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Please correct me if I am wrong. I believe that it is unknown whether the decimal representation of $\pi$ contains all strings, or lists on 10 digits. I would like to know if there is a number that is known that does in fact contain all of them, without the number being constructed explicitly to do so. Is there a number that can be expressed without explicitly making it contain all strings that does contain them?

Regards.

Asinomás
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  • What do you mean with contain all strings? – Umberto Jan 07 '14 at 16:04
  • Choosing the sequence of digits randomly and independently produces such a number with full probability. – Did Jan 07 '14 at 16:08
  • So for example there is a part in the decimal representations that contains "01245345" for example, that is a string. Any "word" made up of digits is a string. – Asinomás Jan 07 '14 at 16:08
  • @did oh, interesting. So then any "randomly" generated irrational would be an option? Are there numbers which are not created reffering to the digits though? – Asinomás Jan 07 '14 at 16:09
  • You seem to be asking about numbers that are normal in base 10. In fact, almost all real numbers are normal (thus in particular normal in base 10). – fuglede Jan 07 '14 at 16:11
  • Any "randomly" generated number would be irrational and would contain every finite string infinitely many times, yes, with probability 1. // "Are there numbers which are not created reffering to the digits though?" Not sure I get the question. – Did Jan 07 '14 at 16:13
  • @Did oh, sorry, I misunderstood. But is there an actual number? – Asinomás Jan 07 '14 at 16:15
  • Please define "actual number". – Did Jan 07 '14 at 16:22
  • @did I'm sorry, I'm sort of out of my depth here. But I want a number, which can be defined without creating it by talking about its decimal expansion. And which is unique. So I can know what the number is. You say a random number works with probability 1. But I want a number that is uniquely determined. I don't know if this makes sense. It seems to me like the random number could be any number. – Asinomás Jan 07 '14 at 16:26
  • @Did: Irrational numbers do not have that property. – fuglede Jan 07 '14 at 16:36
  • I think Did means if you pick a random number the probability it is irrational is 1. – Asinomás Jan 07 '14 at 16:38
  • @user4140: Yes, but you're asking for a condition which is (strictly) stronger than being irrational. – fuglede Jan 07 '14 at 16:41
  • @fuglede How is the comment that you saw fit to address to me, related to what I wrote? – Did Jan 07 '14 at 16:42
  • @Did: Semantics, perhaps. I read your "and would" as "and would thus", and not the "and" as separating two different statements. – fuglede Jan 07 '14 at 16:46
  • @fuglede Then let me suggest that you hesitate a little more before "refuting" statements which were not actually written. – Did Jan 07 '14 at 16:49

3 Answers3

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The Champernowne constant $0.0123456789101112131415\dots$ includes all finite digit strings. Any normal real, and most are, will include all finite digit strings, but that is often hard to prove for a given number.

Ross Millikan
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  • +1, but can we find a number without alluding to the decimal construcion? – Asinomás Jan 07 '14 at 16:26
  • If you read Wikipedia on normal numbers it talks about how difficult it is to prove that a given number is normal. It shows various ways to construct one along the lines of the Champernowne constant, all of which allude to the decimal construction. We don't know how to prove that $\sqrt 2, \pi, e$ are normal, but most strongly believe they are. – Ross Millikan Jan 07 '14 at 16:35
  • Oh, I see. So then the answer to my question would be no? There is no number which can be constructed without alluding to decimal construction which is normal base 10? Thanks. – Asinomás Jan 07 '14 at 16:38
  • I don't know of one. – Ross Millikan Jan 07 '14 at 16:41
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The are countably many (finite) strings of digits. Enumerate them as $s_1, s_2, \dotsc, s_n, \dotsc$ and then concatenate them into $x=0.s_1 s_2 \dots s_n \dots$ The number $x$ is the number you seek.

Igor Rivin
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  • Then you are called Champernowne (but this is a number the OP seeks, not the number). – Did Jan 07 '14 at 16:22
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    @DId No, Champernowne is only my MIDDLE name :) – Igor Rivin Jan 07 '14 at 16:29
  • @did no, I am sorry I am not able to convey what I want. I don't want the number. I want a number, like this one. Which can be expressed without constructing it by alluding to the decimal representation. When you tell me almost all random numbers work, I find that really interesting. But what I want is a number which is uniquely determined. If you where able to provide a unique number that worked I would be content. – Asinomás Jan 07 '14 at 16:33
  • @user4140 I am not sure I understand the concept of a unique random number... – Igor Rivin Jan 07 '14 at 16:35
  • Hmmm, I see is there a number which can be constructed which is known to be normal base 10?? – Asinomás Jan 07 '14 at 16:39
  • The Champernowne number is known to be normal. – Igor Rivin Jan 07 '14 at 16:43
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Let $n\gt1$ denote an integer coprime to $10$, then the real number $$ \sum_{k=1}^\infty n^{-k}\,10^{-n^k} $$ is normal in base $10$. These are called Stoneham numbers, as explained in the must-read introduction of the linked paper.

Did
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