Exploring the Infinity of Pi: A Scientific Perspective

[pallidin]

Is pi Infinite?

 

[chroot]

No, $\pi < \infty$.

- Warren

 

[Integral]

Further

$$3.14 < \pi < 3.15$$

 

[luther_paul]

What do u mean pi is infinite? the value itself or the digit? if u are talking about the digits, its infinite because it is an irrational number, meaning unending digits. the value itself is not infinite because it has an exact value (22/7).

$$3.14 < \pi < 3.15$$

its common sense to say that!peace!

 

[chroot]

Originally posted by luther_paul
the value itself is not infinite because it has an exact value (22/7).

22/7 is a rational number, and thus has a repeating or terminating decimal expansion. 22/7 is not pi. 22/7 is just a crude approximation sometimes used in place of pi.

- Warren

 

[pnaj]

luther_paul

Sorry to be picky, but ...

it is an irrational number, meaning unending digits

... is not actually true (just in case pallidin gets the wrong idea).

A rational number is any number that can be written m/n where m,n are integers.
So,
1/7 = 0.142857142857142857142857..(142857 recurring)
is rational, but has 'unending ' digits.

In fact, technically,
1/1 = 1.000000000000000000000000..(0 recurring)
has infinitely many digits.

But, for an irrational, there will never be a recurring pattern in the expansion, as for the rationals.

Another way to look at it is this.

Suppose you trying to calculate the decimal expansion of 1/7 by hand (or by any other method). Once you'd got to the first recurrence of the sequence (142857) you would stop, because it's for certain that the pattern will recur infinitely.

So far, over 200 billion digits of PI have been calculated ... no pattern yet.

 

[pnaj]

There's a BBC programme that discusses infinity ...

In Our Time - Infinity

 

[master_coda]

Originally posted by pnaj
So far, over 200 billion digits of PI have been calculated ... no pattern yet.

And there never will be a pattern. Just in case anybody wasn't clear about that from the above posts.

 

[luther_paul]

pnaj
thanks for the info... i got that wrong, it must've been nonterminating and non-recurring/nonrepeating...

warren
ur right 22/7 is just a crude estimation of pi, but i used 22/7 to show that pi has an exact value, not infinity.. the topic is about pi being an infinite.

thanks, by the way..

 

[mouseonmoon]

Originally posted by master_coda
And there never will be a pattern. Just in case anybody wasn't clear about that from the above posts.

well, there
seems to be a pattern of PI repeating here..(smile)

anyway-what's the 'fartherest out' that's been found for a 'repeater'
(hope this isn't an 'irrational' question--i'm simply guessing someone's played around
with the idea ie. recurring patterns)

or,in different words, what 'number' begins its pattern sequence with the greatest expansion?
or/and, --what rational number has been found with the 'longest pattern'?

what am i asking?...how long can a sequence be?...is there any formula that 'predicts' these patterns etc...anything along these lines-who's written about this? (or 'wierd' math in general--who was the guy who 'proved' 1+1 didn't equal 2 ?--did anyone ever prove he was wrong??)

why am i asking?...well, why has PI been taken to over 200 billion digits?
Can one really say, 'it will never repeat'?
seriously...

 

[pnaj]

No, you can't say there will never be a pattern, just that you cannot predict that there will be one, even after calculating the first 200 billion digits.

One could positively show that it is terminating ... if you found the pattern.

One cannot positively show that there is no pattern.


Apart from the massively small chance that a pattern does emerge, the main reason for looking seems to be bench-testing of more-and-more powerful computers.

P.

 

[krab]

Originally posted by pnaj
No, you can't say there will never be a pattern, just that you cannot predict that there will be one, even after calculating the first 200 billion digits.

One could positively show that it is terminating ... if you found the pattern.

One cannot positively show that there is no pattern.

That ain't right. Pi was proved to be a transcendental number already way back in 1882. Therefore, no pattern. More info.

