Is Pi Infinitely Long?

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Jimadybobalon
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Is Pi Infinitely Long?

Postby Jimadybobalon » Wed Oct 28, 2009 7:28 pm UTC

I've been arguing with my friend in advanced math class about whether the decimal of Pi goes on forever or not, I say it does, he says it doesn't and ends /somewhere/. Obviously it does go on forever and he's just an idiot :P but his very mundane argument is,

Him - "If it goes on forever, prove it."
Me - "I can't show you that it goes on forever because it goes on forever!"
Him - "Well then it doesn't go on forever"
Me - "Okay, I may not be able to show it goes on forever but you /should/ be able to show that it eventually stops, otherwise it goes on forever"
Him - "Noooo, it might go on for a very very long time - so much so that I can't measure it - but it still comes to an end at some point"

As I have no definitive was to prove this I was wondering if anyone here knew of some way to prove that Pi is indeed Infinite? Any explanations no matter how complicated will be appreciated :P

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Re: Is Pi Infinitely Long?

Postby intimidat0r » Wed Oct 28, 2009 7:31 pm UTC

If a decimal number d terminates, it is clearly rational, because there exists a natural number n high enough such that d*10^n is an integer, and d=d*10^n / 10^n. However...

http://en.wikipedia.org/wiki/Proof_that ... irrational

The proof is kind of advanced and involves calculus, but if you have any familiarity with calculus you shouldn't have an issue working through it.
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Re: Is Pi Infinitely Long?

Postby Macbi » Wed Oct 28, 2009 7:44 pm UTC

Or if you can convince him that:
[math]\pi =\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{7}-\frac{4}{9} \cdots\![/math]
Then pi can't be a fraction since the denominator would be infinitely large.
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Re: Is Pi Infinitely Long?

Postby mike-l » Wed Oct 28, 2009 8:00 pm UTC

Macbi wrote:Or if you can convince him that:
[math]\pi =\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{7}-\frac{4}{9} \cdots\![/math]
Then pi can't be a fraction since the denominator would be infinitely large.


Not sure I buy that...

[math]\sum _{n=1}^{\infty}\frac{1}{2^n}[/math]

is rational.
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Re: Is Pi Infinitely Long?

Postby Klotz » Wed Oct 28, 2009 9:05 pm UTC

If pi was a fraction then you'd get a contradiction.

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Re: Is Pi Infinitely Long?

Postby Jimadybobalon » Wed Oct 28, 2009 9:51 pm UTC

Klotz wrote:If pi was a fraction then you'd get a contradiction.


how so?

Macbi wrote:Or if you can convince him that:
[math]\pi =\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{7}-\frac{4}{9} \cdots\![/math]
Then pi can't be a fraction since the denominator would be infinitely large.


did you mean to have two sevens there? :P otherwise I like that way of calculating Pi :P

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Re: Is Pi Infinitely Long?

Postby voidPtr » Wed Oct 28, 2009 10:27 pm UTC

Jimadybobalon wrote:Him - "If it goes on forever, prove it."
Me - "I can't show you that it goes on forever because it goes on forever!"
...



You can prove that Pi's decimal representation goes on forever (see the wiki proof someone else already mentioned), but since you're having this bet and brought up that statement, I think that proof will be too difficult for you to understand.

The way to prove Pi goes on forever is by trying to prove the opposite -- Pi doesn't go on forever -- and showing the consequences of this would lead to a mathematically absurd result, and therefore it's false. This type of proof is called Proof By Contradiction which is simular to playing Devil's Advocate but mathematically rigorous.

If you're up for it, here's a simpler exercise for you to try using the same method of proof. Try to prove there is no biggest number.

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Re: Is Pi Infinitely Long?

Postby gmalivuk » Thu Oct 29, 2009 1:17 am UTC

voidPtr wrote:The way to prove Pi goes on forever is by trying to prove the opposite -- Pi doesn't go on forever -- and showing the consequences of this would lead to a mathematically absurd result

You don't try to prove the opposite. You start off by assuming the opposite, and then try to show why that leads to contradiction.
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Re: Is Pi Infinitely Long?

