The Endless Pi

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The Endless Pi Is it just an artefact of decimal notation Rate Topic:

#21 20 November 2011 - 03:09 PM the tree

Primate

The idea of a statement (such as 'x is(not) rational') being true but unprovable is getting a little Gödelesque for the sake of this thread, I would contend that truth and provability are one and the same though many would disagree and this certainly very far off of topic.

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#22 20 November 2011 - 09:26 PM DrRocket

Primate

the tree, on 20 November 2011 - 03:09 PM, said:

We were not talking about truth vs provability.

Your assertion -- that a number for which the rationality or irrationality is unproven is actually then neither rational nor irrational -- is just flat wrong, and therefore extremely misleading to a newbie trying to understand mathematics.

Moreover truth and provability ARE NOT the same thing. The people who disagree with you are called mathematicians.
The distinction between "true" and "provable" is critical to understanding the Godel incompleteness theorems, and some of Paul Cohen's work. So, if you contend differently you probably need to do some more study.

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#23 20 November 2011 - 10:55 PM the tree

Primate

DrRocket, on 20 November 2011 - 09:26 PM, said:

Your assertion -- that a number for which the rationality or irrationality is unproven is actually then neither rational nor irrational -- is just flat wrong...

I never made such an assertion. The question was about numbers for which a proof exists.

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#24 20 November 2011 - 11:31 PM DrRocket

Primate

the tree, on 19 November 2011 - 05:32 PM, said:

If a proof does not exist (in the ethereally mathematical sense of existence), then technically speaking it would be neither rational nor irrational, I suppose some non-computable numbers fall between the gaps in that sense.

the tree, on 20 November 2011 - 10:55 PM, said:

I never made such an assertion. The question was about numbers for which a proof exists.

Yes, you did.

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#25 5 January 2012 - 12:10 AM khaled

Meson

What I know about pi ...

$\pi \approx 3.14$

$\pi \approx \frac{22}{7}$

those are approximations, but what do you think about this one ?

$\frac{{\pi}^{2}}{6} = \sum_{n=1}^{\infty}{\frac{1}{{n}^{2}}}$

This post has been edited by khaled: 5 January 2012 - 12:26 AM

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#26 5 January 2012 - 12:55 AM the tree

Primate

What's there to think about it? It's a well known equality.

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#27 30 January 2012 - 05:40 PM The french tourist

Quark

Dekan, on 2 November 2011 - 04:23 PM, said:

Could the seeming lack of precision in Pi, be due to our human custom of using decimal notation. For example, if we divide something into 3 parts, each part is then, in decimal notation, an imprecise 0.333333333.... of the original.

Those kinds of notations are not imprecise. You can easely prove that 0,33333... = 1/3 or 0,99999...=1 (without any imprecision). For example, let x=0,999999... Then we have 10x=9,99999.... Hence 9x=9 (Writting the soustraction of 10x and x, you can skip the decimal part). Finally x=1. You can do the same with 0,33333... (let y=0,33333... And, at the end, you will find 3y=1. There's no lack of precision in these notations, they're just heavy and useless).

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#28 30 January 2012 - 08:09 PM John Cuthber

Chemistry Expert

the tree, on 19 November 2011 - 05:32 PM, said:

I'm fairly sure that a proof always exists for the rationality of any rational number.

If it is rational number then it can be expressed as a ratio of two numbers a and b where a and b are integers. So, for any rational number the proof is simple.

Find a and b.

It may not be practical since it may take an (almost) infinite time.

This post has been edited by John Cuthber: 30 January 2012 - 08:11 PM

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#29 30 January 2012 - 08:36 PM DrRocket

Primate

John Cuthber, on 30 January 2012 - 08:09 PM, said:

There are numbers for which it is not know whether they are rational or irrational. $e + \pi and e \pi$ are such numbers. That does not mean that no proof exists, only that no one has found a proof either way thus far. A proof that no proof exists would be a proof of undecidability of the question. No proof of undecidability has been found either.

Your suggestion as to a proof won't work. It won't work because if a number is presented as a ratio orf integers, there is nothing left to do, and if it is not so presented you have no test to determine if any given ratio is the number in question -- try your method on $e+ \pi$ for instance and you will find that you have no way to make a comparison.

Note that while we do not know whether $e + \pi$ or $e \pi$ are rational, it is known that at most one of them can be rational. The proof is in fact quite simple.

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#30 7 February 2012 - 03:31 PM puffy japanisthebest

Lepton

khaled, on 12 November 2011 - 08:32 AM, said:

I know that calculating Pi requires more complex mathematical operations, as we go farther away from the decimal point,

I'd ask .. is it true that $\pi = \frac{22}{7}$

I mean, is it only true for number of digits after the decimal point .. or is it the exact answer ?

$\pi = \frac{22}{7}$ is probably only true for a little bit.. i think.. :confused:

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#31 7 February 2012 - 09:02 PM DrRocket

Primate

*puffy* japanisthebest, on 7 February 2012 - 03:31 PM, said:

$\pi = \frac{22}{7}$ is probably only true for a little bit.. i think.. :confused:

$\pi$ is irrational, and in fact transcendental.

$\frac {22}{7}$ is manifestly rational.

They are never equal. But $\frac {22}{7}$ is an acceptable approximation to $\pi$ in some simple applications.

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#32 24 February 2012 - 09:08 PM ewmon

Baryon

π is the ratio of the circumference of a circle to its diameter, and yet, it is the irrational ratio. :blink:

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#33 27 February 2012 - 11:48 PM D H

Physics Expert

ewmon, on 24 February 2012 - 09:08 PM, said:

π is the ratio of the circumference of a circle to its diameter, and yet, it is the irrational ratio. :blink:

Don't get hung up on the label "irrational". Mathematicians, until recently, were a rather stodgy lot. Look at the names they gave things. Irrational numbers: Numbers that don't make a bit of sense. The symbol preceding 2 in $\surd 2$ , is the surd symbol: Short for absurd. Negative numbers: Things that aren't numbers. Imaginary numbers: Numbers that aren't quite "real". The designation of the real numbers as the real numbers was a back-formation to contrast the numbers that truly are "real" from those that are not.

Nowadays mathematicians have embraced the rainbow and they have all kinds of numbers. These more recent inventions tend not have some derogatory label such as irrational, absurd, negative, imaginary, or complex. Instead they are merely given descriptive names such as transfinite numbers, p-adic numbers, quaternions, octonions, etc.

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#34 28 February 2012 - 01:49 AM swansont

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connector's "pi solved" tangent has been moved http://www.sciencefo...4686-pi-solved/

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#35 28 February 2012 - 07:29 AM JohnStu

Baryon

It is endless because how you derive pi depend on how much you "zoom in" in the curve. The more accurate and detailed to measure the circumference, the more correct decimal digits. In construction of circles, some architects avoided calculation of pi by simply using a fine brush tied to a stick and span it on the sheet, and viola, a perfect circle. No need to calculate the area.

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#36 1 March 2012 - 12:14 AM phillip1882

Quark

while pi is indeed transcendental, there are some nice representations of it none the less.
one of my personal favorites....
3 + 1/(7 +1/(15 +1/(31 +1/63....
edit: this is wrong I'm afraid.
a much more correct version is...
4/(1 +1/(3 +4/(5 +9/(7 +16/(9 +25/...

This post has been edited by phillip1882: 1 March 2012 - 03:37 AM