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Inspired by one of the top posts I was wondering if $\pi = 3.14159\dots$ were normal--as in you could find every finite string of numbers within $\pi$'s digits--would that mean we could find every finite sequence of the digits of $\pi$ in $\pi$'s digits. Would we find

$$3141592653...\left(\text{$n$-th digit of $\pi$} \right)$$

For any $n$? Also is there an example of a number that is normal that is easy to prove to be normal? I'm unfamiliar with the property.

BDN
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Zduff
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    you can read about normal numbers here. It is not known whether or not $\pi$ is normal. Indeed, it is not known whether every digit occurs infinitely often in the decimal expansion of $\pi$.. The wiki article linked to provides some fairly natural examples. – lulu Jun 20 '18 at 20:12
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    @lulu: Yes. Note, however, that the question is about what it would mean if $\pi$ were normal, not about whether it's normal. – joriki Jun 20 '18 at 20:25
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    @joriki Ah, you are correct. Read too quickly. I'll leave the comment up for the sake of the link. – lulu Jun 20 '18 at 20:33
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    You can, of course, find any finite sequence of digits of $\pi$ somewhere, at least once, among the digits of $\pi$, namely where the string first occurs; that's true of any number, normal or not. Normality would imply that you can find any such string infinitely often, with asymptotic density equal to $1/10^n$, where $n$ is the length of the string. – Barry Cipra Jun 20 '18 at 20:33
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    What do you mean by "one of the top posts"? – Rob Arthan Jun 20 '18 at 20:52
  • Sort questions by votes. One of those first 20 or so. – Zduff Jun 20 '18 at 23:30
  • The normality of numbers is not only difficult to verify. Currently, we know almost nothing concerning whether a given number is normal or not. $\pi$ , $e$ and the algebraic irrational numbers are conjectured to be normal to every base. But not a single of those numbers has been proven to be normal to any base. In fact, we cannot even rule out for any of those numbers that eventually only two distinct digits appear. – Peter Jun 21 '18 at 09:28
  • @BarryCipra We do not know whether every string occurs in $\pi$. This is an open problem as well. What we can say that every $11$-digit string appears somewhere in $\pi$ – Peter Jun 21 '18 at 09:32
  • @BarryCipra In the case of rational numbers, we can even be sure that not every digit-sequence appears. Also, an irrational number like $0.01001000100001\cdots$ does not contain every digit-sequence. – Peter Jun 21 '18 at 09:35
  • @Zduff We can construct numbers from which we know that they must be normal, for example : https://en.wikipedia.org/wiki/Stoneham_number – Peter Jun 21 '18 at 09:38
  • @BarryCipra I just noticed that you probably referred to the specific string the author searches. – Peter Jun 21 '18 at 09:41
  • @Zduff If a number is normal , every finite digit string occurs infinite many often in that number , including the first , lets say , $10^{20}$ digits of the number. It is widely believed that $\pi$ has this property (among many other numbers), but we do not know it. – Peter Jun 21 '18 at 09:43
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    @Peter, I didn't say that you can find every finite string somewhere among the digits of $\pi$, I said that you can find any finite sequence of digits of $\pi$ somewhere among the digits of $\pi$, which is what the OP technically asked. I was really just making a minor point about the wording of the question. What the OP presumably means is, Do the first $n$ digits of $\pi$ (for each $n$) ever appear, in sequence, again? If $\pi$ is normal, the answer is definitely Yes, repeatedly. – Barry Cipra Jun 21 '18 at 11:36

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The answer to your question is yes: the first billion digits of $\pi$ forms a finite block of base-$10$ digits, and every finite block of base-$10$ digits must occur infinitely often in any normal-to-base-$10$ number. (I wonder if you asked exactly what you wanted to?)

A quick comment at this point: it looks from your question like you've gotten the definition of "normal" incorrectly. Normality and its relatives don't just assert that lots of variety occurs, they assert that it occurs statistically: e.g. simple normality to base $10$ implies that asymptotically $1\over 10$th of the number's digits in base $10$ are $2$, not just that infinitely many digits are $2$. Precisely:

  • A number is simply normal to base $b$ if each digit in base $b$ occupies $1\over b$th of the places of the base-$b$ expansion of the number (asymptotically speaking). Note that the number whose binary expansion is $0.01001000100001..._2$ is not simply normal to base $2$, even though both $0$ and $1$ occur infinitely often in its expansion, since their distribution isn't right: the asymptotic probability of a digit being $0$ is $1$, not ${1\over 2}$.

  • A number is normal to base $b$ if it is simply normal to base $b^n$ for all $n$. Intuitively, this means that not only are the individual base-$b$ digits stochastically distributed, but so are the finite blocks of digits; e.g. $1\over 3$ is simply normal to base $2$ but not normal to base $2$ (since e.g. the block "$00$" doesn't appear anywhere in the binary expansion of $1\over 3$, let alone the required "$1\over 4$th of the time").

  • A number is normal (or absolutely normal) if it is simply normal to all bases.

Now let me address your secondary question:

Also is there an example of a number that is normal that is easy to prove to be normal?

Despite their objective commonality (in a precise sense, "most" real numbers are normal), verified examples of naturally-occurring normal numbers are hard to come by. Champernowne's constant is easy to prove "normal to base $10$," but isn't exactly "naturally occurring" and it is not known to be normal to all bases. Examples of absolutely normal numbers can be explicitly constructed; however, these examples are even less naturally-occurring than Champernowne's constant.

Meanwhile, it's conjectured that all irrational algebraic numbers are absolutely normal, but at the same time there is no known irrational algebraic absolutely normal number.

In general, these slides (and the sources they mention) are worth looking at. It's all a very intriguing situation, where we have heuristic evidence suggesting that a truly vast class of naturally-occurring numbers are absolutely normal but we currently lack the tools to verify any of those cases, and even the construction of explicit non-natural examples faces deep mathematical problems/questions.

Noah Schweber
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