Postby phlip » Thu Oct 29, 2009 1:42 am UTC

Is your friend's objection to do with pi in particular, or just infinite decimals in general?

Because if it's the latter, you might have better luck with a simpler number, like 1/3... you can show that 0.33333 < 1/3 < 0.33334, for any finite number of digits (because if you multiply them by three you get 0.99999 < 1 < 1.00002). So any finite number of digits is not enough.

Proving that a countably-infinite number of digits is enough, and that 0.333... = 1/3 exactly, could be trickier... intuition sometimes gets in the way (see also: every thread about 0.999... = 1). But even without that, you've shown that "it has to end eventually" isn't the case.

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Re: Is Pi Infinitely Long?

Postby polymer » Thu Oct 29, 2009 1:47 am UTC

Do you know calculus? All the proofs I'm familiar with involve some degree of calculus :/

Also if you don't care about being too mathematically rigorous, the point mentioned in the previous post can be demonstrated briefly with algebra

[imath]x = 0.333333333.....[/imath]
[imath]10x = 3.33333333.....[/imath]

[imath]-9x=-3, x=1/3[/imath]

Again though, I'm not familiar with a rigorous argument that doesn't involve some calculus.
Last edited by polymer on Thu Oct 29, 2009 2:44 am UTC, edited 3 times in total.

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Re: Is Pi Infinitely Long?

Postby BlackSails » Thu Oct 29, 2009 1:56 am UTC

Why is that argument not rigorous?

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Re: Is Pi Infinitely Long?

Postby polymer » Thu Oct 29, 2009 2:00 am UTC

[imath]x = 10 + 100 + 1000 + 10000+ ...[/imath]
[imath]10x = 100 + 1000+ 10000 + 100000+...[/imath]

[imath]-9x = 10, x=-10/9[/imath]

Which is obviously absurd. The equation has to be determined as convergent if you're going to make the argument.
Last edited by polymer on Thu Oct 29, 2009 2:43 am UTC, edited 2 times in total.

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Re: Is Pi Infinitely Long?

Postby antonfire » Thu Oct 29, 2009 2:09 am UTC

It's not that absurd.
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Re: Is Pi Infinitely Long?

Postby phlip » Thu Oct 29, 2009 2:14 am UTC

BlackSails wrote:Why is that argument not rigorous?

Well, it handwaves over the limits that are inherently involved whenever you're working with an infinite string of digits. You can handwave it away, but the problem with intuitive arguments is that if someone's intuition is incorrect then an intuitive argument often won't be convincing. For instance, intuition wants to say that there're more threes after the decimal point in x = 0.333... than 10x = 3.333... so when you subtract them you should get 2.999...997 or something equally actually-meaningless. Also, as polymer points out, it implicitly assumes that the value exists, as part of calculating that value.

Now, there are ways to back up the "10x - x = 3" step from limits... lemmas to prove something like [imath]10 \times \sum_{i=1}^{\infty} 10^{-i} = 1 + \sum_{i=1}^{\infty} 10^{-i}[/imath] can be proven pretty easily, with the tools for infinite series... but might be a bit beyond the target audience here.

But really, it's because it uses an intuitive and non-rigorous definition of what "0.333..." actually means... because infinite sequences and series are kinda complicated, it's pretty much handwavily defined by giving a rigorous definition of finite decimals, and then saying "that, but infinite somehow". And trying to take patterns learned in finite contexts and apply them intuitively in infinite ones tends to end in pain.

... But all of this is besides the point, as we're trying to show that 1/3 (as an analogous example to pi) cannot have a finite representation, not that this particular infinite representation is exact (even though that will probably be his next question).
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Re: Is Pi Infinitely Long?

