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Thread: Does Pi contain Pi?

  1. #31
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    Suppose a random number generator outputs random digits from 0 to 9. It is obvious that any given finite sequence of digits has a probability of greater than 0 to occur and it will occur infinite times in infinite number of trials.
    Suppose a 2nd machine parses the output of the above mentioned machine for occurrences of the sequence 1919 and flips one of the 4 digits to 1 or 9 in random such that sequence 1919 never occurs (obviously keeping track of the immediately before digits not forming 1919). It would be logical that the probability of occurring any of the digits on the output of the second machine will be uniform and 1 in 10. So with exception of the sequence 1919 and is derivatives the sequence would satisfy the definition of a normal number, but would never include the sequence 1919.
    Just trying to wrap my head around the concept of infinite random sequences.

  2. #32
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    Quote Originally Posted by a1call View Post
    Suppose a 2nd machine parses the output of the above mentioned machine for occurrences of the sequence 1919 and flips one of the 4 digits to 1 or 9 in random such that sequence 1919 never occurs (obviously keeping track of the immediately before digits not forming 1919). It would be logical that the probability of occurring any of the digits on the output of the second machine will be uniform and 1 in 10. So with exception of the sequence 1919 and is derivatives the sequence would satisfy the definition of a normal number, but would never include the sequence 1919.
    Not normal.

    Wikipedia: Normal number

    In mathematics, a normal number is a real number whose infinite sequence of digits in every base b is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b, also all possible b^2 pairs of digits are equally likely with density b^−2, all b^3 triplets of digits equally likely with density b^−3, etc.
    Your proposed sequence would fail to produce all 10^4 quads with equal likelihood.
    0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 ...
    Skepticism enables us to distinguish fancy from fact, to test our speculations. --Carl Sagan

  3. #33
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    Bad wording on my side.
    Not normal and not truly random but with equal distribution density of any of the 10 digits, and with no repeating pattern.
    My point is that the definition of normal is to vague to eliminate the possibility of missing sequences. The 1919 example can be generalized indefinitely to make it possible to have equal/normal density in absence of true randomness (hence possibility of missing sequences in an arbitrarily near normal number.).

    ETA To clarify, if an infinite sequence includes every finite sequence, possible, an equal sequence with any given finite sequence altered in a way not to change the density of any other (non inclusive and non complementary of the above given finite sequence), will be identical in density of all other sequences. The definition is somewhat circular. It's like designing defining a red ball as any ball that is red.
    Last edited by a1call; 2017-Oct-04 at 03:02 AM.

  4. #34
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    Quote Originally Posted by a1call View Post
    My point is that the definition of normal is to vague to eliminate the possibility of missing sequences. The 1919 example can be generalized indefinitely to make it possible to have equal/normal density in absence of true randomness (hence possibility of missing sequences in an arbitrarily near normal number.).

    ETA To clarify, if an infinite sequence includes every finite sequence, possible, an equal sequence with any given finite sequence altered in a way not to change the density of any other (non inclusive and non complementary of the above given finite sequence), will be identical in density of all other sequences. The definition is somewhat circular. It's like designing defining a red ball as any ball that is red.
    I am struggling. What I see you doing:
    1) using a relaxed definition of normal, perhaps paraphrased from an earlier casual posting, that appears to really be a definition for simply-normal (roughly: a tendency toward 1/10 chance of each digit 0-9).
    2) generating a 1919-free sequence that is simply-normal, but is not normal.
    3) and then concluding that the definition of normal is vague, or not useful, or circular.

    Your clarification doesn't clarify, for me. In your contrived sequence, 1919 does not appear, and that would make the sequence not normal -- even though it is simply-normal as you intended. How does that make the definition of normal undesirable?

    (Definitions from Wikipedia: Normal number, quoted posts back. Simply-normal is defined here, too.)

