Proof that pi is rational - lol

[superluminal]

I am insulting you because:

1) You are not making interesting suggestions regarding math and such and leaving it at that, but you are stating outright that the entire body of math is flawed and provide no evidence of this whatsoever. And,

2) You are stupidly arguing definitions that in and of themselves mean nothing. And,

3) You have the unmitigated gall to think you have some view that none of us have thought of before. And,

4) You are condescending in a way that only cranks with a selfrighteous confidence can be.

At least I'm insulting you clearly, with no pretense.

 

[shmoe]

This is just logic. Mathematicians can arbitrarily define terms all day long, but they don't get to define what is logically dictated by something that you claim. To begin with you are giving infinite series a seperation from finite series that they are not afforded by the definition of infinite used often in mathematics.

This is the last time I'm going to bother replying to you, I'm getting tired of repeating myself and you just don't listen (or aren't capable, or are just trolling, or...).

Let's just talk about the real numbers for a second here. Under the axioms for the reals numbers (go look up 'complete ordered field' axioms if you like) addition is an operator that takes two real numbers to another real number. This is what is assumed. Nothing at all is said about applying addition to an infinite number of summands when we are describing the properties of addition. In fact addition isn't even defined on an infinite number of summands, got that?

We get around this by defining an infinite series as the limit of the sequence of partial sums. Never do we apply addition to infinitely many summands, got that? From the addition axioms and this definition of an 'infinite sum' we notice a few things pretty fast, noticeably this doesn't behave exactly how we would hope our sums over a finite number of terms do. If you take a finite sequence of real numbers, their sum is always defined, not so for an infinite one (divergent series). Another is the order of the terms in the sum will matter in the case of a conditionally convergent series, but not if it's absolutely convergent-sometimes we have as much commutivity as we like, other times we don't. With our new definition of infinite sums, we lose some properties we would have liked to have, but we have extended this summation operator to infinite sequences in a way that is somewhat familiar with the finite case, but not exactly the same.

Do you understand this? This is key- our definition (and yea, in some sense our definition is 'arbitrary') of these infinite sums can be shown to share some properties of finite sums and definitely does not share others. If you want to know what it does or does not share, go back to the definition of infinite sum and go from there. For the nth time you have assumed that an infinite series consisting of rationals should end up as a rational (if the series converges). You have made no attempts whatsoever to deduce this from the definition of an infinite series. I have already said it's false in general, my evidence was Niven's proof that pi is irrational, or look at one of the many proofs that sqrt(2) is irrational. Just look at sqrt(2) if you like, proof of it's irrationality is very simple. Proof of existence of a real number x satisfying x^2=2 follows from the least upper bound axiom of the real numbers (this is the 'complete' part of the complete ordered field).

In short-an infinite sum and a finite sum are quite different, they have some very different properties (many the same though) that follow from the definitions mathematicians use and you will never see a mathematician arbitrarily claim they have certain properties that they haven't seen logically deduced from the definitions. You've claimed otherwise which tells me you've never even bothered to look at these definitions. Everything I've said above (including what a mathematician means by a limit, something I think you missed out on in your calculus class as you said you didn't see any epsilons and deltas) should be in full detail in any real analysis text (such as Rudin, Royden, Pffafenberger, or even Spivak's Calculus, or many other books). Do your self a favour and pick up a text and try to understand what mathematicians are actually saying before you try to claim they're wrong.

But whatever. You want to sit about in your ignorance and spout of things like "(mathematicians) don't get to define what is logically dictated by something that you claim" when this is exactly what you are doing without even knowing what it is they've claimed in the first place.

 

[James R]

kriminal99:

I might just add my 2 cents to what shmoe has just said, too.

In reality nothing changes regarding a summation of rationals just because it doesn't end. At what point do you think a summation suddenly becomes irrational? If you add a million terms its still necessarily rational. A billion etc... A summation being infinite simply means you don't stop adding more terms. It doesn't magically change the properties of the things being added or how.

A billion terms, or a googol terms is still a finite number of terms. Infinity is a whole different ball game. A summation doesn't "suddenly become irrational". Not after a billion terms, or after a googleplex terms. But it can converge to an irrational number after an infinite number of terms.

Note, I wrote "converge to", not "add up to". As shmoe said, the important concept here is convergence of a series. A series converges if the related sequence of partial sums converges. To get the "limiting value" of a series, we don't actually add up an infinite number of terms, even if that is what it may superficially look like we're doing. Actually, we calculate a sequence of partial sums, then examine the limit of that sequence (if it exists).

This is just flat out wrong, as hallsofivy pointed out. 3/10 + 3/100 + 3/1000... is a counter example (1/3)

It seems you're confused. I did not assert that every infinite series gives a non-rational result. What I asserted was that some infinite series converge to an irrational limit.

I am wondering... Do you dispute the existence of irrational numbers completely, or are you simply asserting that no series of rational numbers can converge to an irrational number?

It doesn't matter what path you take, there are going to be large numbers of contradictions as long as you are arbitrarily defining terms.

