View Full Version : Infinite expansion of pi?

caveman1917

2010-Jun-01, 02:38 PM

This question may be a bit removed from physics, but, as i see it, it's a basic scientific/math question. So sorry if it doesn't belong here :)

Suppose - for the sake of clarity - we express pi in base-26 (A->Z). Also suppose we get any finite number N expansion of digits/characters of pi we ask for instantaneously.

I always thought, as a consequence of pi being irrational, from this follows we can recover any imaginable text (A-Z) inside the expansion of pi. Or put differently, the distribution of the different digits/characters of pi is ~uniform.

1) Is this correct?

Consider the irrational number Q (in decimal base) 0.101001000100001... (add an extra '0' in between every time). Q clearly doesn't give every (decimal) text in its expansion (no 2,3,4,..).

2) Is this correct?

3) What is the defining difference between pi and Q which governs these different qualities?

Strange

2010-Jun-01, 02:56 PM

This (apparently) depends on it being normal: http://en.wikipedia.org/wiki/Pi#Open_questions

ravens_cry

2010-Jun-01, 09:44 PM

I have thought along a similar lien, and came to the conclusion, if this is so, then you could also find anything that can be reduced to numbers, everything that can digitized, is in pi, somewhere. If pi is the infinite monkeys, then yes, the complete works of Shakespeare are within its infinite randomness. In any format from ASCII to PDF, to even JPEG. And that another thing. Any picture, any program, any music and video possible would be in pi. Every possible genome is in pi. In fact, if the universe is reducible to numbers, then a program to run the universe is lurking in its depths, an infinite number of times in infinite variation.

jlhredshift

2010-Jun-01, 10:09 PM

Erdos would say that the proof is in the Supreme Fascists Big Book.

DrRocket

2010-Jun-01, 10:25 PM

I have thought along a similar lien, and came to the conclusion, if this is so, then you could also find anything that can be reduced to numbers, everything that can digitized, is in pi, somewhere. If pi is the infinite monkeys, then yes, the complete works of Shakespeare are within its infinite randomness. In any format from ASCII to PDF, to even JPEG. And that another thing. Any picture, any program, any music and video possible would be in pi. Every possible genome is in pi. In fact, if the universe is reducible to numbers, then a program to run the universe is lurking in its depths, an infinite number of times in infinite variation.

The digits in the decimal expansion for pi are not random.

They are completely predictable -- they are expressed, in order, in the decimal expansion for pi.

They are sometimes called "pseudo random" because there is no simple expressin for them, but they are not random.

Although this tells you very little with respect to the decimal expansion, pi is not only irrational, it is transcendental. That means that pi is not the a root of any polynomial with rational coefficients. That distinguishes pi from algebraic numbers such as sqrt(2).

caveman1917

2010-Jun-01, 10:55 PM

The digits in the decimal expansion for pi are not random.

They are completely predictable -- they are expressed, in order, in the decimal expansion for pi.

They are sometimes called "pseudo random" because there is no simple expressin for them, but they are not random.

Although this tells you very little with respect to the decimal expansion, pi is not only irrational, it is transcendental. That means that pi is not the a root of any polynomial with rational coefficients. That distinguishes pi from algebraic numbers such as sqrt(2).

But the fact that they aren't random doesn't imply there exists a finite ordered set of decimal symbols which is not a subset of pi, right?

So even being non-random, it can still contain all possible finite patterns?

cjameshuff

2010-Jun-01, 10:57 PM

I have thought along a similar lien, and came to the conclusion, if this is so, then you could also find anything that can be reduced to numbers, everything that can digitized, is in pi, somewhere. If pi is the infinite monkeys, then yes, the complete works of Shakespeare are within its infinite randomness. In any format from ASCII to PDF, to even JPEG. And that another thing. Any picture, any program, any music and video possible would be in pi. Every possible genome is in pi. In fact, if the universe is reducible to numbers, then a program to run the universe is lurking in its depths, an infinite number of times in infinite variation.

As DrRocket points out, being irrational does not make the digits random, and doesn't imply that all patterns exist.

This reminds me of a data storage method I once read about though...a rather thoroughly impractical one, not something that was ever actually used. The storage medium was a single wire with a single notch in it. The data was encoded as the ratio between the wire lengths on either side of the notch.

pzkpfw

2010-Jun-02, 12:10 AM

For fun: http://www.dr-mikes-maths.com/pisearch.html

ravens_cry

2010-Jun-02, 12:44 AM

As DrRocket points out, being irrational does not make the digits random, and doesn't imply that all patterns exist.

This reminds me of a data storage method I once read about though...a rather thoroughly impractical one, not something that was ever actually used. The storage medium was a single wire with a single notch in it. The data was encoded as the ratio between the wire lengths on either side of the notch.

