Digits of an irrational fraction
dkomo
1. Are the digits of any irrational fraction unncorrelated? Has this
been proven or only statistically tested? What about pi and e?
2. Express the digits of an irrational fraction as a binary bit string
(i.e. write the number in base 2). Take the library of all the books
ever written and convert it to a bit string. Does any arbitrary
irrational fraction contain this library? If so, does a copy of the
library appear an infinite number of times within the number?
Gerald
"dkomo" <dkom...@comcast.net> wrote in message
news:VdSdnc_7S7FYjUzY...@comcast.com...
> Some questions regarding irrational fractions:
>
> 1. Are the digits of any irrational fraction unncorrelated? Has this been
> proven or only statistically tested? What about pi and e?
what do you mean by "uncorrelated"? Random? Corrolation requires a frame
size.
> 2. Express the digits of an irrational fraction as a binary bit string
> (i.e. write the number in base 2). Take the library of all the books ever
> written and convert it to a bit string. Does any arbitrary irrational
> fraction contain this library? If so, does a copy of the library appear
> an infinite number of times within the number?
no.
[Mr.] Lynn Kurtz
wrote:
If you mean does every irrational decimal contain this library, the
answer is no. Just take .01001000100001...
If you mean does there exist an irrational decimal which does, the
answer is yes. Just append the above string to the (finite) library
string.
--Lynn
Dik T. Winter
> 1. Are the digits of any irrational fraction unncorrelated? Has this
> been proven or only statistically tested? What about pi and e?
Irrational fraction? I think you mean irrational number. But most about
that is unproven. Have a look at "normal numbers". For instance:
<http://mathworld.wolfram.com/NormalNumber.html>.
> 2. Express the digits of an irrational fraction as a binary bit string
> (i.e. write the number in base 2). Take the library of all the books
> ever written and convert it to a bit string. Does any arbitrary
> irrational fraction contain this library? If so, does a copy of the
> library appear an infinite number of times within the number?
If a number is normal to base 2, the answer to both questions is yes.
Champernowne's number will contain it, and will contain it infinitely
many times.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
matt271...@yahoo.co.uk
"Irrational fraction" sounds like a contradiction. If you just mean
"irrational number" then you could have something like
0.101001000100001000001... (either decimal or binary), the digits of
which have a very strong pattern.
James Waldby
> Some questions regarding irrational fractions:
>
> 1. Are the digits of any irrational fraction unncorrelated? Has this
> been proven or only statistically tested? What about pi and e?
The digits of irrational numbers like 0.122333444455555... and
some numbers other posters mentioned obviously are autocorrelated.
Look up the "normal number" concept, eg at
http://mathworld.wolfram.com/NormalNumber.html or
http://www.mathpages.com/home/kmath519.htm and you
will see that while pi and e and many other well-known irrational
constants haven't been proved normal, there is little or no evidence
to the contrary within their first few million digits.
> 2. Express the digits of an irrational fraction as a binary bit string
> (i.e. write the number in base 2). Take the library of all the books
> ever written and convert it to a bit string. Does any arbitrary
> irrational fraction contain this library? If so, does a copy of the
> library appear an infinite number of times within the number?
It probably is the case that almost all [*] irrationals contain
infinitely many times any arbitrary bit string B, but of course there
still are uncountably many irrationals that don't contain B, if B is
long enough.
[*] In the 3rd sense given at http://en.wikipedia.org/wiki/Almost_all
-jiw
Virgil
James Waldby <n...@no.no> wrote:
> It probably is the case that almost all [*] irrationals contain
> infinitely many times any arbitrary bit string B, but of course there
> still are uncountably many irrationals that don't contain B, if B is
> long enough.
It is necessarily the case that for any given finite sequence of decimal
digits, "most" irrational numbers do not contain that sequence anywhere
in their decimal expansions.
r.e.s.
> It is necessarily the case that for any given finite sequence of
decimal
> digits, "most" irrational numbers do not contain that sequence
anywhere
> in their decimal expansions.
Just the opposite is true, because almost all reals are normal
(i.e., the set of non-normal reals has Lebesgue measure zero).
Peter Webb
"r.e.s." <r...@ZZmindspring.com> wrote in message
news:a%9Ah.938$x74...@newsread4.news.pas.earthlink.net...
However, and slightly more naively, if we use the number of sequences to
mean the cardinality of the set of sequences, the number of sequences
containing a string is the same as the number of sequences not containing
the string, as both can be bijected with R.
r.e.s.
> "r.e.s." <r...@ZZmindspring.com> wrote ...
>> "Virgil" <vir...@comcast.net> wrote ...