Right! That's why consistent systems of mathematics are based on non-contradictory sets of axioms. The "terms" defined are far from arbitrary. Rather, they are very carefully chosen.

 

[kriminal99]

Faith by definition requires no physical evidence or logic. Faith is based on the unquestiong BELIEF in the truth of a thing.

Proof by definition consists of concrete evidence or logical reasoning that is consistent with everything currently postulated about a particular subject.

Theologians are cranks.

As I already pointed out, theologians would categorize their proof of god as "concrete evidence or logical reasoning with everything currently postulated about a particular subject" And you certainly haven't demonstrated the ability to tell the difference. Also as I have already pointed out if your beliefs were only motivated by truth you wouldn't be on here whining like a little baby and insulting me. You are doing it because you know damn well that I am right and it is beyond your current ability to argue with me.

This is the last time I'm going to bother replying to you, I'm getting tired of repeating myself and you just don't listen (or aren't capable, or are just trolling, or...).

Let's just talk about the real numbers for a second here. Under the axioms for the reals numbers (go look up 'complete ordered field' axioms if you like) addition is an operator that takes two real numbers to another real number. This is what is assumed. Nothing at all is said about applying addition to an infinite number of summands when we are describing the properties of addition. In fact addition isn't even defined on an infinite number of summands, got that?

We get around this by defining an infinite series as the limit of the sequence of partial sums. Never do we apply addition to infinitely many summands, got that? From the addition axioms and this definition of an 'infinite sum' we notice a few things pretty fast, noticeably this doesn't behave exactly how we would hope our sums over a finite number of terms do. If you take a finite sequence of real numbers, their sum is always defined, not so for an infinite one (divergent series). Another is the order of the terms in the sum will matter in the case of a conditionally convergent series, but not if it's absolutely convergent-sometimes we have as much commutivity as we like, other times we don't. With our new definition of infinite sums, we lose some properties we would have liked to have, but we have extended this summation operator to infinite sequences in a way that is somewhat familiar with the finite case, but not exactly the same.

Do you understand this? This is key- our definition (and yea, in some sense our definition is 'arbitrary') of these infinite sums can be shown to share some properties of finite sums and definitely does not share others. If you want to know what it does or does not share, go back to the definition of infinite sum and go from there. For the nth time you have assumed that an infinite series consisting of rationals should end up as a rational (if the series converges). You have made no attempts whatsoever to deduce this from the definition of an infinite series. I have already said it's false in general, my evidence was Niven's proof that pi is irrational, or look at one of the many proofs that sqrt(2) is irrational. Just look at sqrt(2) if you like, proof of it's irrationality is very simple. Proof of existence of a real number x satisfying x^2=2 follows from the least upper bound axiom of the real numbers (this is the 'complete' part of the complete ordered field).

In short-an infinite sum and a finite sum are quite different, they have some very different properties (many the same though) that follow from the definitions mathematicians use and you will never see a mathematician arbitrarily claim they have certain properties that they haven't seen logically deduced from the definitions. You've claimed otherwise which tells me you've never even bothered to look at these definitions. Everything I've said above (including what a mathematician means by a limit, something I think you missed out on in your calculus class as you said you didn't see any epsilons and deltas) should be in full detail in any real analysis text (such as Rudin, Royden, Pffafenberger, or even Spivak's Calculus, or many other books). Do your self a favour and pick up a text and try to understand what mathematicians are actually saying before you try to claim they're wrong.

But whatever. You want to sit about in your ignorance and spout of things like "(mathematicians) don't get to define what is logically dictated by something that you claim" when this is exactly what you are doing without even knowing what it is they've claimed in the first place.

Im not the one who is ignorant here. Mathematicians use the real definition of infinity all the time in their reasoning, because it doesn't make sense otherwise etc. AS I CLEARLY STATED AND YOU DID NOT EVEN ADDRESS:
Just because a sum becomes infinite doesn't mean it suddenly looses all its properties. Infinite only means that it keeps going forever. Infinite summations are COMPOSED of finite terms for which addition is cleary defined. If you stop it at any point int the process it still has all the same traits. If infinity were to "end" it would have the same traits. If it converges on a point, due to other things specified it would have to be trivially different from a rational value (rationals are infinitely dense). You have provided 0 reasoning for anything you have claimed. You are basically sitting here arguing about something you know absolutely about and you entire arguments say: "Mathematicians don't think this" Well your assesment of the mathematics community is WRONG and IRRELEVANT. Some mathematicians may believe certain things but there is not unanomous agreement. And its irrelevant anyways because whatever is the best way to look at things is the way things SHOULD be looked at. You admit YOUR definition is arbitrary. What you do not understand is that it contradicts other ways you treat the concept of reality that are based on reality. Im glad you are leaving. I don't think you have a basic understanding of debate, reasoning or argument itself to consider and argue on these ideas.

kriminal99:

I might just add my 2 cents to what shmoe has just said, too.

A billion terms, or a googol terms is still a finite number of terms. Infinity is a whole different ball game. A summation doesn't "suddenly become irrational". Not after a billion terms, or after a googleplex terms. But it can converge to an irrational number after an infinite number of terms.