 

[HallsofIvy]

Originally posted by mouseonmoon
what rational number has been found with the 'longest pattern'?

what am i asking?...how long can a sequence be?...is there any formula that 'predicts' these patterns etc...

Assuming that by "pattern" you mean section of digits that then repeats, the rational number m/p where p is prime and m< p will have repeating section with p digits. Since there exists arbitrarily large prime numbers, the repeating section can be as large as you please.

Originally posted by pnaj
No, you can't say there will never be a pattern, just that you cannot predict that there will be one, even after calculating the first 200 billion digits.

What exactly do you mean by "pattern"? Most of the responses here are using it to mean a section that repeats.
It is fairly easy to show that if the decimal expansion of a number eventually consists of a section that then repeats, that number is rational. It is also fairly easy to show that pi is not rational and so never repeats.

 

[master_coda]

When I said that pi doesn't have a "pattern" I meant that it does not repeat. Almost any other notion of pattern is difficult to define mathematically.

 

[chroot]

Originally posted by HallsofIvy
Assuming that by "pattern" you mean section of digits that then repeats, the rational number m/p where p is prime and m< p will have repeating section with p digits.

Is this true? How about 1/3, which has only one repeating digit? Or 1/7, which has six?

- Warren

 

[lethe]

Originally posted by master_coda
When I said that pi doesn't have a "pattern" I meant that it does not repeat. Almost any other notion of pattern is difficult to define mathematically.

there are lots of irrational numbers that do have patterns. my favorite example is this one:

0.1101001000100001000001...

one way to define whether a number has a pattern is to show that the number is normal. a number is normal if, loosely speaking, any finite sequence of digits appears about as frequently as it would for a random sequence of digits

it is probable that pi is normal, but not proved, so in fact, there may be a pattern. in fact, people sometimes look for patterns in the digits of pi. we simply don t know if it has any. remember the movie Pi? i think that guy found the torah in the digits of pi, or something (of course, it was a work of fiction).

 

[pnaj]

HallsofIvy,
Did you see the original question?

We were all trying to help someone understand the nature of the difference between a rational number and a irrational one.

So, I wouldn't dispute what you say, of course ... you're right ... but I'm not sure how much help ...

Originally posted by HallsofIvy
It is also fairly easy to show that pi is not rational and so never repeats.

... would really be to someone who might not really know what 'rational' is.


================
Also, krab and master_coda, thanks for pointing out that PI has been proved to be transcendental (and thus not rational), I could have misled someone there ... I wasn't precise enough. What I should have said was ...

Suppose you didn't know whether PI was rational or not.

Then ...

Originally posted by pnaj
No, you can't say there will never be a pattern, just that you cannot predict that there will be one, even after calculating the first 200 billion digits.

One could positively show that it is terminating ... if you found the pattern.

One cannot positively show that there is no pattern.

... makes more sense.

What I trying to get our friend to get the gist of was this:

If you look at the decimal expansion of any particular number upto to squillionth digit, and not see a terminating-repeating sequence of digits, that does not mean that is not rational. You gain no information that will help you predict whether it is rational or not.


Cheers,
P.

 

[master_coda]

Originally posted by lethe
there are lots of irrational numbers that do have patterns. my favorite example is this one:

0.1101001000100001000001...

one way to define whether a number has a pattern is to show that the number is normal. a number is normal if, loosely speaking, any finite sequence of digits appears about as frequently as it would for a random sequence of digits

it is probable that pi is normal, but not proved, so in fact, there may be a pattern. in fact, people sometimes look for patterns in the digits of pi. we simply don t know if it has any. remember the movie Pi? i think that guy found the torah in the digits of pi, or something (of course, it was a work of fiction).

But I could argue that a number has a pattern as long as we have an algorithm for generating its digits.

Perhaps it's best to avoid the use of the word "pattern", since it seems hard to define properly.

 

[chroot]

Originally posted by master_coda
But I could argue that a number has a pattern as long as we have an algorithm for generating its digits.

