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From: Shepherd Moon on 22 May 2010 21:12
Hello,

I'm a math layman with a question about pi. I don't want to clutter
the list with an inane question (I have seen proposals for a
sci.math.moderated group intended to weed out such questions, so I am
sensitive to this issue), but I hope my question will at least be
worthy of a response. I've provided citations where possible, not
because I pretend to any kind of expertise in mathematical research,
but just to show readers where I obtained my information in case the
information is wrong or (more likely) in case I have interpreted the
information incorrectly.

With that caveat, here goes.

Is it possible to prove whether or not there is a maximum possible run
of one digit within the decimal expansion of pi? For example, whether
or not it is possible to prove that a run of n 0s either cannot
appear, or on the contrary must appear, in pi.

I understand from basic Google searches (Wikipedia, etc.) that pi has
been proven to be an irrational number[1]. Because pi is irrational,
it should have the property of irrational numbers that its decimal
expansion neither terminates (as in 5/8 = 0.625000...) nor repeats (as
in 1/9 = 0.111...).[2]

I can see that in the case of 0.111.. there is no limit to the run of
1s since the sequence goes on without end.

But I guess I see even as I write this that although 0.625000...
terminates, it also has no maximum run of 0s, since (if I'm correct)
there is an infinite number of 0s in that decimal expansion. It's not
as if the first 3 digits, 6, 2, and 5, can be "subtracted" from the
infinity of 0s that follows. As a side question, is 2.5/3 = 0.8333...
considered a terminating decimal (i.e., can a decimal terminate in any
digit, not just 0)? It seems to me that 0.8333... is indeed a
terminating decimal, but I just want to make sure my thinking is
correct.

So perhaps pi is closer to 0.625000... in that it has more than one
type of digit in it (all 10 decimal digits), yet may also have within
it an infinite run of one (or maybe all) of those 10 digits.

Yet that is where my head starts to hurt, as the saying goes. Why?
Because if pi neither terminates nor repeats, then it obviously
doesn't "end" (as well as begin) with the same single digit repeating
endlessly (as in 0.111...) but it also doesn't terminate in a single
digit after a sequence of other digits (as in 0.625000...). So it
seems to me, speaking as a naive layman, of course, that one should
always expect a run of any single digit in pi to be interrupted by one
of the other digits.

At this point, I can think of two possibilities. One is a kind of
probabilistic explanation, to the effect that as one samples larger
and larger expansions of pi, the probability of finding a longer run
of a digit increases without bound. I know that sometimes these types
of number properties are proven with reductio ad absurdum (i.e.,
assume there is a maximum run of n 0s, then show that a run of n+1 0s
exists). But I don't know if anyone has done that with regard to pi.
Another possibility is based on one I read on another web site (I
can't recall the link now) that employs the transfinite numbers of
Cantor. I forget the exact explanation, but I think it involved levels
of infinity, but I can't recall the details. Perhaps others know this
approach and can explain it and let me know whether it addresses the
issue of a maximum possible run of one digit within pi.

I suppose there is a third possibility - that each digit does indeed
have a maximum possible run in pi. But my math intuition, feeble as it
is, tells me that such a maximum run seems unlikely in an irrational
number.

Well, I hope this post hasn't seemed too much like rubbish or a waste
of time. I look forward to any help others can give in helping me
understand whether or not this type of proof is (a) impossible, (b)
possible but not yet done, or (c) possible and already done. I think
it is a fascinating question, though maybe that's because I haven't
learned enough of the necessary math to answer the question.

Thanks in advance for your time and for any help.

Shepherdmoon

[1] http://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational

[2] "Irrational numbers have decimal expansions that neither terminate
nor become periodic."
http://mathworld.wolfram.com/IrrationalNumber.html
From: Jim Burns on 22 May 2010 21:40
Shepherd Moon wrote:
> Hello,
>
> I'm a math layman with a question about pi. I don't want to clutter
> the list with an inane question (I have seen proposals for a
> sci.math.moderated group intended to weed out such questions, so I am
> sensitive to this issue), but I hope my question will at least be
> worthy of a response. I've provided citations where possible, not
> because I pretend to any kind of expertise in mathematical research,
> but just to show readers where I obtained my information in case the
> information is wrong or (more likely) in case I have interpreted the
> information incorrectly.
>
> With that caveat, here goes.
>
> Is it possible to prove whether or not there is a maximum possible run
> of one digit within the decimal expansion of pi? For example, whether
> or not it is possible to prove that a run of n 0s either cannot
> appear, or on the contrary must appear, in pi.

