The Endless Pi

[Dekan]

When we try to write Pi in decimal notation, we get an endless stream of digits. 3.141592653589723.....and so on forever.

 

This seems counter-intuitive. You'd think that a simple thing, like the ratio between the radius and circumference of a circle, ought to be a simple, precise number. After all, isn't the Universe made almost entirely of circular and round things.

 

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.

 

But suppose instead, we use Fractional notation. Then we can write each part as a precise 1/3.

 

And applying fractional notation to Pi, we can easily get a very good value. 355/113, gives in decimal 3.141592 - which is accurate to the 6th decimal place.

 

An excellent result, for a fraction which uses only 3 figures in its numerator and denominator! Could fractions with more figures, achieve even more accuracy - and perhaps finally pin Pi down?

Edited November 2, 2011 by Dekan

[imatfaal]

No - pi is irrational and cannot be defined as a ratio of two integers. It is also transcendental so cannot be the root of a polynomial with rational coefficients.

 

Irrational Numbers

Proof that pi is irrational

 

Transcendetal Numbers

Proof that pi is transcendental

Edited November 2, 2011 by imatfaal

[Dekan]

No - pi is irrational and cannot be defined as a ratio of two integers. It is also transcendental so cannot be the root of a polynomial with rational coefficients.

 

Irrational Numbers

Proof that pi is irrational

 

Transcendetal Numbers

Proof that pi is transcendental

 

Thanks imatfaal. I've tried to study the links you kindly provided. But they contain mathematics so sophisticated, that to my simple brain, they might as well be written in cuneiform! Hence I can only dumbly accept their conclusion: that Pi is indeed irrational and transcendental.

 

I have some further thoughts about the implications of this. But they would be more suited to "Speculations".

 

Thanks again for your reply!

[imatfaal]

That hideous strength of mathematics - it able to convince both of us, neither of whom can understand it, that something is true

[Dekan]

That hideous strength of mathematics - it able to convince both of us, neither of whom can understand it, that something is true

 

It is indeed a Dark Tower.

[imatfaal]

Mathematics - unlike other academic pursuits comes out of the silent planet and commands attention, it would be a personal heresy (and the last battle of science) to deny its force.

 

OK enough - before I try an shoehorn the L the W and the W into a text about maths

[the tree]

Hence I can only dumbly accept their conclusion: that Pi is indeed irrational and transcendental.

Okay really now, if something is transcendental then it has to be irrational. The transcendentals are a subset of the irrationals.

 

Not wanting to scare you too much, but the irrationals are infinitely more numerous than the rationals - numbers don't fit into our systems as neatly as we'd presumed they would.

[DrRocket]

Not wanting to scare you too much, but the irrationals are infinitely more numerous than the rationals - numbers don't fit into our systems as neatly as we'd presumed they would.

 

The rationals, and the set of algebraic numbers are both countable, hence of Lebesgue measure 0. The transcendentals, being the complement of the algebraic numbers in are uncountable and of full measure -- you can do this in either the real or complex numbers as you please.

 

Not sure exactly what you mean by "infinitely more numerous", but the above ought to cover it.

Edited November 6, 2011 by DrRocket

[the tree]

That is what I meant, I was sort of deliberately avoiding technical language there.

[DrRocket]

That is what I meant, I was sort of deliberately avoiding technical language there.

 

That's what the technical language is for. There is so much BS on the board surrounding the word "infinity" that God only knows how some people might interpret your statement.

[khaled]

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 [math]\pi = \frac{22}{7}[/math]

 

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

[the tree]

I'd ask .. is it true that [math]\pi = \frac{22}{7}[/math]

No. It's about 0.04% off. And for that matter, we've already covered that pi is an irrational number - so the answer there should have more than a little obvious. Edited November 12, 2011 by the tree

[DrRocket]

 

I'd ask .. is it true that [math]\pi = \frac{22}{7}[/math]

 

 

Folklore has it that it is true, by act of the legislature, in Mississippi and/or Iowa.

 

Elsewhere pi is transcendental, hence irrational.

[the tree]

Some things in mathematics you have to accept. Pi being an irrational number is one of them.

No, no, no - there are proofs and detailed explanations for pi being irrational, you don't just have to accept it.

[imatfaal]

No, no, no - there are proofs and detailed explanations for pi being irrational, you don't just have to accept it.

 

I would recommend reading some of the proofs of irrationality on the wikipedia page to further bolster the tree's post above. whilst they don't go through the proof of pi irrationality (which you can find here but is not for the faint-hearted), the proofs for root 2 and log₂3 are beautifully followable.

 

One thing occurred to me - and I couldn't fathom an answer nor find one; must there exist a proof of irrationality/rationality for every number. There are numbers for which no proof exists whether they are irrational (2^e) - is it possible that there are numbers for which it can never be determined if they are irrational or rational.

[the tree]

One thing occurred to me - and I couldn't fathom an answer nor find one; must there exist a proof of irrationality/rationality for every number. There are numbers for which no proof exists whether they are irrational (2^e) - is it possible that there are numbers for which it can never be determined if they are irrational or rational.

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.

[DrRocket]

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.

 

Absolutely not.

 

Any real number is unequivocally either rational or irrational. It may not be known which is the case, but the answer is never "neither".

 

I seem to recall, not positive though, that it is an open problem whether [math] e + \pi[/math] is rational or irrational.

[imatfaal]

Absolutely not.

 

Any real number is unequivocally either rational or irrational. It may not be known which is the case, but the answer is never "neither".

 

I seem to recall, not positive though, that it is an open problem whether [math] e + \pi[/math] is rational or irrational.

 

It is not known whether π + e or π − e is irrational or not. In fact, there is no pair of non-zero integers m and n for which it is known whether mπ + ne is irrational or not. Moreover, it is not known whether the set {π, e} is algebraically independent over Q.

 

I still cannot find mention of whether (ir)rationality must be provable for all numbers - or if some numbers (whilst they must be either irrational or rational) can never be shown to be one or the other

Edited November 20, 2011 by imatfaal

[the tree]

Any real number is unequivocally either rational or irrational. It may not be known which is the case, but the answer is never "neither".

I know, my point was that a proof must exist, known or otherwise, for every computable number.

[imatfaal]

I know, my point was that a proof must exist, known or otherwise, for every computable number.

 

That was my question - must a proof exist? I cannot find anywhere that says this (or not ) on line. Mathworld gives many forms that will/will not be irrational - but does not mention a theory that states that a proof must exists

[the tree]

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.

[DrRocket]

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.

 

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.

[the tree]

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.

[DrRocket]

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.

 

 

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

 

Yes, you did.

[khaled]

What I know about pi ...

 

[math]\pi \approx 3.14[/math]

 

[math]\pi \approx \frac{22}{7}[/math]

 

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

 

[math]\frac{{\pi}^{2}}{6} = \sum_{n=1}^{\infty}{\frac{1}{{n}^{2}}}[/math]

Edited January 5, 2012 by khaled