(2). The infinite summation involved with doing this mathematically would represent the marginal length added/subtracted by increasing the number of stairs.

However there is no number sqrt(2) that exists at the end of this calculation that we just can't get to because it never ends that is not rational. Whatever the result of that sequence is it differs trivially from a rational number. If you give me any definition of square root 2, I will eventually be able to reduce it to this one.

I don't need to show that the expansion terminates or repeats, because this does not rule out trivial difference from a rational, nor do any of the proofs that sqrt(2) is not rational.