    Granted the definition of normal does not give you a method to determine if a given sequence is normal. It can make some non-normal sequences detectable. That all makes it interesting. Is pi normal? Who knows?
    0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 ...
    Skepticism enables us to distinguish fancy from fact, to test our speculations. --Carl Sagan

  5. #35
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    Quote Originally Posted by a1call View Post
    Bad wording on my side.
    Not normal and not truly random but with equal distribution density of any of the 10 digits, and with no repeating pattern.
    My point is that the definition of normal is to vague to eliminate the possibility of missing sequences. The 1919 example can be generalized indefinitely to make it possible to have equal/normal density in absence of true randomness (hence possibility of missing sequences in an arbitrarily near normal number.).

    ETA To clarify, if an infinite sequence includes every finite sequence, possible, an equal sequence with any given finite sequence altered in a way not to change the density of any other (non inclusive and non complementary of the above given finite sequence), will be identical in density of all other sequences. The definition is somewhat circular. It's like designing defining a red ball as any ball that is red.
    I too struggled with what you were getting at. Your definition of the modified sequence would mean that the sequence 31919 would also not occur, right? Or any sequence of the form S1919T where S and T are finite sequences of any possible length.

  6. #36
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    Unfortunately, all I can do is to just repeat what was already mentioned, to express my admittedly insufficient comprehension of the relevance of "Normal Number" concept to existence or lack thereof any given finite sequence in pi. I will try to do a more organized recap.


    • Consider the Real Number A:
      • It has infinite number of decimal digits
      • Its decimal expansion includes all the digits 0-9
      • It is not a Normal-Number
        • It has a bias such that some digits occur significantly more often than others

      • It includes all the possible finite combinations of digits 0-9
      • The Die equivalent example would be a loaded Die landing 100 consecutive 1's in-spite of this being a more rare occurrence of it landing 100 consecutive 6's
      • Number A is not a Normal-Number, yet it includes all the possible finite sequences
        • Not being a Normal-Number is not sufficient condition for it not to include all possible finite sequences of digits from 0-9


    • My understating of the definition of a Normal-Number is a Real Number B Which includes all the possible sequences of digits in all bases. Additionally all these sequences occur with equal frequency to their counter parts with equal number of digits in any particular base
      • To me, this seems like a vague/insufficient/circular definition
      • It does not identify/express any characteristics of the Normal-Numbers which would distinguish an infinite sequence which includes a given finite sequence from one which does not, other than the fact that it does not.
      • https://en.wikipedia.org/wiki/Normal_number
        • In mathematics, a normal number is a real number whose infinite sequence of digits in every base b[1] is distributed uniformly
          • I think there exists a base for any given number (Hence "Normal" or lack thereof), where there is a bias towards the occurrence of some digits over others
            • For any given number there exists a base where finite and/or infinite sequence of digits is-not distributed uniformly
              • In case of finite digits any base of a root of the number greater than 1
                • Base square, cube, ... root of any number is, 100, 1000, .....

              • In case of infinite digits, i think it can be shown that there are polynomials of infinite order whose roots would be bases where the digits of a given number would not be distributed uniformly






  7. #37
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    Quote Originally Posted by Jens View Post
    I'm going to sue you for making my brain hurt.
    No need. Just take a couple of 'pi'lls. As'pi'rtn shoud do nicely. And if it upsets your stomach, have a slice of . . .

    Alright, alright. I promise I won't posst again tonight, pi golly.

  8. #38
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    There is some resolution.
    Please see axn's post number 9:
    http://www.mersenneforum.org/showthr...261#post469261

  9. #39
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    Did anyone say only a normal number includes all possible finite sequences, that normal was necessary? I think it was only offered as being sufficient, and that pi, like almost all irrationals, was likely, but not proven to be, normal. I'm pretty sure I did not say it was necessary.

    I can imagine many ways to construct an infinite sequence that contained all possible finite sequences, and not be normal: say, just make methodically sure it has all 10 1-digit sequences, a string of 10 zeroes, then all 100 2-digit sequences, 100 zeroes, all 1000 3-digit sequences, as many zeroes, etc. So if you go far enough any given finite sequence is there (if not earlier). Because it contains so many 0 sequences, it is not normal.