Postby achan1058 » Thu Oct 29, 2009 2:41 am UTC

polymer wrote:[imath]x = 10 + 100 + 1000 + 10000+ ...[/imath]
[imath]-10x = 100 + 1000+ 10000 + 100000+...[/imath]

[imath]-9x = 10, x=-10/9[/imath]

Which is obviously absurd. The equation has to be determined as convergent if you're going to make the argument.
I think the fact that it is an increasing sequence with an upper bound makes it convergent, which is obviously what your example lack. No calculus necessary, you just have to take 1 fact from real analysis for granted. (or do you take real analysis under calculus as well?)

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Re: Is Pi Infinitely Long?

Postby BlackSails » Thu Oct 29, 2009 2:46 am UTC

antonfire wrote:It's not that absurd.


Not absurd in the same way that 1+2+3......=-1/12?

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Re: Is Pi Infinitely Long?

Postby polymer » Thu Oct 29, 2009 2:54 am UTC

phlip wrote:Now, there are ways to back up the "10x - x = 3" step from limits... lemmas to prove something like [imath]10 \times \sum_{i=1}^{\infty} 3 \times 10^{-i} = 1 + \sum_{i=1}^{\infty} 3 \times 10^{-i}[/imath]

you can proves lemmas that suggest that 3.33333.... = 1 + 0.3333333? Or did I misunderstand?

achan1058 wrote:I think the fact that it is an increasing sequence with an upper bound makes it convergent, which is obviously what your example lack. No calculus necessary, you just have to take 1 fact from real analysis for granted. (or do you take real analysis under calculus as well?)

Fair point, I guess this doesn't directly answer the question in question.

Also I didn't get my first real taste of analysis until preparation for polynomial series, so I figured the op wouldn't have been introduced to it either unless they had taken calculus.

antonfire wrote:It's not that absurd.

These obscure math things are making it very difficult for me to pretend to know what I'm talking about XD. Excuse me while I lurk another year before I embarrass myself further :P.
Last edited by polymer on Thu Oct 29, 2009 3:08 am UTC, edited 1 time in total.

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Re: Is Pi Infinitely Long?

Postby phlip » Thu Oct 29, 2009 3:02 am UTC

polymer wrote:you can proves lemmas that suggest that 3.33333.... = 1 + 0.3333333? Or did I misunderstand?
My bad, I didn't mean to have those 3s in there. I've fixed it in the post.

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Re: Is Pi Infinitely Long?

Postby Tirian » Thu Oct 29, 2009 3:04 am UTC

I'm not sure what this argument has to do with pi. Being the sum of an infinite series is neither necessary nor sufficient for being rational or irrational.

Unfortunately, the proof that pi is irrational is beyond the OP's current level. This is the most elementary proof that I know of, but you'd need a semester of calculus and a solid appreciation of the Fundamental Theorem of Calculus. Lambert's original proof (and Lindemann's later proof that pi can't even be the root of a rational polynomial) is quite a bit more unpleasant.

Perhaps your friend could be goaded into claiming that the square root of 2 also doesn't go on forever, because you'd have no trouble shooting that down.

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Re: Is Pi Infinitely Long?

Postby t0rajir0u » Thu Oct 29, 2009 3:28 am UTC

antonfire wrote:It's not that absurd.

What anton means is that there are senses in which 10 + 100 + ... does converge. For example, changing to either the 5-adic or the 2-adic metric will make the "absurd" statement a legitimate equality.

Another sense in which 10 + 100 + 1000 + ... = -10/9 is a perfectly legitimate statement is that the function [imath]f(x) = \sum_{n \ge 1} x^n[/imath] has an analytic continuation to the entire complex plane minus [imath]1[/imath].

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Re: Is Pi Infinitely Long?

Postby phlip » Thu Oct 29, 2009 3:46 am UTC

Tirian wrote:I'm not sure what this argument has to do with pi. Being the sum of an infinite series is neither necessary nor sufficient for being rational or irrational.

Well, the point isn't rational vs irrational, but terminating vs non-terminating decimal expansions... ie decadic-rational vs others. The OP's friend either thinks that pi is one of the former, or that the latter simply doesn't exist. If they think that non-terminating decimal expansions simply don't exist, then 1/3 is a much easier counterexample to prove than pi is.