Note, I wrote "converge to", not "add up to". As shmoe said, the important concept here is convergence of a series. A series converges if the related sequence of partial sums converges. To get the "limiting value" of a series, we don't actually add up an infinite number of terms, even if that is what it may superficially look like we're doing. Actually, we calculate a sequence of partial sums, then examine the limit of that sequence (if it exists).

It seems you're confused. I did not assert that every infinite series gives a non-rational result. What I asserted was that some infinite series converge to an irrational limit.

I am wondering... Do you dispute the existence of irrational numbers completely, or are you simply asserting that no series of rational numbers can converge to an irrational number?

Right! That's why consistent systems of mathematics are based on non-contradictory sets of axioms. The "terms" defined are far from arbitrary. Rather, they are very carefully chosen.

"converge to an irrational number" Well the point here I have been trying to make is that they don't really converge on an "irrational number" but rather we call them irrational because we cant pin a set rational value on them. There is no irrational number for them to converge on, but rather whatever this infinite chain goes to is what we understand the rational number to be. All the irrationals that we have created through various means are not rational because they are constructed by some infinite effort algorithm that causes us to add together infinite chains of fractions.

What is thought, and what I am arguing is wrong, is that these infinite effort algorithms which are used to create irrational numbers have some claim to reality so the irrational numbers really exist. The problem with this is that there is not a good reason to believe this in any of the cases when you look at them carefully and objectively. There is the "perfect circle" that motivates pi... There is the trying to measure lengths that aren't parallel to either axis of the coordinate plane that motivates square roots of numbers that don't have them.

Now as some limits have a rather well defined meaning, there is a problem related to limits of an infinite effort algorithm, aka, infinite summation of fractions. This problem is equivalent to the statement "there is no irrational value for these chains to converge on". That is you can only come up with a limit for a limited number of decimal points.

Take pi for example. If you have the supposed limit of a pi sequence that gives you 50 decimal places, it might tell you that convex polygons with 500 sides or more are always going to have the ratio of perimeter to diagonol involving what we construe as pi for 50 decimal places. 100 decimal places maybe 1000 sides (just guessing here at the ratio of sides to decimal places) You can go as far as you want but you don't have a circle until you have a polygon with an infinite number of sides and the limit of the pi sequence to an infinite number of decimal places. And then there is no reason to believe the result wouldn't be trivially different from a rational value (except it doesn't make sense to think of infinity ending).

Our math does not allow for circles OR diagonol lines WITHOUT infinite effort. We know these things exist in the real world, so we use these infinite sequences to try and model them. However, in the real world it could be anything that allows these things to exist. Like dimensions not defined by lines for example.

Actually you clearly said "ONLY FINITE SUMMATIONS OF RATIONALS RESULT IN RATIONALS" This statement is not really subject to interpretation in this regard its meaning is clear. Perhaps you made a typo or something but nevertheless...

Unfortunately in reality it is far beyond the capacity of a human being to "carefully choose" how to define terms in a belief set like mathematics such that they will not ever contradict other things in the set. If a human being was capable of seeing every logical consequence of defining something a certain way they wouldn't need mathematics to begin with... So what may be "carefully chosen" can cause contradictions immediately outside what the person who "carefully chose" could consider.

 

[superluminal]

K99

As I already pointed out, theologians would categorize their proof of god as "concrete evidence or logical reasoning with everything currently postulated about a particular subject" And you certainly haven't demonstrated the ability to tell the difference.

No my sadly misinformed friend. Theologians do not claim their faith is based on concrete evidence of any kind. Ask a theologian for repeatable test results regarding his faith. See what happens.

Also as I have already pointed out if your beliefs were only motivated by truth you wouldn't be on here whining like a little baby and insulting me. You are doing it because you know damn well that I am right and it is beyond your current ability to argue with me.

Once again grasshopper, no. Most here subscribe to the idea of politeness, or at least restrained ridicule (everyone here is doing it to you. Even our mod does it.) regarding, hmmm... let's say... people with curious predilictions for unsupported and inconsistent reasoning. I, on the other hand, have no problem calling the handle on the door to the loony-bin a crank.

People can be misinformed, curious, seeking enlightenment... and there is a way to go about it. Analyze your posts to see why you are not one of those.

 

[AndersHermansson]

All the irrationals that we have created through various means are not rational because they are constructed by some infinite effort algorithm that causes us to add together infinite chains of fractions.

It has been mentioned many times that you don't need infinite sequences to construct an irrational number.

 

[James R]

kriminal99:

Do you agree that sqrt(2) is irrational? If not, consider this proof:

First assume that sqrt(2) is rational, and can be written as p/q, where p and q are integers with no common factors (apart from 1).

Then p^2 = 2 q^2, which means that p^2 is an even number. Since the square of an even number is even, and the square of an odd number is odd, p must be even.

Since p is even, we can write p = 2r, where r is an integer. Then, it follows that:

(2r)^2 = 4 r^2 = 2 q^2

or q^2 = 2 r^2

Therefore, q^2, and therefore q, is even.