There are lots of algorithms to generate the digits of pi...

- Warren

 

[pnaj]

Sorry, probably my fault for introducing the dreaded 'pattern' word.

 

[mouseonmoon]

first, let me thank krab for the Wikipedia link-
love it and been ' lost in delight ' ever since!

(and speaking of logarithms-John Napier invented 'em-what a character! and math was 'just a hobby' for him-
http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Napier.html

and speakin' of patterns:
The infinite continued fraction expansion of e contains an interesting pattern that can be written as follows:
e= {1;1,1,2,1,1,4,1,1,6,1,1,8,1,1 ...}
(which was described by Richard Feynman as "The most remarkable formula in mathematics"!)

But back to the original post in this thread-is Pi infinite?-

the mind boggles-my does anyway...

seems there 'could be' an 'infinite' number of 'patterns' in the 'expansion' of Pi (are there absolutely no patterns within the 200 billion digits so far generated?-seems impossible)--yet it 'cannot' 'in fact' 'repeat' (("repeat": as it is proven to be transcendent))...holding in thought the computer
laboring on with the 'expansion' of Pi to 'infinity'...

Has anyone put this question to one of those gifted individuals known as 'idiot savants'? I've heard where some are able to 'instantly' perform calculations of immense complexity--i am not aware of anyone with this ability today...(perhaps it would be inhumane to attempt such a thing-let me express my vote against such an 'experiment'!)

i've reached my limit...sparkplugs burnt out, confused astoundment, brain cells fused...
excuse me, i need to find the 'Dear Santa' forums...oh Dorothy!

 

[krab]

Originally posted by pnaj
Sorry, probably my fault for introducing the dreaded 'pattern' word.

Don't be sorry. As Warren points out, with a sufficiently broad definition of pattern, pi does have a pattern. Just no repeating digits.

 

[lethe]

Originally posted by krab
Don't be sorry. As Warren points out, with a sufficiently broad definition of pattern, pi does have a pattern. Just no repeating digits.

yes, if you make the definition of "pattern" so broad as to be trivial and useless, so that all things are patterns, then pi has a pattern. on the other hand, there exist patterns which are not trivial and are mathematically well defined. one example is repeating sequence of digits, which pi does not contain. there are other patterns which pi may have. it is not known whether it does or does not.

 

[selfAdjoint]

ANd of course pi escapes patterns more general than repeating digits, since it's transcendental, i.e. not the solution of any polynomial equation with rational coefficients.

So both square root of two and pi lack repeating digits, but square root of two will be found to have a more subtle pattern deriving from its definition as a solution of x² = 2, but you will never be able to find such a pattern in pi.

 

[moshek]

pnaj,

With the eye's of non Euclidian mathematics
that see alway's mathematics as
"One whole organism" ( Hilbert vision 1900)
your original question does pai infinity?"
became significant when we present pi in base 16
and this close a circle of 5,000 years voyager to find pai.


Moshek

 

[lethe]

Originally posted by selfAdjoint
ANd of course pi escapes patterns more general than repeating digits, since it's transcendental, i.e. not the solution of any polynomial equation with rational coefficients.

So both square root of two and pi lack repeating digits, but square root of two will be found to have a more subtle pattern deriving from its definition as a solution of x² = 2, but you will never be able to find such a pattern in pi.

for example, root 2 repeats digits in its continued fraction representation, whereas pi does not.

 

[krab]

Originally posted by lethe
yes, if you make the definition of "pattern" so broad as to be trivial and useless, so that all things are patterns, then pi has a pattern.