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Re: Is Pi Infinitely Long?

Postby Tirian » Thu Oct 29, 2009 5:45 am UTC

I'm not convinced that irrational is harder to prove than non-terminating. It is obviously sufficient, and you can prove that [imath]\sqrt 2[/imath] is irrational without getting folks talking about p-adics in a high school math thread. That's got to be worth something.

And I also think it's not impossible that this guy is thinking specifically about [imath]\pi[/imath]. I certainly hope that someone doesn't get into an advanced math class while needing to be convinced that 1/3 never terminates. But I can imagine how someone might get it in his head that with a scanning electron microscope and enough time you could find out exactly how many atoms were in the diameter and circumference of a perfect circular crystal the size of a galaxy. I know, I know, but 250 years ago he would have had cause to be skeptical, and my understanding is that people weren't quick to approve of Lambert's proof.

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Re: Is Pi Infinitely Long?

Postby t0rajir0u » Thu Oct 29, 2009 2:34 pm UTC

One thing you might try explaining to your friend is that circles don't exist. Circles are a mathematical abstraction and pi is defined in terms of them. Pi just happens to be useful.

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Re: Is Pi Infinitely Long?

Postby Token » Thu Oct 29, 2009 5:57 pm UTC

t0rajir0u wrote:One thing you might try explaining to your friend is that circles don't exist.

I reject your assertion as meaningless in the absence of a decent universal ontology.
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Re: Is Pi Infinitely Long?

Postby Random832 » Thu Oct 29, 2009 6:27 pm UTC

polymer wrote:[imath]x = 10 + 100 + 1000 + 10000+ ...[/imath]
[imath]10x = 100 + 1000+ 10000 + 100000+...[/imath]

[imath]-9x = 10, x=-10/9[/imath]

Which is obviously absurd.


O RLY?

...9999999 = -1 (explained at that article)
...1111111 = -1/9
...11111110 = -10/9

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Re: Is Pi Infinitely Long?

Postby Jimadybobalon » Mon Nov 02, 2009 10:46 pm UTC

Okay just to specify: My friend does know that there are infinite numbers (eg: 1/3) and this is just a thing about Pi. More of his argument went:

Him: "Okay lets just say you have a circle with a radius of 0.5, then the circumference of that circle is Pi, as [math]2 * Pi * r = 2 * Pi * 0.5 = Pi[/math], right? So then if the circumference of that circle is Pi then if you lay that out as a straight line then you have a straight line of Pi. So technically that straight line has to finish somewhere, as it is just a straight line with a start and a finish"
Me: "Yes the straight line will obviously end but since the it is the decimal that goes on forever then it eventually wouldn't make a difference as you can't measure that precise, it wouldn't continuously get bigger, it would just be too precise to measure"
Him: "No, but it has to end somewhere!"

He really is quite stubborn :P And I've used the argument:

Me: "lets just say you have [math]y = 1/x[/math], now what is Y if x is 0.1? Yes, 10. What about 0.01? Yeees, 100. Now you agree that that X can get infinitely smaller and therefore Y will get infinitely bigger?"
Eventually he grudgingly agreed, as he knew where I was going with this.
Me: "Okay then if Y can be an infinite number as X has infinite amount of decimals then why can't Pi's decimal go on forever?"
At this point he usually just refers back to his straight line thing and won't accept anything else. Anyone got an irrefutable proof which is also simple enough for my idiot friend to understand? :P

t0rajir0u wrote:One thing you might try explaining to your friend is that circles don't exist. Circles are a mathematical abstraction and pi is defined in terms of them. Pi just happens to be useful.

Do go on about that :P I'm intrigued and have never heard that before :P

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Re: Is Pi Infinitely Long?

Postby Cosmologicon » Mon Nov 02, 2009 11:05 pm UTC

So let me get this straight. He thinks that 1/3 = 0.333... goes on forever. He thinks that a number that describes the length of a line segment cannot go on forever. Does he not believe in segments of length 1/3 or what?