But if p and q are both even, then they have a common factor of 2. This is in contradiction to the initial assumption that they had no common factors. The only thing which can be wrong in this argument is that sqrt(2) can be written as the fraction p/q (reduced to "lowest terms").

Since sqrt(2) cannot be written as a fraction, it must be irrational.

Do you agree? Have we therefore shown that at least one irrational number exists?

What is thought, and what I am arguing is wrong, is that these infinite effort algorithms which are used to create irrational numbers have some claim to reality so the irrational numbers really exist.

Note that the proof that sqrt(2) is irrational does not involve an infinite summation of any kind. sqrt(2) is not "constructed" that way.

Take pi for example. If you have the supposed limit of a pi sequence that gives you 50 decimal places, it might tell you that convex polygons with 500 sides or more are always going to have the ratio of perimeter to diagonol involving what we construe as pi for 50 decimal places. [etc]... And then there is no reason to believe the result wouldn't be trivially different from a rational value (except it doesn't make sense to think of infinity ending).

ALL irrational numbers are "trivially different" from SOME rational number. In other words, it is always possible to find a rational number as close as you want to specify to a given irrational number. But that doesn't change the fact that some numbers are irrational.

Actually you clearly said "ONLY FINITE SUMMATIONS OF RATIONALS RESULT IN RATIONALS" This statement is not really subject to interpretation in this regard its meaning is clear. Perhaps you made a typo or something but nevertheless...

There's not much point going on with this, is there? There are two possibilities:
1. I wasn't clear.
2. I was clear, and it's your misinterpretation.
Either way, I've now made it perfectly clear what I wanted to say, so there should be no more confusion. Right?

 

[oxymoron]

This is what I don't get James R. The definition of the word irrational, means exactly what you said in your proof. Yet kriminal99 has been shown this proof but he will not accept it!? He believes that the actual definition of the word rational is somehow flawed. If it is, then isn't claiming that pi is rational a flawed statement because you are using the word 'rational', a word that, which you have already stated, has a flawed defintion. And we go round and round and round...

If a number is rational then it MUST be able to be written as p/q where p and q are relatively prime integers. This is the definition of rational. So why can't we use the very meaning of the word to convince kriminal99 that π is irrational, ie. the converse of this proof???

 

[James R]

My very first response to this thread was to say that if kriminal99 wants to use his own peculiar definition of the term "rational number", then he won't get far in a discussion with mathematicians. And he hasn't.

 

[Robust]

Since we all know my mathematics is limited, I am not joining this debate but wanted to post a note as an observer.

I believe the issue is one of symantics and that most here are right in that Kriminal99 seems to be forming his own definition.

I get the impression he does so because he sees the existance of a perfect circle as requiring a rational number for pi since a circle couldn't be formed otherwise.

It seems to me that is incorrect.

Pi is the ratio of two numbers. Those two numbers, if based on the existance of a perfect circle, must indeed be rational and have a finite number of decimals, etc.

That would seem to be required or one must consider there is no such thing as a perfect circle.

Have I got this even close?

Mat M, Pi is not the ratio of two numbers - it is the ratio of line to arc

 

[Naomi]

Let me repeat the obvious for the asshumpteenth time.

The definition of a rational number: Any number which can be written in the form a/b, where both a and b are integers, and b is not zero.

Now this, you learned in junior high school at the very latest.

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-

K99, you could just avoid this hassle if you'd stop trying to yell at everyone and just do the following.

1) Write pi in abovementioned a/b form, and find a and b for us.

2) Show that it works out to be pi.

3) Gloat.

This is the only way for you to prove that pi is rational.

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-

However, you've obviously never succeeded, since you're still sitting here yelling. You're claiming you're right, when we know you're not with the official definition of "rational number". Thus, the only thing to conclude is that you have a different definition of "rational number" in mind.

You seem to include "any sum of any number of rational numbers" in the definition of "rational number", and you're trying to stretch that to say that an infinite summation of rational numbers is rational.

In that case, you're only correct under your own definition. You are still incorrect under the official definition, which you aren't using.

Do you get it now? You're still wrong if you use the real, actual, official definition of "rational number", but not if you use some personally invented definition of "rational number", which no one cares about.

You can stop trying to persuade us here, because no one cares about your weird personal concept of 'rational'. If you want people to listen, use the proper definitions. I have a feeling I could be wasting my breath on someone who won't bother to listen, but I'm saying it anyway.

Otherwise you won't get far.

 

[Robust]

I asked this in the other thread, but I'll ask again here. Do you believe there are any irrational numbers at all?

Have you considered the possibility that maybe, just maybe, your intuition about limits is flawed?

Can you define what you mean by "infintessimally small"? For that matter, can you define what it means for a number to be the limit of a sequence?

Sorkus, I have to take exception to your stating that pi can only be described by the sum of an infinite sequence of fractions to arrive at the rational answer. What yiu imply is that pi is sacrosanct.
It has recently been shown where it is not. History records a number of authenticated pi values including the Euler irrational pi....and more recently a finite pi value. It is shown, however, that none is sacrosanct in defining area of the circle.