    Sure. Some infinite sequences include all finite sequences, yet are not normal.

    Does that rehabilitate the definition of normal?
    0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 ...
    Skepticism enables us to distinguish fancy from fact, to test our speculations. --Carl Sagan

  10. #40
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    Quote Originally Posted by 01101001 View Post
    Did anyone say only a normal number includes all possible finite sequences, that normal was necessary? I think it was only offered as being sufficient, and that pi, like almost all irrationals, was likely, but not proven to be, normal. I'm pretty sure I did not say it was necessary.
    Exactly correct. If a number is normal, you can be certain that it contains all possible sequences.

    Quote Originally Posted by a1call View Post
    • I think there exists a base for any given number (Hence "Normal" or lack thereof), where there is a bias towards the occurrence of some digits over others
      • For any given number there exists a base where finite and/or infinite sequence of digits is-not distributed uniformly
        • In case of finite digits any base of a root of the number greater than 1
          • Base square, cube, ... root of any number is, 100, 1000, .....

        • In case of infinite digits, i think it can be shown that there are polynomials of infinite order whose roots would be bases where the digits of a given number would not be distributed uniformly


    I'm reasonably certain that when talking about different bases there, the author of that section of the Wiki page is assuming that the bases are integers. In most cases dealing with alternative bases, we assume a positive integer; it's technically possible to have fractional or irrational bases, but it's not generally done except when specifically exploring that. So, yes, in principle, you can say that pi is just 1.0 in base pi, and you can always pick some arbitrary real number as a base such that any other real number can be represented as any value you want, including a terminating decimal, but that's not what they're referring to here.

    It might be possible to show that a real number contains all possible sequences even if not normal (such a number can certainly exist), but in most cases, unless it's been carefully constructed (like 01101001's example), such a proof would probably be even more elusive than showing normality. In the case of pi, we think that it is probably normal, so that would make it certain that it contains all possible finite sequences, but that has not been proven to be the case.
    Conserve energy. Commute with the Hamiltonian.

  11. #41
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    Quote Originally Posted by 01101001 View Post
    Did anyone say only a normal number includes all possible finite sequences, that normal was necessary? I think it was only offered as being sufficient, and that pi, like almost all irrationals, was likely, but not proven to be, normal. I'm pretty sure I did not say it was necessary.
    That was a recap of my post 30. I did not claim or meant to imply that you or anyone else claimed otherwise. I brought it up because I thought it was relevant to the discussion, and if I am not mistaking no one else had pointed it out plainly, before.

  12. #42
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    Quote Originally Posted by Grey View Post

    I'm reasonably certain that when talking about different bases there, the author of that section of the Wiki page is assuming that the bases are integers.
    Yes you are correct as was pointed out by axn on the other board. However the wilki article is problematic. The information seems to be only included as a tooltip (Which would be hidden from tablet/smart-phone visitors). it is very unusual for a mathematical article to refer to bases of only natural numbers as "or any other base", I have generally found mathematicians to be a very exhaustive (not to mention exhausting) individuals.

    Since your link was my 1st encounter of the concept, It is not out of place, that I took the text literally.


    Intuitively this means that no digit, or (finite) combination of digits, occurs more frequently than any other, and this is true whether the number is written in base 10, binary, or any other base.
    Last edited by a1call; 2017-Oct-06 at 12:07 AM.

  13. #43
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    Quote Originally Posted by a1call View Post
    Yes you are correct as was pointed out by axn on the other board. However the wilki article is problematic. The information seems to be only included as a tooltip (Which would be hidden from tablet/smart-phone visitors). it is very unusual for a mathematical article to refer to bases of only natural numbers as "or any other base", I have generally found mathematicians to be a very exhaustive (not to mention exhausting) individuals.
    Clearly, you must be referring to wiki authors, not mathematicians!
    Since your link was my 1st encounter of the concept, It is not out of place, that I took the text literally.
    A lesson that has been repeated nearly as often as 1919, wikis should never be taken as a primary source!