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Re: Is Pi Infinitely Long?

Postby Certhas » Tue Nov 03, 2009 2:23 am UTC

t0rajir0u wrote:Another sense in which 10 + 100 + 1000 + ... = -10/9 is a perfectly legitimate statement is that the function [imath]f(x) = \sum_{n \ge 1} x^n[/imath] has an analytic continuation to the entire complex plane minus [imath]1[/imath].


Which, BTW, is maths that is heavily used in physics to describe real systems measured in real accelerators. My favourite way to put this is 1+2+4+8+16+...=-1. I love that fact, way more intuitive than -1/12, too.
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Re: Is Pi Infinitely Long?

Postby Certhas » Tue Nov 03, 2009 2:30 am UTC

Jimadybobalon wrote:Him: "Okay lets just say you have a circle with a radius of 0.5, then the circumference of that circle is Pi, as [math]2 * Pi * r = 2 * Pi * 0.5 = Pi[/math], right? So then if the circumference of that circle is Pi then if you lay that out as a straight line then you have a straight line of Pi. So technically that straight line has to finish somewhere, as it is just a straight line with a start and a finish"
Me: "Yes the straight line will obviously end but since the it is the decimal that goes on forever then it eventually wouldn't make a difference as you can't measure that precise, it wouldn't continuously get bigger, it would just be too precise to measure"
Him: "No, but it has to end somewhere!"


That sounds more like http://en.wikipedia.org/wiki/Zeno%27s_p ... e_tortoise

Ask him whether he thinks that if I add half a liter to a liter bottle, and then 1/4th of a litre, and then 1/8th and then 1/16th, and so on, and I kept going forever, whether the bottle would overflow.

The line is the same, you keep adding infinitely many bits to it with each additional digit, but they all end up to something finite that ends at a finite distance after infinitely many steps. In fact no matter how many digits of pi you add you will never get past 4. This already constitutes a proof of the convergence of adding up all those decimal places (technically it's bounded and monotonous)
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Re: Is Pi Infinitely Long?

Postby phlip » Tue Nov 03, 2009 2:33 am UTC

Certhas wrote:monotonous

That too, but I think the word you're after is "monotonic".

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Re: Is Pi Infinitely Long?

Postby kevmus » Tue Nov 03, 2009 4:01 am UTC

If he's into line segments, show him a right isoceles triangle with two sides measuring one.

The other side must measure sqrt(2) by pythagoras

It's a line so it's gotta end somewhere, right?

Then show him the proof that sqrt(2) is irrational (which should be at his level)
Spoiler:
1- Assume sqrt(2) is rational
2- There exist x and y (both integers) such that x/y = sqrt(2), where this fraction is completely reduced.
3- square both sides, (x/y)^2=2
4- so x^2 = 2 y^2
5- so x^2 is even.
6- since x^2 is even, x must be even (odd * odd = odd, even*even=even)
7- so there is an integer, k, so that x=2k
8- plug that into equation 4, you get (2k)^2 = 2 y^2
9- that means that 4k^2=2y^2
10- so 2k^2 = y^2
11- so y^2 is even
12- which means that y is even
13- but wait, y and x are both even, but we said the fraction was completely reduced. Clearly it isn't because we can divide both sides by 2, so there must be something wrong with one of our assumptions
14- but our only assumption was that sqrt(2) is rational, so sqrt(2) is irrational.

QED


That isn't enough to prove that pi is irrational, but it will at least make him think twice about his "It's a line segment, it's gotta end somewhere!"


-
Another way to see it is that pi is an infinite sum.
pi = 3 + .1 + .04 + ....

or pi = 3*10^0 + 1*10^-1 + 4*10^-2 ...

so pi must be less than 9* 10 ^0 + 9*10^-1 + 9*10^-2
so pi must be less than 9( 10^0 + 10^-1 + 10^-2 ...)
but the sum on the right comes out to 10/9, (1+.1+.01+.001 ... means 1.1111... so 10/9)
so pi must be less than 90/9
so pi must be less than 10. This seems obvious, but it is a way to poke another hole in his logic of "But then it gets to infinity, so how can I see the line segment stop!"