Aw dang, it would be so. . .poetic if it did. Is there any kinds of numbers that do?

DrRocket

2010-Jun-02, 12:50 AM

But the fact that they aren't random doesn't imply there exists a finite ordered set of decimal symbols which is not a subset of pi, right?

So even being non-random, it can still contain all possible finite patterns?

Possibly. Not much is actually known in this regard.

The fact that the digits are not random (and there is no good definition for "random" outside of formal probability theory) means that one cannot use probability theory to reach any conclusions.

cjameshuff

2010-Jun-02, 01:48 AM

Aw dang, it would be so. . .poetic if it did. Is there any kinds of numbers that do?

As far as I'm aware, it's not known that pi doesn't qualify. My point was that it it's not necessarily true that it does, simply by being infinite and non-repeating.

ravens_cry

2010-Jun-02, 11:01 AM

As far as I'm aware, it's not known that pi doesn't qualify. My point was that it it's not necessarily true that it does, simply by being infinite and non-repeating.

Oh, OK then. In my opinion,it has a nice symmetry if it does though.

uncommonsense

2010-Jun-02, 04:24 PM

Indeed. It's my understanding though that we have not come to any main stream answers to the OP's question. I find it fun to imagine the diameter of circle is the shortest one dimensional distance between the farthest 2 points on a circle.

But the distance between the same two points taking a two dimensional path that, starting from first point, makes the highest available number of turns towards the second point (the smoothest turn), which, according to what we know about pi, is a great deal of turns - how many? we don't know yet. Maybe it means one small turn at every smallest universal distance. Maybe it means one turn that accounts for every two dimensional direction in the universe. Maybe it means there are an infinite number of two dimensional directions between the two points. Who knows. But I think the nature of pi has some yet unknown meaning.

DrRocket

2010-Jun-02, 04:34 PM

But the distance between the same two points taking a two dimensional path that, starting from first point, makes the highest available number of turns towards the second point (the smoothest turn), which, according to what we know about pi, is a great deal of turns - how many? we don't know yet. Maybe it means one small turn at every smallest universal distance. Maybe it means one turn that accounts for every two dimensional direction in the universe. Maybe it means there are an infinite number of two dimensional directions between the two points. Who knows. But I think the nature of pi has some yet unknown meaning.

This is completely meaningless.

uncommonsense

2010-Jun-02, 04:47 PM

And it there is no proof to support it. It's just the kind of thing one imagines when you find it hard to "just round it off cause we don't understand the exact ratio".

grapes

2010-Jun-02, 05:01 PM

This (apparently) depends on it being normal: http://en.wikipedia.org/wiki/Pi#Open_questionsThat is the answer to the OP, but here is a more direct link to the discussion of normal numbers: http://en.wikipedia.org/wiki/Normal_number

A normal number is what the OP was looking for, but we don't know if pi is normal. We don't know if any particular number is normal, except the ones constructed specially to be normal, like Champernowne's number, mentioned at that link, which is just a decimal point followed in order by every single number: .01234567891011121314151617... Obviously, that works. :)

This is completely meaningless.So true.

DrRocket

2010-Jun-02, 05:45 PM

That is the answer to the OP, but here is a more direct link to the discussion of normal numbers: http://en.wikipedia.org/wiki/Normal_number

A normal number is what the OP was looking for, but we don't know if pi is normal. We don't know if any particular number is normal, except the ones constructed specially to be normal, like Champernowne's number, mentioned at that link, which is just a decimal point followed in order by every single number: .01234567891011121314151617... Obviously, that works. :)

One has to be very careful in reading that Wiki piece. The definition of a normal number does NOT involve probabilities. One can, as the article says, construe the meaning very roughly as a probability, but that interpretation is not rigorous, and there is no probability space involved. It is simply that the definition can be interpreted formally as analogous to the law of large numbers. One cannot use probability theory to make inferences here.

Second note the theorem that the measure of the set of non-normal numbers is zero, so that almost all (in the sense of measure theory) numbers are normal. That theorem is non-constructive, as are many such theorems and while it tells you that there are a great many normal numbers (there must be an uncountable number if the complement has measure zero) it tells you nothing about how to determine if a particular number is normal or not.

The situation is similar with transcendental numbers. The algebraic numbers are countable, hence of Lebesgue measure zero, and hence almost all real numbers are transcendental. Pi is transcendental, and so is e. The proof is not obvious.

So, the situation is that the problem has been given a name "normal numbers" and some theorems of the measure-theoretic type are known, but there is no good general way to determine if a specific given irrational number is normal.

