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Is pi a normal number?

  1. Dec 17, 2013 #1
    In mathematics, a normal number is a real number whose infinite sequence of digits is distributed uniformly in the sense that each of the digit values has the same frequency, also all digits are equally likely. Wikipedia says that it is widely believed that pi is normal, but a proof remains elusive.

    What about my "proof"? It seems impossible to me that by counting the endless digits of pi we could end up at the following result:
    pi=3.1415926........................9999999999999999999999999.....endless string of 9's

    if this would be possible then pi could have an exact value, because we could round up the endless
    string of 9's into 100000....... And pi would have an exact value, containing a finite number
    of decimals, because we could ignore the last 0's, in other words we could cut the decimals of pi
    without making an error.

    It is rather more likely that pi really is a normal number, and the digits of pi are "random":
    http://info.sjc.ox.ac.uk/users/gualtieri/Is Pi normal.htm
     
  2. jcsd
  3. Dec 17, 2013 #2

    adjacent

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    Pi is a type of number called an irrational number.(In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers and b is non-zero.)The wikipedia topic about Irrational numbers exactly states that:
     
  4. Dec 17, 2013 #3

    adjacent

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    What's the need of poof for pi being a normal number?Pi is already an irrational number.It can't be a normal number.Finding a proof that Pi is not irrational seems absurd
     
  5. Dec 17, 2013 #4

    D H

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    7777777, you appear to be conflating non-normal with rational. Just because a number is not normal does not mean it is rational. Consider the following base 10 number: 1.01001000100001000001 … This is not rational, but it also is not normal.


    adjacent, you have it exactly backwards. All normal numbers are not rational.
     
  6. Dec 17, 2013 #5

    jbunniii

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    It is unknown whether ##\pi##, ##e##, or even ##\sqrt{2}## are normal numbers. Almost all real numbers are normal numbers (in the sense that the set of real numbers that are not normal has Lebesgue measure zero), but as far as I know, the only known specific examples are numbers that were specially constructed to be normal.
     
  7. Dec 18, 2013 #6

    adjacent

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    Err.Sorry I thought normal numbers are something else.
     
  8. Dec 18, 2013 #7
    This number seems to be rational and normal : 0.17586934201758693420....?
     
  9. Dec 18, 2013 #8

    D H

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    That number is not normal. All ten digits do appear in that number with equal probabilities. However, only ten two digit strings appear: 01, 17, 20, 34, 42, 58, 69, 75, 86, and 93. Where are the other ninety two digit strings? For any string length n, a number that is normal in base 10 has all 10n possible n-digit strings appear with equal probabilities.

    To be truly normal, a number has to satisfy these properties in all bases. A rational cannot be normal.
     
  10. Dec 18, 2013 #9

    pwsnafu

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    What about two-digit combinations? You need every two-digit combinations to occur with equal probability. And then three-digit combinations and so on.
     
  11. Dec 18, 2013 #10
    You've shown that pi does not have a sequence of recurring 9's (ignoring your use of the phrase "round up") which means exactly nothing for your goal.

    Furthermore, pi already has an exact value.
     
  12. Dec 18, 2013 #11

    AlephZero

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    The OP only quoted part of the definition of "normal number" from the web page. Maybe that has some connection with his/her other wrong ideas.

    Your example would be "normal" using the OP's incomplete definition of course, but that would make "normal numbers" rather uninteresting.
     
  13. Dec 18, 2013 #12

    jbunniii

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    Also, a number such as 0.17586934201758693420.... will generally lose the property that every digit is equally represented if you express it in some other base. So it fails the one-digit criterion, too.
     
  14. Dec 18, 2013 #13

    jbriggs444

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    In particular, if you have a pattern that repeats with a n digit cycle in base 10 then in base 10n it will have a one digit cycle. In base 10n - 1 the representation will terminate.

    Of course this means that the number is rational. The same manipulation works in bases other than ten.

    http://www.wolframalpha.com/input/?i=0.175869342/9999999999
     
  15. Dec 19, 2013 #14
    I did not mean to confuse pi with rational numbers. I found an interesting article with several
    formulas showing how pi is approximated by rational numbers, for example:
    3.14159 < pi < 3.14160
    It is an interesting question if there is a limit of how accurately pi can be approximated
    in this way.

    http://educ.jmu.edu/~lucassk/Papers/more on pi.pdf

    What I was trying to say was that my "proof" is based on the silly idea of an exact value of pi, and why it does not exist, thus meaning that pi is a normal number. I am not sure if my reasoning
    is correct, though.
     
  16. Dec 19, 2013 #15

    D H

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    What do you mean by "the silly idea of an exact value of pi"? It's a very specific number, so of course it has an exact value. Here's a smallish subset of the infinitely many different ways to represent pi: http://www.wolframalpha.com/input/?i=Representations+of+Pi.

    You appear to have two key misconceptions:
    1. That irrational numbers don't have an exact value.
      This is not the case. All that irrational means is that they cannot be represented as the ratio of two integers. It does not means that they don't have an exact value.

    2. That all irrational numbers are normal.
      This too is not the case. I gave a specific example of a non-normal irrational number in post #4.
     
  17. Dec 19, 2013 #16

    jbriggs444

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    Pi is a real number. All real numbers are exact in the sense that for any non-zero error bound, there is an approximation which can be written as a decimal fraction and which is accurate to within that error bound.

    [Providing that one does not quibble about how many decimal places can be computed within the lifetime of the universe or written down on the contents of the universe. Or quibble about whether the definition for a particular number allows its decimals to be computed at all]

    If you want to reason about whether pi has an exact value, you should probably start by defining "exact value".
     