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Re: Is Pi Infinitely Long?

Postby t0rajir0u » Tue Nov 03, 2009 2:51 pm UTC

Jimadybobalon wrote:
t0rajir0u wrote:One thing you might try explaining to your friend is that circles don't exist. Circles are a mathematical abstraction and pi is defined in terms of them. Pi just happens to be useful.

Do go on about that :P I'm intrigued and have never heard that before :P

If your friend insists on his physical intuition, tell him that it is impossible to create a perfect circle because at the molecular level, any circle is just a fuzzy collection of things that looks like a circle. If you wanted to pick out a few molecules it would look like a polygon with a huge number of sides (but even this is an idealization). You won't find pi anywhere in real life.

Nevertheless, it is useful to pretend that pi does exist because a lot of behavior in the physical world is well-approximated by computations involving pi.

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Re: Is Pi Infinitely Long?

Postby Token » Tue Nov 03, 2009 3:13 pm UTC

t0rajir0u wrote:If your friend insists on his physical intuition, tell him that it is impossible to create a perfect circle because at the molecular level, any circle is just a fuzzy collection of things that looks like a circle. If you wanted to pick out a few molecules it would look like a polygon with a huge number of sides (but even this is an idealization). You won't find pi anywhere in real life.

So, since the universe contains finitely many particles*, and since perfect circles contain more than finitely many points, pi doesn't exist? If the universe contained uncountably many particles which could assume arbitrary positions, would pi exist then? Or would the particles actually have to form themselves into a perfect circle? What if the universe consisted only of two particles of unit mass and a gravitational force - would pi exist then?

* If you want, turn "the universe contains" into "the visible universe is widely believed to contain only".
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Re: Is Pi Infinitely Long?

Postby cogman » Tue Nov 03, 2009 3:47 pm UTC

I would start by talking about [imath]\sqrt{2}[/imath] first. Since your friends argument is that it doesn't exist physically therefore it can't exist in the universe. Use a different, easier to prove irrational number, and prove that it is irrational.

Though, saying that. No matter what proof you give, your friend is just going to say "Well, math is broken and stupid, and you just proved it". I've met several people like your friend, and that is generally the position they take when they can't counter argue (.999... = 1 for example)

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Re: Is Pi Infinitely Long?

Postby Eebster the Great » Tue Nov 03, 2009 11:51 pm UTC

t0rajir0u wrote:
Jimadybobalon wrote:
t0rajir0u wrote:One thing you might try explaining to your friend is that circles don't exist. Circles are a mathematical abstraction and pi is defined in terms of them. Pi just happens to be useful.

Do go on about that :P I'm intrigued and have never heard that before :P

If your friend insists on his physical intuition, tell him that it is impossible to create a perfect circle because at the molecular level, any circle is just a fuzzy collection of things that looks like a circle. If you wanted to pick out a few molecules it would look like a polygon with a huge number of sides (but even this is an idealization). You won't find pi anywhere in real life.

Nevertheless, it is useful to pretend that pi does exist because a lot of behavior in the physical world is well-approximated by computations involving pi.


While your last statement is impossible to refute, since we will never have perfectly accurate calculations, many physical equations logically should include constants which are exactly equal to pi. For example, the ratio of Planck's constant to the reduced Planck's constant should be exactly 2 pi. In fact, pi is ubiquitous in physics and in math. You seem to assume that pi is only relevant with respect to circles (and perhaps circular trig functions), which is obviously not true.

And as for perfect circles . . . in theory, a body in an inertial reference frame in the absence of other bodies should produce circular gravitational waves. Admittedly, no bodies are isolated from other bodies, but even then, the net effect should be a sum of circular waves. Even bodies in motion create waves that rely on pi as a constant. So the universe DOES produce fields that rely on the exact value of pi. In fact, even if one takes quantum fluctuations into account, the average net effect should still rely on an exact value of pi.