 

[James R]

What do you mean by "finite pi value", Robust?

 

[kriminal99]

No my sadly misinformed friend. Theologians do not claim their faith is based on concrete evidence of any kind. Ask a theologian for repeatable test results regarding his faith. See what happens.

Once again grasshopper, no. Most here subscribe to the idea of politeness, or at least restrained ridicule (everyone here is doing it to you. Even our mod does it.) regarding, hmmm... let's say... people with curious predilictions for unsupported and inconsistent reasoning. I, on the other hand, have no problem calling the handle on the door to the loony-bin a crank.

People can be misinformed, curious, seeking enlightenment... and there is a way to go about it. Analyze your posts to see why you are not one of those.

They would refer to some time when a disaster was averted in their life and claim it as proof of god. At best what you may be referring to is that a theologian in modern society, in the face of the unending stream of scientific dogma socially accepted by the world, might externally differentiate their idea of proof of god from what a scientist might claim is proof. However even then in their own mind they would consider it as actual proof.

Further more you have demonstrated a complete lack of ability to even know what "proof" is. The mod has shown no more ability to objectively consider different ideas or approaches then anyone else. As a mod of the Physyics and math community, he accurately represents his community. Refusal to admit when you made a mistake is an obvious sign of lack of objective reasoning ability.

Other people refrain from childishly insulting other people because they know that all aggresive acts are borne of insecurity. If you actually believed that I was a "crank" as opposed to realizing your own deeply ingrained inability to reason with the ideas that I present, then you hardly feel the need to act in such a manner.

It has been mentioned many times that you don't need infinite sequences to construct an irrational number.

Yes and has been done so ERRONEOUSLY.

kriminal99:

Do you agree that sqrt(2) is irrational? If not, consider this proof:

First assume that sqrt(2) is rational, and can be written as p/q, where p and q are integers with no common factors (apart from 1).

Then p^2 = 2 q^2, which means that p^2 is an even number. Since the square of an even number is even, and the square of an odd number is odd, p must be even.

Since p is even, we can write p = 2r, where r is an integer. Then, it follows that:

(2r)^2 = 4 r^2 = 2 q^2

or q^2 = 2 r^2

Therefore, q^2, and therefore q, is even.

But if p and q are both even, then they have a common factor of 2. This is in contradiction to the initial assumption that they had no common factors. The only thing which can be wrong in this argument is that sqrt(2) can be written as the fraction p/q (reduced to "lowest terms").

Since sqrt(2) cannot be written as a fraction, it must be irrational.

Do you agree? Have we therefore shown that at least one irrational number exists?

Note that the proof that sqrt(2) is irrational does not involve an infinite summation of any kind. sqrt(2) is not "constructed" that way.

ALL irrational numbers are "trivially different" from SOME rational number. In other words, it is always possible to find a rational number as close as you want to specify to a given irrational number. But that doesn't change the fact that some numbers are irrational.

There's not much point going on with this, is there? There are two possibilities:
1. I wasn't clear.
2. I was clear, and it's your misinterpretation.
Either way, I've now made it perfectly clear what I wanted to say, so there should be no more confusion. Right?

I have seen many proofs that irrational numbers cannot be written as fractions.

I have seen no proofs that not being able to be written as a fraction means that an irrational number is not trivially different from some rational value. Simply saying "whatever number squared gives us 2" does not require an infinite effort. It doesn't give any existence value to it either.

Trying to come up with a well defined number does require an infinite value. Consider the method of finding square roots by hand for example, where you take the square roots of perfect squares close to the number in question and then get closer and closer to the square root by getting closer and closer to the square. This is one example of an infinite effort algorithm required to define a value on sqrt(2).

Mathematicians do not believe that all irrationals are trivially different from some rational value. They do believe that values which are trivially different are equal. Cantor's diagonol argument is interpretted to mean that "the reals can not be put into one to one correspondence with the natural numbers" This means that the infinite number of irrationals between 0 and 1 is somehow greater than the infinite number of rationals and it is this type of reasoning that prevents them from accepting that irrationals are trivially different from rationals. Notice how other people in this thread were speaking of "ordered infinities" and other such nonsense.

Basically this gets at the heart of my point anyways so this is a good place to argue at.

The very fact that we can put a decimal value on an irrational value means that it is trivially different from a rational value. We can see that we can find an infinite number of decimal places of any irrational number. There is nothing stopping us from finding an infinite number of decimal places of pi or sqrt(2). If infinity were to somehow run its course, one should rest assured that every point on the number line would be labeled as a rational value. "Irrational" is a naive name for an algorithm for making an infinite summation of rationals.

One might try to use a definition of irrational such as "not motivated by division of one integer by another". This would differentiate .33... from other infinite summations of rationals. However then if you were to make a summation formula to motivate the same value it would no longer be rational.