  18. Dec 19, 2013 #17
    Exact value is the opposite of an approximate value. You can of course represent pi in many ways, as D H said, but these are just formulas for calculating the approximate values of pi. And using these formulas you will always end up at an approximate value. Because no-one cannot write down an infinite numbers of digits which the exact value of pi has.

    Doubtless there are exact formulas for calculating the value of pi. But it is another thing to use
    these formulas and arrive at the exact value. I think you will always obtain only an approximate
    value. In some sense, these formulas create pi and its value. Maybe these formulas could
    also create an exact value, but it takes forever to calculate and write down its value.
    The infinite number of digits in pi makes me wonder if it really exists at all. Maybe pi is
    just invented by humans. I don't even know if pi is a very useful concept.

    My argument was that because pi is a normal number, it cannot have an exact value. I may
    be also wrong.
     
  19. Dec 19, 2013 #18

    jbriggs444

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    To rephrase what you are saying...

    A representation can be inexact if the value that it represents does not match the value that it is supposed to approximate.

    A representation can be exact if the value that it represents is equal to the value that it is supposed to approximate.

    But in this sense, "pi" is a representation of the ideal ratio of the circumference of a circle to its diameter. It is exact.

    So possibly what you are saying is that "pi" is not a valid representation and that pi is inexact since all valid representations [using some unspecified notion of what constitutes a valid representation] are inexact.

    So you really need to start by defining what it means for a representation to validly represent a number.
     
  20. Dec 19, 2013 #19
    I admit that the definition of [itex]\pi[/itex] is exact. But [itex]\pi[/itex] is different than its value, which is always approximate, unless you write down all its infinite digits. Only then you can say that this is
    an exact value of [itex]\pi[/itex]. I have never seen exact value of [itex]\pi[/itex], but it does not mean it does not
    exist. I told already that I am just making a hypothesis concerning [itex]\pi[/itex] and I may be wrong.
    Either [itex]\pi[/itex] has an exact value or it does not. It cannot at the same time both have an
    exact value and have it not.

    [itex]\pi[/itex]=[itex]\pi[/itex] and this is certainly exact. But [itex]\pi[/itex] is not the value
    of [itex]\pi[/itex].
     
  21. Dec 19, 2013 #20
    You really need to state precisely what you mean when you use this word.
     
  22. Dec 19, 2013 #21
    True. But note that the following is also true:

    So ##\pi## is no less exact than ##1/3##. But nobody seems to make a deal out of that...
     
  23. Dec 21, 2013 #22
    I began to wonder if I did not already state precisely in my above post what did I mean by the value of [itex]\pi[/itex]. So I did some google search with the words "value of pi" and "exact
    value of pi" to see what is the reason of the confusion I made. The consensus online seems to be
    that "[itex]\pi[/itex] is a transcendental number. It can be proven mathematically that its
    exact value can never be written down with digits", direct wiki.answers quote.
    I adhere to this consensus and this is what I mean when I use the the words "value of [itex]\pi[/itex]"

    Only very few people tend to say or define that the exact value of [itex]\pi[/itex] is [itex]\pi[/itex],
    and I think this is reason for the confusion.

    There are even those who have arrived at various "proofs" and calculated the exact value of [itex]\pi[/itex]. For example here is a "proof" that
    [itex]\pi[/itex] = 17 - 8[itex]\sqrt{3}[/itex] <crackpot link deleted>.
    I think there must be some error, wikipedia states that "[itex]\pi[/itex] cannot be expressed
    using any combination of rational numbers and square roots or n-th roots".
     
    Last edited by a moderator: Dec 21, 2013
  24. Dec 21, 2013 #23

    Mentallic

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    Oh but they have...

    That isn't a proof because the author has made at least one mistake. On page 20, the diagram labelled "1st modification in basic figure" is wrong, hence any conclusions he draws from it are invalid.
     
    Last edited by a moderator: Dec 21, 2013
  25. Dec 21, 2013 #24
    You'll have to forgive me for not accepting "Exact value is the opposite of an approximate value" as a proper definition.

    I apologize in advance if this comes off as rude or belligerent. That is not my intent. But you really should have some idea about what the words that you use mean. It's fine if your intended meaning turns out to be "wrong" in the sense that it's not the same as the consensus meaning. But if you really mean what you say, then what you say has to have meaning to you.

    In other words, I'm asking you what you mean when you use the word value. That's not something that you need Google for. Again, I'm not trying be belligerent here. I'm just trying to understand you.

    A. ##\pi## is transcendental. That is a fact.
    B. No transcendental number has a finite decimal representation. That is a fact.

    Neither of these facts help me understand what you mean by value other than (possibly) (1) a number has a value and therefore is not itself a value, (2) a number's value is something different than its decimal expansion, (3) a number's value is something that can be (sometimes/usually/always) written down. Also, this particular quote seems to imply that ##\pi## has an exact value and leaves open the possibility that it can be written down, just not with digits.

    Key questions that I still have are (1) do all numbers have a value, (2) how does a number's value differ from the number and its representations (if at all), (3) is a number's value always something that can be written down either in reality or in theory.

    What is the exact value of 2? If the exact value of 2 is 2, then why can't the exact value of ##\pi## be ##\pi##?

    That proof is most definitely bogus. A cursory glance at the front page of that "journal" demonstrates that it is less than legitimate.
     
    Last edited by a moderator: Dec 21, 2013
  26. Dec 21, 2013 #25

    CompuChip

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    Just skimmed the thread, if I get it correctly your definition of an "exact value" is a real number with a terminating expansion in base 10?
     
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