  14. #44
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    [Duplicated and enhanced below. Lost track of state.]
    Last edited by 01101001; 2017-Oct-06 at 04:19 PM.
    0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 ...
    Skepticism enables us to distinguish fancy from fact, to test our speculations. --Carl Sagan

  15. #45
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    I had to reread the article because I never considered non-integer bases in the context. I thinks it's because, even neglecting the footnote, that b is immediately used in a counting sense "b digit values", then "b^2 pairs" and "b^3 triples".

    But, yeah, it would have been clearer writing to start out with "positive integer base b" instead of using the footnote to clarify, since it clearly caused at least one reader to go astray. Wait -- it has to be "integer-greater-than-1 base b". Eww. I'm beginning to appreciate the use of a footnote.

    Writing is hard.

    Quote Originally Posted by grapes View Post
    A lesson that has been repeated nearly as often as 1919, wikis should never be taken as a primary source!
    Hah! You know, there was no year 1919. It went: 1918, 1119, 1920. True fact.
    0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 ...
    Skepticism enables us to distinguish fancy from fact, to test our speculations. --Carl Sagan

  16. #46
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    Honestly, I'm okay with the omission there. Even when I took a course specifically on number theory (because it was fun!), we only touched on non-integral bases just to point out that it was technically possible. But I really don't think they are played with often, even by mathematicians. That said, grapes is right: Wikipedia can often provide a nice initial overview of a subject you're unfamiliar with, but it's a good idea to not assume that all the details are always precise.
    Conserve energy. Commute with the Hamiltonian.

  17. #47
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    Quote Originally Posted by Grey View Post
    Honestly, I'm okay with the omission there. Even when I took a course specifically on number theory (because it was fun!), we only touched on non-integral bases just to point out that it was technically possible. But I really don't think they are played with often, even by mathematicians. That said, grapes is right: Wikipedia can often provide a nice initial overview of a subject you're unfamiliar with, but it's a good idea to not assume that all the details are always precise.
    Well, my antipathy to Wikipedia has perhaps been aired enough around these parts, but I think this is an important point.
    One often sees surveys of this or that group of experts who have pronounced this or that group of Wikipedia pages as being textbook-accurate. But experts are the wrong people to ask, because they know the subject already, and they simply make the necessary assumptions and interpretations as they go along. Writing good unambiguous textbook or encyclopaedia text is really, really hard, having multiple people editing each other's work ad lib. makes it even harder, and the right metric by which to judge Wikipedia pages is how often beginners get the wrong end of the stick, or leave no better informed than when they arrived.

    Grant Hutchison

  18. #48
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    Quote Originally Posted by Grey View Post
    Honestly, I'm okay with the omission there.
    Well I'm not. I consider the article erogenous.
    In number theory exotic bases is quite valid. Even factorial and primorial bases have been researched.

    http://mathworld.wolfram.com/Base.html

    Wikipedia article error can be traced March 2010 as "correction" made by Trovatore:


    22:32, 5 March 2010Trovatore (talk | contribs)‎ . . (21,631 bytes) (+69)‎ . . ("integer base" is misleading -- we're not considering negative or zero bases, or even unary. "Base" by default is a natural number at least 2 -- put this in a footnote.) (undo)

    So it's not that the Mathematician authors are non existent on wikipedia, they just happen to be a minority.


    Wikipedia is obviously a very valuable reference source, but it is not without problems, Sadly, Wikipedia is responsible for many erroneous concepts considered as absolute facts by the masses. It is considered infallible by many, even when there are oblivious errors as there is in this case.

  19. #49
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    Quote Originally Posted by a1call View Post
    Well I'm not. I consider the article erogenous.
    Damn you, predictive text?

    Grant Hutchison

  20. #50
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    Quote Originally Posted by grant hutchison View Post
    Damn you, predictive text?

    Grant Hutchison


    Was just going to proofread my post.

    It could and has been worst.