DrRocket

2010-Jun-02, 05:52 PM

And it there is no proof to support it. It's just the kind of thing one imagines when you find it hard to "just round it off cause we don't understand the exact ratio".

Not quite you don't understand.

There is no way to construct a proof to either support your statement or refute. It doesn't not mean anything -- it is mathematical gibberish. There is no statement there to be proved.

On the other had we do, most assuredly understand the ratio that we call pi. It is extremely well understood.

1) It is the ration of the circumference of a circle to its diameter in Euclidean geometry.

2) It is half the period of the sine and cosine functions, which can be defined completely rigorously in terms of power series.

3) e^(i pi) = -1 (the exponential function is defined by a power series and its value at 1 is e)

4) pi is known to be transcendental

The fact that pi has no finite or repeating decimal, nor a discernible pattern in an infinite decimal representation is not important. Pi is understood extremely well.

Strange

2010-Jun-02, 06:14 PM

But the distance between the same two points taking a two dimensional path that, starting from first point, makes the highest available number of turns towards the second point (the smoothest turn), which, according to what we know about pi, is a great deal of turns - how many? we don't know yet. Maybe it means one small turn at every smallest universal distance. Maybe it means one turn that accounts for every two dimensional direction in the universe. Maybe it means there are an infinite number of two dimensional directions between the two points. Who knows. But I think the nature of pi has some yet unknown meaning.

If you think this means something (and I can't disern anything in it) perhaps you could explain. The following bits don't mean anything to me:

what does "makes the highest available number of turns towards the second point" mean? Isn't that just an infinite number of turns (a spiral or random walk?)

what does "the smoothest turn" mean? Can you define "smooth" in this context?

what does "smallest universal distance" mean?

what does "every two dimensional direction in the universe" mean?

what is a "two dimensional direction"?

what does the "number of two dimensional directions between the two points" mean?

And then, I don't understand all the bits that join them together :) Or your conclusion :(

caveman1917

2010-Jun-02, 06:43 PM

So, the situation is that the problem has been given a name "normal numbers" and some theorems of the measure-theoretic type are known, but there is no good general way to determine if a specific given irrational number is normal.

Yes so irrationality isn't enough, however calling them normal numbers doesn't elucidate much.

I just posted the question here for some more insight into what that actually 'means'. Why some are normal and why some are not, why pi is/isn't, and what lines of thought there exist that point to (semi)answers to these questions?

It's difficult to get that sort of answers from wikipedia, or having to skim through a whole math book for a relevant footnote :)

uncommonsense

2010-Jun-02, 06:48 PM

What I'm seeing probably has no application in math. But from what I see of a circle:

Take circle with diameter of 1, so circumference is Pi;

Make diameter a line of distance 1, between point A and point B (points A and B are on the circumference).

Distance AB is the greatest distance of a straight line within a circle. It is a 1 dimensional path.

However, distance from A to B along the circumference is 1/2 Pi. This is a 2 dimensional path that, although a curve, an object moving along this path can said to have moved equally (even if for the smallest distance) in every 2 dimensional direction while going from A to B. What was this distance between A and B? 1/2 Pi.

What I'm saying is the ends of the diameter, point A and B, are EXACTLY at points that are 1 away from each other. However, If circumference starts EXACTLY from point A, it will not end up EXACTLY at point B because there is no exact ratio that says it will. Thats all.

DrRocket

2010-Jun-02, 06:51 PM

Yes so irrationality isn't enough, however calling them normal numbers doesn't elucidate much.

I just posted the question here for some more insight into what that actually 'means'. Why some are normal and why some are not, why pi is/isn't, and what lines of thought there exist that point to (semi)answers to these questions?

It's difficult to get that sort of answers from wikipedia, or having to skim through a whole math book for a relevant footnote :)

Find a copy of An Introduction to the Theory of Numbers by Hardy and Wright. Chapter IX considers problems of decimal expansions and section 9.12 specifically addresses normal number.

caveman1917

2010-Jun-02, 07:00 PM

What I'm seeing probably has no application in math. But from what I see of a circle:

Take circle with diameter of 1, so circumference is Pi;

Make diameter a line of distance 1, between point A and point B (points A and B are on the circumference).

Distance AB is the greatest distance of a straight line within a circle. It is a 1 dimensional path.

However, distance from A to B along the circumference is 1/2 Pi. This is a 2 dimensional path that, although a curve, an object moving along this path can said to have moved equally (even if for the smallest distance) in every 2 dimensional direction while going from A to B.

No, because for example sin30° != cos30°. So if sin is your x-component and cos your y-component, they will in general not be equal as you move over the curve.

What was this distance between A and B? 1/2 Pi.

The distance remains 1, you defined it that way. Perhaps you mean the length of the circle-curve from A to B?