If the value is rational now, as in every time we use it, and it would be rational in the end, then what is worth the confusion and problems caused by calling it irrational? Its just mathematical history that causes us to do this.

The point of going on about your earlier statement is 2 fold.

1) It shows you have a lot of trouble admitting when you make a mistake, communicating or otherwise. This says alot about your character, and since everyone here cannot seem to refrain from making propaganda statements this is relevant.

2) I said MAYBE you meant that to begin with, only as a gesture of good faith. What is far more likely is that you believed that all infinite summations of rationals are irrational at the time you said it. As anyone who reads through all the posts in the thread would see, the various people and even mathematicians arguing here all have incompatible views regarding the concepts of infinity, rationals, irrationals, etc. They are all still however trying to maintain the illusion that there is some consensus and no debate to be had, because while careful nonmathematical analysis is beyond their ability they still want to maintain the appearance of the unrealistic degree of intelligence usually falsely associated with mathematicians.

Let me repeat the obvious for the asshumpteenth time.

The definition of a rational number: Any number which can be written in the form a/b, where both a and b are integers, and b is not zero.

Now this, you learned in junior high school at the very latest.

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-

K99, you could just avoid this hassle if you'd stop trying to yell at everyone and just do the following.

1) Write pi in abovementioned a/b form, and find a and b for us.

2) Show that it works out to be pi.

3) Gloat.

This is the only way for you to prove that pi is rational.

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-

However, you've obviously never succeeded, since you're still sitting here yelling. You're claiming you're right, when we know you're not with the official definition of "rational number". Thus, the only thing to conclude is that you have a different definition of "rational number" in mind.

You seem to include "any sum of any number of rational numbers" in the definition of "rational number", and you're trying to stretch that to say that an infinite summation of rational numbers is rational.

In that case, you're only correct under your own definition. You are still incorrect under the official definition, which you aren't using.

Do you get it now? You're still wrong if you use the real, actual, official definition of "rational number", but not if you use some personally invented definition of "rational number", which no one cares about.

You can stop trying to persuade us here, because no one cares about your weird personal concept of 'rational'. If you want people to listen, use the proper definitions. I have a feeling I could be wasting my breath on someone who won't bother to listen, but I'm saying it anyway.

Otherwise you won't get far.

To begin with Im not yelling at anyone, just pointing out mistakes in reasoning. To place integer values on a and b would take an infinite amount of time, as the integers would be infinitely large.

The discussion is regarding what the definition of rational SHOULD BE and what irrationals SHOULD BE CALLED. If a person was to arbitrarilly name a rational number a bumbleshoot, then it would make no sense for that person to ask the world to use their name.

If on the other hand someone were to point out that defining irrationals and rationals as they are ultimately results in self contradiction then of course they are going to use a "different definition". Math is all about DEFINITIONS. When you define natural numbers you are simultaneously defining everything that will ever be "prooven" to be a direct consequence in mathematics. Its not a non argument to debate over the definition of something. Thats what mathematics is all about... The ideas that math is founded on, like infinity, must be argued in language not in formal mathematical notation. This is the problem that the people here are having, because reasoning in logic and language is much harder than reasoning with numbers.

I am not "defining" rational numbers as infinite sums of rational numbers. Defining rational numbers as they are simultaneously defines sums of rationals as rational. If people were capable of making of few more relations at once, then saying infinite summations of rationals are not rational would sound as stupid as saying " a skinny fat person" is now. Its just a blatantly obvious logical consequence that noone has made an argument for mathematically.

Look at it this way. The proof that an irrational number cannot be written as a fraction is evidence of something. Consequently calling these numbers irrational is misinterpretation of this evidence. What this evidence means is that irrational number isn't finished being defined because it takes an infinite effort to define it. Not that if it were somehow defined that it wouldn't be able to be written as a fraction.

The logical evidence that pi is rational, or should be considered rational, is simple. As I said in last post, we can calculate decimal places of pi indefinitely. decimals are equivalent to rational values. There is nothing to pi over and above the rational values we attribute to it. The only reason to call them irrational is history.

By the way, when you metaphorically stomp your feet and pound your fist on tables like a monkey in a debate, when you don't even understand the other person's point, you just end up looking silly....

 

[Silas]

I have seen many proofs that irrational numbers cannot be written as fractions.

Trying to come up with a well defined number does require an infinite value. Consider the method of finding square roots by hand for example, where you take the square roots of perfect squares close to the number in question and then get closer and closer to the square root by getting closer and closer to the square. This is one example of an infinite effort algorithm required to define a value on sqrt(2).

You don't need "proofs" that irrational numbers cannot be written as fractions. They cannot, by fiat. What you have seen are proofs that there exist numbers that cannot be written as fractions. These numbers are called irrational numbers.

I have seen no proofs that not being able to be written as a fraction means that an irrational number is not trivially different from some rational value.

I quote the rational number 3.141592653589793. This is (for engineering purposes) trivially different from π. This page displays the first 10,000 digits of π. As displayed that is a rational number (multiply it by 10^10,000 to get a whole number), which is considerably more trivially different from π. Nontheless, π is a proven irrational number. But your definition of trivial means "differs by an infinitesimal". Any number that differs by an infinitesimal from π cannot be rational itself. It could only be rational if it became cyclic, at which point the numbers that repeated would no longer be trivially different (in your definition) from π.