    At least most people wouldn't have known what it meant, until you pointed it out. I had to Google it myself.

    Anywho,

    Here is the version just before the "correction":
    https://en.wikipedia.org/w/index.php?title=Normal_number&diff=347989600&oldid =347987529



    m (we can always find non-integer bases in which the digits aren't distributed uniformly (a trivial example is base-that-number-itself))


    ("integer base" is misleading -- we're not considering negative or zero bases, or even unary. "Base" by default is a natural number at least 2 -- put this in a footnote.)
    Last edited by a1call; 2017-Oct-06 at 11:59 PM.

  21. #51
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    Quote Originally Posted by a1call View Post
    Sadly, Wikipedia is responsible for many erroneous concepts considered as absolute facts by the masses. It is considered infallible by many, even when there are oblivious errors as there is in this case.
    I hate those oblivious errors, they're the worst!

    But actually, I really don't think Wiki is responsible for widespread misconceptions by the masses-- usually Wiki entries are so expert they are hardly even accessible to the masses. And the articles that are accessible are generally better researched than newspaper or magazine articles. But they do make mistakes, of course. Still, even when I'm looking up astronomy information, I go there first, and am rarely disappointed, it's typically better than university course websites.

  22. #52
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    Quote Originally Posted by Ken G View Post
    I hate those oblivious errors, they're the worst!
    I resemble that remark!
    But actually, I really don't think Wiki is responsible for widespread misconceptions by the masses-- usually Wiki entries are so expert they are hardly even accessible to the masses. And the articles that are accessible are generally better researched than newspaper or magazine articles. But they do make mistakes, of course. Still, even when I'm looking up astronomy information, I go there first, and am rarely disappointed, it's typically better than university course websites.
    I grant you, wiki pediatric articles are at least as un misleading as facebook articles, especially headlines

  23. #53
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    The Wikipedia article on Normal-Numbers seems to be an article based on Alternative-Facts. it fails to recognize that "Any other base" includes non integer bases. Furthermore it fails to distinguish between Normal-numbers to a given base with Absolutely-Normal-Numbers which have 2 different (But not necessarily exclusive) definitions.

    On the same subject from a more reliable Math reference source (but by no means more infallible, since it too contains contradictory definitions):



    A normal number is an irrational number for which any finite pattern of numbers occurs with the expected limiting frequency in the expansion in a given base (or all bases).
    ....
    A number that is normal in base- is often called -normal.
    ....
    A number that is -normal for every , 3, ... is said to be absolutely normal (Bailey and Crandall 2003).
    ...
    Determining if numbers are normal is an unresolved problem. It is not even known if fundamental mathematical constants such as pi (Wagon 1985, Bailey and Crandall 2003), the natural logarithm of 2 (Bailey and Crandall 2003), Apéry's constant (Bailey and Crandall 2003), Pythagoras's constant (Bailey and Crandall 2003), and e are normal, although the first 30 million digits of are very uniformly distributed (Bailey 1988).

    ....

    http://mathworld.wolfram.com/NormalNumber.html

    A real number that is -normal for every base 2, 3, 4, ... is said to be absolutely normal. As proved by Borel (1922, p. 198), almost all real numbers in are absolutely normal (Niven 1956, p. 103; Stoneham 1970; Kuipers and Niederreiter 1974, p. 71; Bailey and Crandall 2002).


    http://mathworld.wolfram.com/AbsolutelyNormal.html

    It leaves one scratching one's head if a number that is b-normal to some integer base b, will or will not be always normal in all other positive integer bases greater than 1.

    It seems to me the authors don't really know either.
    Last edited by a1call; 2017-Oct-08 at 11:46 PM.