I think you will find the idea of a normal number useful.
http://en.wikipedia.org/wiki/Normal_number

In a normal number, every 1-sequence of digits appears as
often as every other 1-sequence, every 2-sequence as often
as every other 2-sequence, and every n-sequence of digits
as often as every other n-sequence, for every natural n.

(This sounds like I'm saying that every sequence is
"just as probable" as every other sequence of the same
length. That makes no sense, though, because pi
(and every other single real number) /is not random/.
There is a 100% probability that the tenths digit
is '1', and so on. The definition of "normal number"
applies a test to a sequence of digits (from a real)
that a random sequence of digits (equi-probable and
independent) would pass, but it doesn't care whether
the digits came from some random process or, for
example, a calculation of pi.)

It seems to me that, if pi is normal, then there is no
maximum run of any digit. (Nor is there any finite
sequence of digits that does not appear.)

Wikipedia gives a couple of examples of numbers proven
to be normal, but it says that sqrt(2), pi, ln(2), and e
have not been proven normal, although it is "strongly
conjectured" that they are.

Jim Burns

From: Rick Decker on 22 May 2010 22:39
On 5/22/10 9:12 PM, Shepherd Moon wrote:
> Hello,
>
[snip]
>
> Is it possible to prove whether or not there is a maximum possible run
> of one digit within the decimal expansion of pi? For example, whether
> or not it is possible to prove that a run of n 0s either cannot
> appear, or on the contrary must appear, in pi.

Simple answer: at present we don't know. Most mathematicians who
have an opinion would likely say that the answer to "is there a maximum
possible run of a given digit in the decimal expansion of pi?" is no.
A more restrictive form of this question is "for every positive
integer k, and every integer d, 0 <= d <= 9, is there a run of exactly k
d's in the decimal expansion of pi?" I'd guess that most mathematicians
would be agnostic on that question.

That's not what you asked, though. You asked whether there is a
proof of the assertion above. I suspect that most mathematicians
who have an opinion of that more general question would say "yes,
there is a proof (that it holds or not) but we haven't found it."

In any case, a definitive answer to either of these questions would
guarantee a modicum of fame to the solver.


Regards,

Rick

p.s. Thanks, by the way, for lurking long enough to get a sense of
what constitutes a reasonable query for this ng. Your post was
a refreshing change from the off-topic and/or incoherent trolls that
waste much of our bandwidth.
From: Shepherd Moon on 22 May 2010 23:15
On May 22, 9:40 pm, Jim Burns <burns...(a)osu.edu> wrote:
> Shepherd Moon wrote:
> > Hello,
>
> > I'm a math layman with a question about pi. I don't want to clutter
> > the list with an inane question (I have seen proposals for a
> > sci.math.moderated group intended to weed out such questions, so I am
> > sensitive to this issue), but I hope my question will at least be
> > worthy of a response. I've provided citations where possible, not
> > because I pretend to any kind of expertise in mathematical research,
> > but just to show readers where I obtained my information in case the
> > information is wrong or (more likely) in case I have interpreted the
> > information incorrectly.
>
> > With that caveat, here goes.
>
> > Is it possible to prove whether or not there is a maximum possible run
> > of one digit within the decimal expansion of pi? For example, whether
> > or not it is possible to prove that a run of n 0s either cannot
> > appear, or on the contrary must appear, in pi.
>
> I think you will find the idea of a normal number useful.http://en.wikipedia.org/wiki/Normal_number
>
> In a normal number, every 1-sequence of digits appears as
> often as every other 1-sequence, every 2-sequence as often
> as every other 2-sequence, and every n-sequence of digits
> as often as every other n-sequence, for every natural n.
>
> (This sounds like I'm saying that every sequence is
> "just as probable" as every other sequence of the same
> length. That makes no sense, though, because pi
> (and every other single real number) /is not random/.
> There is a 100% probability that the tenths digit
> is '1', and so on. The definition of "normal number"
> applies a test to a sequence of digits (from a real)
> that a random sequence of digits (equi-probable and
> independent) would pass, but it doesn't care whether
> the digits came from some random process or, for
> example, a calculation of pi.)
>
> It seems to me that, if pi is normal, then there is no
> maximum run of any digit. (Nor is there any finite
> sequence of digits that does not appear.)
>
> Wikipedia gives a couple of examples of numbers proven
> to be normal, but it says that sqrt(2), pi, ln(2), and e
> have not been proven normal, although it is "strongly
> conjectured" that they are.
>
> Jim Burns- Hide quoted text -
>
> - Show quoted text -