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Re: Is Pi Infinitely Long?

Postby antonfire » Wed Nov 04, 2009 12:17 am UTC

Eebster the Great wrote:For example, the ratio of Planck's constant to the reduced Planck's constant should be exactly 2 pi.
This is true by definition. If I defined the "modified planck's constant" to be 132.09182309 times plack's constant, it would hardly mean that 132.09182309 is somehow special.

Eebster the Great wrote:And as for perfect circles . . . in theory, a body in an inertial reference frame in the absence of other bodies should produce circular gravitational waves. Admittedly, no bodies are isolated from other bodies, but even then, the net effect should be a sum of circular waves. Even bodies in motion create waves that rely on pi as a constant. So the universe DOES produce fields that rely on the exact value of pi. In fact, even if one takes quantum fluctuations into account, the average net effect should still rely on an exact value of pi.
Not, however, if you take general relativity into account. Circles' circumferences aren't proportional to their radii in curved space.

Besides, the key phrase you used is "in theory" which means precisely that what you are talking about is an approximation to the real world. I can come up with an equally good (albeit more complicated) theory which makes predictions that are just as consistent with observation as your theory, but makes no mention of pi and does not require its "presence" or whatever to operate.

Your argument amounts to "well pretend the real world is like this, then there are perfect circles", which is not much more convincing than drawing a circle and saying "well pretend this is a perfect circle".


Of course, the same deal applies to every number. "7" is a made-up thing which makes it easier to think and communicate about certain things. That doesn't mean it's not useful or that it doesn't come up pretty frequently in discussions of the real world.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

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Re: Is Pi Infinitely Long?

Postby Eebster the Great » Wed Nov 04, 2009 6:41 am UTC

antonfire wrote:
Eebster the Great wrote:For example, the ratio of Planck's constant to the reduced Planck's constant should be exactly 2 pi.
This is true by definition. If I defined the "modified planck's constant" to be 132.09182309 times plack's constant, it would hardly mean that 132.09182309 is somehow special.

Eebster the Great wrote:And as for perfect circles . . . in theory, a body in an inertial reference frame in the absence of other bodies should produce circular gravitational waves. Admittedly, no bodies are isolated from other bodies, but even then, the net effect should be a sum of circular waves. Even bodies in motion create waves that rely on pi as a constant. So the universe DOES produce fields that rely on the exact value of pi. In fact, even if one takes quantum fluctuations into account, the average net effect should still rely on an exact value of pi.
Not, however, if you take general relativity into account. Circles' circumferences aren't proportional to their radii in curved space.

Besides, the key phrase you used is "in theory" which means precisely that what you are talking about is an approximation to the real world. I can come up with an equally good (albeit more complicated) theory which makes predictions that are just as consistent with observation as your theory, but makes no mention of pi and does not require its "presence" or whatever to operate.

Your argument amounts to "well pretend the real world is like this, then there are perfect circles", which is not much more convincing than drawing a circle and saying "well pretend this is a perfect circle".


Of course, the same deal applies to every number. "7" is a made-up thing which makes it easier to think and communicate about certain things. That doesn't mean it's not useful or that it doesn't come up pretty frequently in discussions of the real world.


You are missing the point of my generalization. I began discussing theory, then described how theory interacts with the real world. Or at least that was my goal.

According to GR, circles' circumferences are not 2pi r, but they are still proportional to pi, so this does not fundamentally change the question. I guess ultimately this comes down to what "reality" is. If you define reality as the layout of particles, I can show you that you can easily extract the exact value of pi from particles' probability waves. If you consider these waves to be mere mathematical abstractions, I question what is real, since it can be proven these particles do not have a definite location and momentum.

But let's suppose we could measure a particle's location with arbitrary accuracy (this IS possible, although not with current technology). Even though we cannot actually do this today, the results we could obtain are still real, are they not? So let's look at what they would show us. They would show particles at actual locations corresponding to the probability of that position calculated in theory. So we let n approach infinity, delta momentum approach infinity, and low and behold, the constant we are calculating approaches pi (or whatever constant we are calculating). It never "reaches" that exact number in the sense that it is, after all, a limit. But real numbers are just Cauchy sequences anyways, so a limit is going to be necessary somewhere. Again, it depends on what you mean by "reality."