>>> It is necessarily the case that for any given finite sequence
>>> of decimal digits, "most" irrational numbers do not contain
>>> that sequence anywhere in their decimal expansions.
>>
>> Just the opposite is true, because almost all reals are normal
>> (i.e., the set of non-normal reals has Lebesgue measure zero).
>
> However, and slightly more naively, if we use the number of sequences to
> mean the cardinality of the set of sequences, the number of sequences
> containing a string is the same as the number of sequences not containing
> the string, as both can be bijected with R.
Yes, it's interesting to look at this from various angles.
From the cardinality POV, in the present case one can't
say "most" or "almost all" have the property of interest,
since *both* the set and its complement are uncountable.
A set of reals in the unit interval such that the digit '7'
(say) occurs in no decimal representation of any element,
is a set comparable to the Cantor ternary set in being
uncountable yet having Lebesgue measure 0.
Dave L. Renfro
> Look up the "normal number" concept, eg at
> http://mathworld.wolfram.com/NormalNumber.html or
> http://www.mathpages.com/home/kmath519.htm and you
> will see that while pi and e and many other well-known
> irrational constants haven't been proved normal, there
> is little or no evidence to the contrary within their
> first few million digits.
[snip part of an earlier poster's comments]
> It probably is the case that almost all [*] irrationals
> contain infinitely many times any arbitrary bit string B,
> but of course there still are uncountably many irrationals
> that don't contain B, if B is long enough.
>
> [*] In the 3rd sense given at http://en.wikipedia.org/wiki/Almost_all
The property you're describing is shared by almost all real
numbers in both the Baire category sense and the Lebesgue
measure sense. In fact, its complement is even smaller
than first-category-and-measure-zero, being a countable
union of lower porous sets. Note there is a huge difference
between the notion of a normal number (to base 10) and the
property you're talking about, since the set of normal numbers
is large in one way (Lebesgue measure) but small in another
way (Baire category), whereas the set you're talking about
(sometimes called the "absolutely disjunctive" real numbers,
for which you can google the phrase I put in quotes) is large
in both the Lebesgue measure sense and in the Baire category
sense (and larger still than what the conjunction of these
two notions could allow for). For more details, see my post at
http://groups.google.com/group/sci.math/msg/4ec315328c1afdb8
Dave L. Renfro
dkomo
> In article <VdSdnc_7S7FYjUzY...@comcast.com> dkomo <dkom...@comcast.net> writes:
> > 1. Are the digits of any irrational fraction unncorrelated? Has this
> > been proven or only statistically tested? What about pi and e?
>
> Irrational fraction? I think you mean irrational number.
Yep, sorry about the confusion. I was using computer science
terminology instead of math terminology, where, for example
x = sqrt(7) = 2.645751...
and 2 is the integer part given by functions such as int(x)
and .645751... is the fractional part given by functions such as frac(x)
But most about
> that is unproven. Have a look at "normal numbers". For instance:
> <http://mathworld.wolfram.com/NormalNumber.html>.
>
> > 2. Express the digits of an irrational fraction as a binary bit string
> > (i.e. write the number in base 2). Take the library of all the books
> > ever written and convert it to a bit string. Does any arbitrary
> > irrational fraction contain this library? If so, does a copy of the
> > library appear an infinite number of times within the number?
>
> If a number is normal to base 2, the answer to both questions is yes.
> Champernowne's number will contain it, and will contain it infinitely
> many times.
Now my next question is does an irrational number *have* to be normal in
order to able to contain any arbitrary finite length string, and contain
infinitely many copies of it?
Obviously, if the number *is* normal, that would rule out such specially
constructed numbers such as 0.101001000100001000001... that don't have
the containment property.
Arturo Magidin
dkomo <dkom...@comcast.net> wrote:
>Now my next question is does an irrational number *have* to be normal in
>order to able to contain any arbitrary finite length string, and contain
>infinitely many copies of it?
No.
The number
0.01234567891011121314151617181920...
made up by concatenating the decimal expressions of the natural
numbers is irrational, and contains any arbitrary finite length string
infinitely many times. However, it is not a normal number.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================
Arturo Magidin
magidin-at-member-ams-org
dkomo
Can we then say that if we pick an irrational number at random, the
probability that its digits contain any arbitrary finite length bit
string is 1?
Dave L. Renfro
> http://en.wikipedia.org/wiki/Almost_all
I glanced at this entry just now, and the use
of the "for all" symbol superscripted with oo
(infinity) is not standard. At least, the usage given
at this web page is not what I've seen the symbol used
for. I've always seen it used for "all but finitely many",
whereas the wiki entry has it being used for a subsequence
of the positive integers that is asymptotically equal to 1.