Thank you for the reference to normal numbers. I will read about it.

Shepherdmoon
From: Peter Webb on 22 May 2010 23:21
These are very good questions. Jim Burns mentioned "normal" numbers, which
is pretty central to the questions you are asking, but has the huge
disadvantage that mathematics currently knows very little about normal
numbers. As he said, the question of whether pi is normal in base 10 is
still open.

You should also be aware that your question concerns the base 10 decimal
expansion of pi. Irrational numbers (in particular) look completely
different in different bases. Most mathematicians (I suspect) would take the
view that it would be very strange if something special happened in base 10;
why should mathematics make pi do special things in a base which corresponds
to the number of fingers human's have?

I have interspersed a few answers to some of your questions ...

"Shepherd Moon" <shepherdmoon(a)yahoo.com> wrote in message
news:55c3af11-24eb-41db-b8a4-e97393f01a19(a)z33g2000vbb.googlegroups.com...
> Hello,
>
> I'm a math layman with a question about pi. I don't want to clutter
> the list with an inane question (I have seen proposals for a
> sci.math.moderated group intended to weed out such questions, so I am
> sensitive to this issue), but I hope my question will at least be
> worthy of a response. I've provided citations where possible, not
> because I pretend to any kind of expertise in mathematical research,
> but just to show readers where I obtained my information in case the
> information is wrong or (more likely) in case I have interpreted the
> information incorrectly.
>
> With that caveat, here goes.
>
> Is it possible to prove whether or not there is a maximum possible run
> of one digit within the decimal expansion of pi?

It could be:

a) True, but impossible to prove even in principle using normal arithmetic
and algebra.
b) True, and provable
c) False.

AFAIK, nobody knows which.


> For example, whether
> or not it is possible to prove that a run of n 0s either cannot
> appear, or on the contrary must appear, in pi.
>
> I understand from basic Google searches (Wikipedia, etc.) that pi has
> been proven to be an irrational number[1]. Because pi is irrational,
> it should have the property of irrational numbers that its decimal
> expansion neither terminates (as in 5/8 = 0.625000...) nor repeats (as
> in 1/9 = 0.111...).[2]
>

Correct. Note that this is true in whatever base you use; 5/8 in base 117
either repeats or terminates.

> I can see that in the case of 0.111.. there is no limit to the run of
> 1s since the sequence goes on without end.
>
> But I guess I see even as I write this that although 0.625000...
> terminates, it also has no maximum run of 0s, since (if I'm correct)
> there is an infinite number of 0s in that decimal expansion.

You can consider 5/8 to either terminate or repeat, depending on whether you
consider it as .625 or .625000...

This is a matter of human definition and our choice of symbols to represent
numbers, and not mathematics.


> It's not
> as if the first 3 digits, 6, 2, and 5, can be "subtracted" from the
> infinity of 0s that follows. As a side question, is 2.5/3 = 0.8333...
> considered a terminating decimal (i.e., can a decimal terminate in any
> digit, not just 0)? It seems to me that 0.8333... is indeed a
> terminating decimal, but I just want to make sure my thinking is
> correct.
>

No, not usually. A terminating number has some string of digits followed by
nothing or all zeroes, in base 10 it must be of the form k/(2^m * 5^n). In
your case, 2.5/3 = 5/6 and 6 cannot be represented as 2^m * 5^n as it is
divisible by 3.