As for integers, I do not see how they are similar. Let me ask you: is it just an approximation to say a certain region of space contains a charge of +7? Because charge is quantized. So as long as we can be reasonably sure it is greater than +6 and less than +8 (actually, for electric charge, we would have to show it is between +20/3 and +22/3), we can be reasonably sure it is exactly +7. I see no approximation necessary.

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Re: Is Pi Infinitely Long?

Postby BlackSails » Wed Nov 04, 2009 6:52 am UTC

Eebster the Great wrote:But let's suppose we could measure a particle's location with arbitrary accuracy (this IS possible, although not with current technology).


Nuh uh.

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Re: Is Pi Infinitely Long?

Postby antonfire » Wed Nov 04, 2009 7:10 am UTC

Eebster the Great wrote:According to GR, circles' circumferences are not 2pi r, but they are still proportional to pi, so this does not fundamentally change the question.
what?

Eebster the Great wrote: If you consider these waves to be mere mathematical abstractions
and I do
Eebster the Great wrote:I question what is real
now you're getting it.

Eebster the Great wrote:But let's suppose we could measure a particle's location with arbitrary accuracy (this IS possible, although not with current technology).
Why do you assume this? Spacetime could very well be quantized for all you know.

Eebster the Great wrote:So we let n approach infinity, delta momentum approach infinity, and low and behold, the constant we are calculating approaches pi (or whatever constant we are calculating).
Lo and behold, your theory stops making accurate predictions because it was developed based on observations at relatively low energies. Or relatively large scales.


Eebster the Great wrote:As for integers, I do not see how they are similar. Let me ask you: is it just an approximation to say a certain region of space contains a charge of +7? Because charge is quantized. So as long as we can be reasonably sure it is greater than +6 and less than +8 (actually, for electric charge, we would have to show it is between +20/3 and +22/3), we can be reasonably sure it is exactly +7. I see no approximation necessary.
What's the net change contained in a small chunk somewhere not too far from a hydrogen nucleus? Even in somewhat unsophisticated models of physics this is not an integer until you "collapse the wavefunction" or whatever. And even if you do that, no measurement tells you with absolute certainty that the charge is greater than +6. There's always the tiny probability that the charge was actually +5 but the needle in your chargeometer twitched at the wrong moment.

Anyway, we've observed charge to be quantized for the most part. We have some pretty good models of the world which happen to require charge to be quantized. Doesn't mean charge is quantized. You're still talking about models. We used to think it was quantized into multiples of 1. Then we made some observations and realized that a better model has them quantized into multiples of 1/3. Who's to say that tomorrow we won't come up with a better model that has them quantized into multiples of 1/15?



This probably seems like a stupid philosophical quibble, and it is, but I get annoyed when people conflate their favorite model of reality with reality. The map is not the territory. The number 7 is no more and no less real than the word "force".

We had a related and equally quibbly discussion on these fora a while ago which dragged on for a bit, you might be interested.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

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Re: Is Pi Infinitely Long?

Postby Kow » Wed Nov 04, 2009 12:57 pm UTC

Tirian wrote: But I can imagine how someone might get it in his head that with a scanning electron microscope and enough time you could find out exactly how many atoms were in the diameter and circumference of a perfect circular crystal the size of a galaxy.


While the value of π has been computed to more than a trillion (10^12) digits, elementary applications, such as calculating the circumference of a circle, will rarely require more than a dozen decimal places. For example, a value truncated to 11 decimal places is accurate enough to calculate the circumference of a circle the size of the earth with a precision of a millimeter, and one truncated to 39 decimal places is sufficient to compute the circumference of any circle that fits in the observable universe to a precision comparable to the size of a hydrogen atom.
-Wikipedia

You wouldn't even get close.
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