Since the irrationality of π has been proved, the statement "an irrational number is not trivially different from some rational value" stands disproved, by mathematical logic, no contradiction.

Simply saying "whatever number squared gives us 2" does not require an infinite effort. It doesn't give any existence value to it either.

What "existence value" does 1 have? What "existence value" does -1 have? And √-1? And a circle? And a hypotenuse? They all have notional mathematical meanings but not one of them exists in reality. However, √2 and π and i are nontheless real and very useful numbers.


Let us return to your original post:

If something is the limit of a sequence that means the difference between the result of that sequence and the limit is infintessimally small (or else something else would be the limit), anotherwords they are equal.

(Proof, .33 repeating equals 1/3, 1/3 * 3 = 1, .33 repeating * 3 equals .99, therefore .99 repeating = 1)

The sequence for calculating π is the limit of an infinite sequence of adding fractions, therefore the result is rational. Therefore π is rational, or math is self contradicting.

.99 repeating = 1, but both .99 repeating and 1 are rational numbers. You've used the wrong example, really. This is an infinite sum of the same term again and again, only multiplied by successive negative powers of ten. In a different number base, the successive powers of that base become zero (as 0.5 really equals 0.50000000....). I don't believe that is true of any sum series which results in π.

Aside: Amending the definition of rationals to exclude infinite summations of rationals makes the new definition necessarily self contradicting because the full definition of rationals before this necessitates that any summation of rationals result in a rational.

It's actually a logical flaw in your final statement. A rational number is a number which can be represented as a fraction. I'm afraid I cannot make the leap which in your view compels the statement that all rational sums make rationals when you are able to include an infinite number of terms. A number that is the sum of an infinite number of (<1 for the sake of argument) terms is mathematically - qualitatively - logically - different from a normal sum like 1/2 + 1/3 + 1/5. This was in point of fact the great discovery of calculus, if you like. At least, that's how I see it.

&pi; is not just irrational, it is also transcendental, which means it cannot be found as the root of an equation (in the way that &radic;2 satisfies x² - 2 = 0). Therefore the only way to approach &pi; is via an infinite series.

 

[James R]

kriminal99:

I have seen many proofs that irrational numbers cannot be written as fractions.

I have seen no proofs that not being able to be written as a fraction means that an irrational number is not trivially different from some rational value.

What do you mean by "trivially different"? Trivially different is still different, isn't it? So, it seems we have a meaningful distinction between rational and irrational numbers after all.

Mathematicians do not believe that all irrationals are trivially different from some rational value. They do believe that values which are trivially different are equal.

You'll need to define "trivially different" for me at this point. Then I might be able to tell you whether mathematicians in general agree with you or not.

The very fact that we can put a decimal value on an irrational value means that it is trivially different from a rational value. We can see that we can find an infinite number of decimal places of any irrational number.

No, we can't. That would require infinite effort. You don't seem to have a very good grasp of the concept of infinity. Infinity never ends.

The point of going on about your earlier statement is 2 fold.

1) It shows you have a lot of trouble admitting when you make a mistake, communicating or otherwise. This says alot about your character, and since everyone here cannot seem to refrain from making propaganda statements this is relevant.

It says a lot about your character, I think, that you put yourself in the position of psychoanalyst. Well, two can play that game.

2) I said MAYBE you meant that to begin with, only as a gesture of good faith. What is far more likely is that you believed that all infinite summations of rationals are irrational at the time you said it.

Interesting. How did you reach that conclusion, exactly? You obviously have amazing insight and knowledge about me and what I know.

As anyone who reads through all the posts in the thread would see, the various people and even mathematicians arguing here all have incompatible views regarding the concepts of infinity, rationals, irrationals, etc.

All of them? Really. Once again, brilliant insight.

The discussion is regarding what the definition of rational SHOULD BE and what irrationals SHOULD BE CALLED.

So, lay it out for all us dumb people, please. What definitions SHOULD we use? So far, you've told us a lot about how NOT to define these things. What is your alternative?

The logical evidence that pi is rational, or should be considered rational, is simple. As I said in last post, we can calculate decimal places of pi indefinitely. decimals are equivalent to rational values. There is nothing to pi over and above the rational values we attribute to it. The only reason to call them irrational is history.

I think you missed the point about infinite summations. Try again.

By the way, when you metaphorically stomp your feet and pound your fist on tables like a monkey in a debate, when you don't even understand the other person's point, you just end up looking silly....

Indeed.

 

[Pete]

kriminal99,
What does 'rational number' mean?

I've always attached the concept of 'the quotient of two integers' to that term, but it seems that it actually means something different. What?

When someone uses the words "rational number", what concept do you think of?

 

[kriminal99]

kriminal99,
What does 'rational number' mean?

I've always attached the concept of 'the quotient of two integers' to that term, but it seems that it actually means something different. What?