What I'm saying is the ends of the diameter, point A and B, are EXACTLY at points that are 1 away from each other.

Yes you defined them to be exactly that distance.

However, If circumference starts EXACTLY from point A, it will not end up EXACTLY at point B because there is no exact ratio that says it will. Thats all.

The ratio is pi, so it will end up exactly on B. You even defined yourself that A and B lie on the circle.

Perhaps you mean that you cannot compute the ratio (pi) exactly, so if you're drawing this semi-circle in practice, you won't end up exactly on your target?

If your argument - as it seems - is purely a practical one, consider this:

the decimal representation of pi truncated to 39 decimal places is sufficient to estimate the circumference of any circle that fits in the observable universe with precision comparable to the radius of a hydrogen atom

(from http://en.wikipedia.org/wiki/Pi#Decimal_representation)

DrRocket

2010-Jun-02, 07:01 PM

What I'm saying is the ends of the diameter, point A and B, are EXACTLY at points that are 1 away from each other. However, If circumference starts EXACTLY from point A, it will not end up EXACTLY at point B because there is no exact ratio that says it will. Thats all.

No.

If you start at A and move along the circle EXACTLY pi units of length you wind up EXACTLY at B.

You see to base your idea on the notion there is no exact value of pi. That is wrong -- the exact value of pi is pi.

Pi is every bit as much a number as is 2.

Strange

2010-Jun-02, 07:32 PM

What I'm saying is the ends of the diameter, point A and B, are EXACTLY at points that are 1 away from each other. However, If circumference starts EXACTLY from point A, it will not end up EXACTLY at point B because there is no exact ratio that says it will. Thats all.

Yes it will, that is the definition of a (semi)circle. What has any ratio got to do with it? And there is a very exact ratio: Pi. The fact you can't measure or enumerate Pi to an unlimited accuracy is irrelevant. If the circumference were exactly 2, you still wouldn't be able to measure or "position" points A and B any more precisely just because the circumference is an integer.

DrRocket

2010-Jun-02, 08:05 PM

Yes it will, that is the definition of a (semi)circle. What has any ratio got to do with it? And there is a very exact ratio: Pi. The fact you can't measure or enumerate Pi to an unlimited accuracy is irrelevant. If the circumference were exactly 2, you still wouldn't be able to measure or "position" points A and B any more precisely just because the circumference is an integer.

Right

This line ------------------ is EXACTLY pi mugulframs long. It may difficult to figure out what 1 mugulfram is, but pi mugulframs is determined by that line.

uncommonsense

2010-Jun-02, 08:35 PM

I do not disagree that that is how math abstracts the situation. Without using numbers I am secure that if I measure points A and B, mark them, then using a compass to draw a circle around this diameter. I am happy that I have a diameter that dissects my circle. I am just a little hazed when the numbers cannot prove the symmetry. Well, yes they can according to accepted math, but in a practical sense, ....... you know what I mean. I'll get over it.

Strange

2010-Jun-02, 08:51 PM

I am just a little hazed when the numbers cannot prove the symmetry.

Are you confusing arithmetic with math, perhaps?

uncommonsense

2010-Jun-02, 09:03 PM

Well, since I was not aware that there was a major categorical distinction, then likely yes.

HenrikOlsen

2010-Jun-02, 09:12 PM

3) e^(i pi) = 1 (the exponential function is defined by a power series and its value at 1 is e)

e^(i pi)=-1 you mean.

HenrikOlsen

2010-Jun-02, 09:17 PM

Well, since I was not aware that there was a major categorical distinction, then likely yes.

Math is what happens once you get away from the idea that counting involves your fingers.

uncommonsense

2010-Jun-02, 09:27 PM

Math is what happens once you get away from the idea that counting involves your fingers.

That's gonna be a hard one to give up...........:think:

DrRocket

2010-Jun-02, 09:41 PM

e^(i pi)=-1 you mean.

right

jfribrg

2010-Jun-08, 02:12 PM

Pi may not be proven to be normal, but the postulated normalcy of Pi figured in an early calculation of the digits of Pi. In 1874, William Shanks calculated the first 727 digits. Others suspected that there was a mistake, but nobody was motivated to devote several years of work in finding it. The reason they suspected the mistake was that the distribution of the last couple of hundred digits did not appear to be normal. It could be statistical variation or it could be a mistake, or it could be that the digits of Pi are not normal. Until someone did the calculations, it was impossible to tell which of the three possibilities were correct. It wasn't until 1946 that D.F. Ferguson determined that the 506'th digit was wrong ( which made the remaining 220 digits meaningless).

Powered by vBulletin® Version 4.2.3 Copyright © 2019 vBulletin Solutions, Inc. All rights reserved.