  24. #54
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    Some more insights:

    “A number which is normal in any scale is called absolutely normal. The
    existence of absolutely normal numbers was proved by E. Borel. His proof is
    based on the measure theory and, being purely existential, it does not provide
    any method for constructing such a number. The first effective example of an
    absolutely normal number was given by me in the year 1916. As was proved by
    Borel almost all (in the sense of measure theory) real numbers are absolutely
    normal. However, as regards most of the commonly used numbers, we either
    know them not to be normal or we are unable to decide whether they are normal
    or not. For example we do not know whether the numbers2, π, e are normal
    in the scale of 10. Therefore, though according to the theorem of Borel almost
    all numbers are absolutely normal, it was by no means easy to construct an
    example of an absolutely normal number. Examples of such numbers are fairly
    complicated.”
    M. W. Sierpinski.
    Elementary Theory of Numbers
    , Warszawa, 1964, p. 277.



    2. Another example is Champernowne’s number 0.123456789101112131415. . .
    which has all natural numbers in their natural order, written in base 10. It can be proved that
    Champernowne’s number is normal to base 10, but not in some other bases

    http://www.glyc.dc.uba.ar/santiago/papers/absnor.pdf

  25. #55
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    Quote Originally Posted by wd40 View Post
    In this universe Pi is an irrational and a transcendental number.

    If Pi had been exactly equal to 3, would the history of the development of mathematics & man's perception of infinity have plaued out significantly differently in any way from OTL?
    It certainly would have seriously affected mail sorting machines...
    "The difference between theory and practice is that in theory, there's no difference."

    "Aikido: the art of hitting people with planets."

  26. #56
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    Quote Originally Posted by a1call View Post
    The Wikipedia article on Normal-Numbers seems to be an article based on Alternative-Facts. it fails to recognize that "Any other base" includes non integer bases. Furthermore it fails to distinguish between Normal-numbers to a given base with Absolutely-Normal-Numbers which have 2 different (But not necessarily exclusive) definitions.
    You can help improve it!
    On the same subject from a more reliable Math reference source (but by no means more infallible, since it too contains contradictory definitions):
    Why do you say it is contradictory?
    It leaves one scratching one's head if a number that is b-normal to some integer base b, will or will not be always normal in all other positive integer bases greater than 1.

    It seems to me the authors don't really know either.
    It looks like you've answered your own question, below.

    Quote Originally Posted by a1call
    2. Another example is Champernowne’s number 0.123456789101112131415. . .
    which has all natural numbers in their natural order, written in base 10. It can be proved that
    Champernowne’s number is normal to base 10, but not in some other bases

  27. #57
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    Quote Originally Posted by grapes View Post
    You can help improve it!

    Why do you say it is contradictory?

    It looks like you've answered your own question, below.
    Well one might be very rightfully inclined to think so, until you encounter quotes like:

    Champernowne's number0.1234567891011121314151617...,obtained by concatenating the decimal representations of the natural numbers in order, is normal in base 10, but it might not be normal in some other bases.
    https://en.m.wikipedia.org/wiki/Normal_number
    Along with numerous similar statements in a very long and vague article.

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    Quote Originally Posted by a1call View Post


    Was just going to proofread my post.

    It could and has been worst.

    At least most people wouldn't have known what it meant, until you pointed it out. I had to Google it myself.

    Anywho,

    Here is the version just before the "correction":
    https://en.wikipedia.org/w/index.php?title=Normal_number&diff=347989600&oldid =347987529







    I think most people here know what "erogenous" means. :blush:
    We won't judge you, though.

    Information about American English usage here and here. Floating point issues? Please read this before posting.

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    Quote Originally Posted by swampyankee View Post
    I think most people here know what "erogenous" means. :blush:
    We won't judge you, though.
    Yeah, I even know a song that uses the word ("Counting Out Time", on The Lamb by Genesis).

    ETA: Dumb song on a great album, IMO.
    As above, so below

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    Forgive my ignorance, but it seems that pi has not been proved to be normal, despite an overwhelmingly strong likelihood of it being so.

    Of course pi isn't the only irrational transcendental. There are heaps of them. How about Euler's number e, or the golden ratio phi? Have these been proven normal to the extent that my assertion that they both contain the entire history of the Earth coded in ASCII characters is correct?

    clop

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