> So perhaps pi is closer to 0.625000... in that it has more than one
> type of digit in it (all 10 decimal digits), yet may also have within
> it an infinite run of one (or maybe all) of those 10 digits.
>

No.

Pi does not contain an infinite run of any digit. No real number can have an
infinite run of more than one digit.

This is a key argument for what you are discussing, so I will give you a
sort-of proof.

Firstly, if pi (or any other Real) contained an infinite run of a single
digit, this must start at some location and continue for the entire
expansion. This would make it rational.

A similar argument will show you that no Real can contain an infinite run of
more than one number. Lets say that the first run commences at position n. A
second run wouyld have to start at some other positin, say m. But that means
that the first run is not infinite, as it goes only from n to m-1.

Making this entirely rigourous requires rigorous definition of what a Real
number is, but I hope you get the idea.


> Yet that is where my head starts to hurt, as the saying goes. Why?
> Because if pi neither terminates nor repeats, then it obviously
> doesn't "end" (as well as begin) with the same single digit repeating
> endlessly (as in 0.111...) but it also doesn't terminate in a single
> digit after a sequence of other digits (as in 0.625000...). So it
> seems to me, speaking as a naive layman, of course, that one should
> always expect a run of any single digit in pi to be interrupted by one
> of the other digits.

Yes, that is the argument I gave above.

>
> At this point, I can think of two possibilities. One is a kind of
> probabilistic explanation,

Unfortunately probabilistic arguments of this form are not proofs. When
applied to infinite sets (such as the digits in the exapnsion of pi) a
statement can be true with probability 1 but in fact be false, or true with
a probability of 0 but still in fact be true. These are not exceptions; this
is is unfortunatelyy all too common.



> to the effect that as one samples larger
> and larger expansions of pi, the probability of finding a longer run
> of a digit increases without bound. I know that sometimes these types
> of number properties are proven with reductio ad absurdum (i.e.,
> assume there is a maximum run of n 0s, then show that a run of n+1 0s
> exists). But I don't know if anyone has done that with regard to pi.
> Another possibility is based on one I read on another web site (I
> can't recall the link now) that employs the transfinite numbers of
> Cantor. I forget the exact explanation, but I think it involved levels
> of infinity, but I can't recall the details. Perhaps others know this
> approach and can explain it and let me know whether it addresses the
> issue of a maximum possible run of one digit within pi.
>

No, it doesn't, at least not yet. This would be speculation on the part of
the authors.

Most (practically all) mathematics can be constructed from a very small
number of axioms, and this sometimes called "normal arithmetic" (and the
word "normal" is unrelated to the usage above and by Jim Burns). There are
some statements which can be shown be true by adding in additional axioms,
but in practice these are very uncommon. Transfinite numbers are not part of
"normal arithmetic", but adding them in allows some additional statements
which can be stated in normal arithmetic and are true in normal arithmetic
but cannot be proved in normal arithmetic, but can be proved with the
additional axioms.

This stuff is well beyond your current level of mathematical sophistication,
but try Googling "Goodstein sequences" or "transfinite recursion" to get an
idea.

BTW, you say this stuff makes your "head spin". Cantor in fact died in an
insane asylum. This is seriously head-spinning stuff

> I suppose there is a third possibility - that each digit does indeed
> have a maximum possible run in pi. But my math intuition, feeble as it
> is, tells me that such a maximum run seems unlikely in an irrational
> number.
>

Many irrational numbers have this property that there is a maximum run of
one or all digits. Despite the fact that the probability of this occuring is
0. Most mathematicians would share your untuition that this is not true of
pi.


> Well, I hope this post hasn't seemed too much like rubbish or a waste
> of time. I look forward to any help others can give in helping me
> understand whether or not this type of proof is

(a) impossible, (- possibly true)

(b) possible but not yet done, (- possibly true)

or (c) possible and already done, (AFAIK, definitely false)

> I think
> it is a fascinating question, though maybe that's because I haven't
> learned enough of the necessary math to answer the question.
>

It is a fascinating question, and nobody else on the planet has learned
enough of the neccessary maths to answer it either, so don't feel too bad.


> Thanks in advance for your time and for any help.
>
> Shepherdmoon
>

My pleasure.

Peter Webb

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