That's not what I mean. A possible non-trivial broad definition of pattern is simply a finite algorithm for finding the digits. The number you gave as your favourite transcendental number can be written

$$\sum_{n=0}^\infty (.1)^{1+n(n+1)/2}$$

and only has a readily discernable pattern because of the "coincidence" that (1/10) appears raised to a power, and we represent our numbers on the base 10. Pi can be defined for example by the following "pattern":

$$\pi=8\,\sum_{n=0}^\infty [(4n+1)(4n+3)]^{-1}$$

$$=8\left[{1\over 1\times 3}+{1\over 5\times 7}+{1\over 9\times 11}+...\right]$$

This, admittedly, won't have a pattern among the digits no matter what the base is; but it's still a pattern in the broad (but non-trivial) sense.

 

[toasty]

Originally posted by krab
That ain't right. Pi was proved to be a transcendental number already way back in 1882. Therefore, no pattern. More info.

Sorry to be reviving this but can you explain exactly what you mean by 'trancendental''?

Can you also give the source, and if possible a short summary of the proof in simple language?

Thank You

 

[matt grime]

transcendental numbers are those that are not the root of any polynomial with integer coefficients (equivalently with rational coeffecients)

e and pi are transcendental. numbers that are not transcendental are called algebraic.

i'm not sure the proofs I've seen are considered nice, especially if this is the first time you've seen the definition of algebraic and transcendental.

almost all numbers are transcendental - there are only a countable number of algebraic numbers so there must be an uncountable number of transcendentals.


Hilbert's proof of the transendence of e runs to about two pages of Le Veque's Fundamentals of number theory and doesn't really involve anything useful. He says 'it is as elegant as it is mysterious' and involves integrating by parts.

a similar proof for pi was also given by hilbert.

there is a 1976 springer-verlag lecture series in mathematics by Mahler which gives several proofs of these facts, though I've never seen it.


If you wish me to I can reproduce the e proof and give you another proof for pi that can be found in a lot of galois theory textbooks under the stuff about transcendental extensions.

 

[toasty]

Thank You

Sorry to bring you back to this one !

I found the page with the links which I neglected to click on before because I did not notice it!

However it does neatly tie in with some other things.

Transcendental as I recall was used about things 'inducted' by limited information. A good example is the proof that 0 may not be a demoninator.

 

[quantumdude]

I'm just glad Donde isn't here.

 

[Integral]

Originally posted by Tom
I'm just glad Donde isn't here.

Funny you should menttion that.. I was thinking of him also..

 

[Jug]

Is pi finite?

Originally posted by Tom
I'm just glad Donde isn't here.

What is the intrinsic value of any number, transcendental, irrational, etc., beyond what it can prove by empirical evaluation? For example, can the degree of arc be better defined by the purely abstract irrational pi than it can by the rational value of 355/113?

 

[selfAdjoint]

Originally posted by Jug
What is the intrinsic value of any number, transcendental, irrational, etc., beyond what it can prove by empirical evaluation? For example, can the degree of arc be better defined by the purely abstract irrational pi than it can by the rational value of 355/113?

Yes it can. And when pi comes up in formulas that physicists use, you don't wnat to complicate the logic by assuming pi is some approximation like 22/7 or 355/113, or 3.1416. Physics assumes space is a continuum and that the values it discusses can take on all real numbers. The exception is action, which can only take on integer multiples of h.

What is true is that physicists can never tell whether some particular value they work with is irrational, unless math tells them it is. So pi and e are known to be irrational (and transcendental), but h and alpha are not sure.

 

[Jug]

Have to disagree, SelfAdjoint. What I hear you saying is that we shouldn't complicate logic with the truth.

But as regards the subject proper, pi is of course infinite. In that regard then, and not being a maths person myself, I can only relate to the basics from which our mathematics derive - the perfect ratios given by Pythagoras: 0:1:1:2:3:4 - and where it is assumed that the cipher was intended by Pythagoras to represent the finite condition; which is to suggest that the cipher in place of denoting a lack of magnitude might in fact denote total magnitude.

My contention then is that the true ratio of pi cannot be arbitrarily determined but must in fact conform exactingly to some full set of ratios describing the finite condition. Just some thoughts on the thing...