When someone uses the words "rational number", what concept do you think of?

How about the "quotient of two integers" oO? Im not sure what your asking here? You mean why am I giving them some kind of priority? Our number system only allows us to write rational numbers in terms of actual numerical digits. Irrationals aren't really numbers with set values, but rather they are infinitely long operations on rational values on which a set value cannot be pinned on because they will never finish being defined.

You might say that I am equating "number" with "rational" by means of the argument that I am making. However when someone makes an argument for something you look at the actual argument, you don't look at the result ignore the argument and then say "mathematician's don't equate those 2"

This could have been done to counter any formal mathematical proof done by any mathematician by a bitter ignorant person who didn't want to accept what was being proposed...

kriminal99:

What do you mean by "trivially different"? Trivially different is still different, isn't it? So, it seems we have a meaningful distinction between rational and irrational numbers after all.

You'll need to define "trivially different" for me at this point. Then I might be able to tell you whether mathematicians in general agree with you or not.

No, we can't. That would require infinite effort. You don't seem to have a very good grasp of the concept of infinity. Infinity never ends.

It says a lot about your character, I think, that you put yourself in the position of psychoanalyst. Well, two can play that game.

Interesting. How did you reach that conclusion, exactly? You obviously have amazing insight and knowledge about me and what I know.

All of them? Really. Once again, brilliant insight.

So, lay it out for all us dumb people, please. What definitions SHOULD we use? So far, you've told us a lot about how NOT to define these things. What is your alternative?

I think you missed the point about infinite summations. Try again.

Indeed.

Trivially means not meaningfully different. As in .999999... = 1.

Why would I need you to tell me what mathematicians in general think? For that matter there is no collective hivemind of all mathematicians that all agree unanomously on certain things. There might be a majority that accept certain ideas but usually such majorities are born of convienience, ignorance, and unwillingness to question whats already believed by others.

Of course infinity doesn't end. My point is simple. Suppose it will only ever be signifigant to talk about pi to 30 decimal places. Suppose defining it numerically past a certain point would equate to coming up with results dealing with molecules rather than the macroscopic appearance of a circle, even if you started out with a large planet, star or etc. The lengths that you were dealing with don't even make sense any more at that level. Then in such a case Pi is EQUAL to some rational value for all practical purposes. Further decimal places would be meaningless.

Psychoanalyst? What I was saying is that you made a mistake and was refusing to admit and dishonestly trying to make it look like I just misinterpreted what you said. Anyone who sees this, while perhaps not considering the reasons for doing so consiously, would have their opinion of you go downhill drastically. Call this whatever you want.

I said that you probably thought it was true at the time you said it because if you simply meant to type something different then you would have no reason to hide it and act dishonestly, rather you would just say "oops typo". People give plenty of insight into what goes through their head if you take the time to deduce it from what is said. Another possibility is that you place a large degree of value on communication of ideas rather than the ideas themselves, such that mis-stating ideas is something you pride yourself on not doing. Typical behavior by such people is to criticize others for spelling, grammar, or in the case of math/physics, perhaps something like jerryrigging symbols when theres no math notation. It's not a very advanced belief set seeing as its incompatible with, yet dependent on, concieving new ideas. If you want to consider new ideas you HAVE to approach things differently, invent new terms, and consdier that things you believed before were wrong.

If your a great at communicating to other mathematicians and scientists (not even good at just communicating in general) but you don't have anything to communicate then your pretty much useless. Obviously these types of people don't typically accomplish very much. I thought it unlikely that you were this type of person because how would such a person end up moderating a discussion board where people were talking about complex ideas. In which case the only explanation for your behavior would have been that you forgot .333 repeating was a rational infinite summation of rationals. But looking back at the types of statements you have been making, I may have been wrong.

Yes all of them. As in one person at first said infinite summations can't be rational, then another defines infinite a different way then others and various other small details. The point is when you study mathematics they don't teach you anything about the foundations of mathematics or assumptions that are being made. Many mathematicians would have difficulty understanding the arguments of the people with alternative views on the foundations of mathematics, and perhaps even the views of the people the foundations of mathematics are based on.

I didn't miss any point because I read people's responses carefully, and my statement regarding immature behavior in debates was directed towards people who constantly feel the need to insult everyone who do not agree with them.

 

[AndersHermansson]

Our number system only allows us to write rational numbers in terms of actual numerical digits.

Not true. A rational number can be written p/q where p and q are integers.

I wonder how you can claim mathematics is inconsistent, and keep a straight face, when you appearantly have a rather incomplete knowledge of it.

 

[kriminal99]

Not true. A rational number can be written p/q where p and q are integers.

I wonder how you can claim mathematics is inconsistent, and keep a straight face, when you appearantly have a rather incomplete knowledge of it.

Although I admit my statement was ambiguous, I can't possibly imagine how anyone keeping track of this thread could think thats what I was saying.

What I was saying was that ONLY RATIONALS can be written in terms of actual digits. Anotherwords, you can't define an irrational value in terms of numerical digits.