However, maybe the symbol "for all ^ oo" means something
different in number theory than in other areas of mathematics.
Any number theorists know about this?
Dave L. Renfro
Dave L. Renfro
> whereas the wiki entry has it being used for a subsequence
> of the positive integers that is asymptotically equal to 1.
"... that has asymptotic density 1" (and yes, I know there
are different notions of this), but I suppose everyone knew
what I meant.
Dave L. Renfro
David Marcus
Not only can we, but we just did. "almost all" = "probability 1". Just
different ways of saying the same thing.
--
David Marcus
r.e.s.
Yet another way ...
If X is an infinite binary sequence of i.i.d. uniform
random variables, and s is a finite binary string, then,
with probability 1, 0.X is irrational and s occurs in X
infinitely often.
r.e.s.
To be clear ... That's another way, that is, of answering,
but not of saying the "same" thing -- because the mere
occurrence of the string is a property much weaker than
normality (as David Renfro mentions).
Herman Rubin
There is no problem in producing an irrational decimal,
binary, whatever which contains all finite libraries
infinitely often.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
David Sweeney
This is uncanny! I was thinking about something similar myself, a few weeks
ago.
Let x be a normal number (such as Champernowne's number). Then if you
convert any book, eg, Hamlet, into a numerical notation, that sequence of
numbers will occur somewhere within the decimal expansion of x. It follows
that the text of every book ever written will occur at least once in the
decimal expansion of x. This is incredible! x is just one real number, and
yet it contains a (countably) infinite amount of information!
It is rather like the short story "The Library of Babel" by Jorge Luis
Broges, which features a library containing every possible book. (Well, all
the books are of the same format, eg, 200 pages, 30 lines per page, 80
characters per line. But it would require an absolutely enormous library to
accommodate them all. The Library would have to be several billion cubic
light-years in volume...)
Frederick Williams
>
> ... [Champernowne's number] is just one real number, and
> yet it contains a (countably) infinite amount of information!
Which, sadly you can't extract.
--
Remove "antispam" and ".invalid" for e-mail address.
We have lingered in the chambers of the sea
By sea-girls wreathed with seaweed red and brown
Till human voices wake us, and we drown.
Mike
>
>
> It is rather like the short story "The Library of Babel" by Jorge Luis
> Broges, which features a library containing every possible book. (Well, all
> the books are of the same format, eg, 200 pages, 30 lines per page, 80
> characters per line. But it would require an absolutely enormous library to
> accommodate them all. The Library would have to be several billion cubic
> light-years in volume...)
>
Mike
David Marcus
> Let x be a normal number (such as Champernowne's number). Then if you
> convert any book, eg, Hamlet, into a numerical notation, that sequence of
> numbers will occur somewhere within the decimal expansion of x. It follows
> that the text of every book ever written will occur at least once in the
> decimal expansion of x. This is incredible! x is just one real number, and
> yet it contains a (countably) infinite amount of information!
Perhaps "infinite amount of information" is well defined, but I doubt
one can apply the adjective "countably" to it.
--
David Marcus
David Sweeney
"David Marcus" <David...@alumdotmit.edu> wrote in message
news:MPG.204419e5...@news.rcn.com...
Yes, good point! I've just read up on information theory and concluded the
following:
Since x is normal, at a given point in its decimal expansion, every digit in
the range 0 to 9 is equally likely to come up. So the information in one
digit is log(10) (where log is to the base 2). This equals approx 3.3 bits
of information. Every digit adds another 3.3 bits of info. Of course, this
tends to infinity, in the way that a real function f(t) = kt tends to
infinity as t tends to infinity.
So yes, it's a case of "infinity" as in "a real variable tending to
infinity", rather than the cardinality of the set of natural numbers that
Cantor was talking about. Thanks for pointing that out.
David Sweeney
"Mike" <m.fee@iirrll..ccrrii..nnzz> wrote in message
news:MPG.2044d0f0...@news.fx.net.nz...
> In article <8c2dnchRpvEdpETY...@bt.com>,
> David.S...@btinternet.com says...
>>
>>
>> It is rather like the short story "The Library of Babel" by Jorge Luis
>> the books are of the same format, eg, 200 pages, 30 lines per page, 80
>> characters per line. But it would require an absolutely enormous library
>> to
>> accommodate them all. The Library would have to be several billion cubic
>> light-years in volume...)
>>
> True only for very large values of 'several'.
>
> Mike
Yes, I suppose so.
Perhaps the Library would be 10^100 cubic light years in volume, or even:
10^(10^100) cubic light years.
I worked out an estimate once, but I've forgotten the exact figure I arrived
at.
PS I meant to put Borges (and not Broges!)