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There is extended decimal notation for hyperreal numbers which was developed by A.H. Lightstone:

$d.d_1d_2d_3...;...d_{H-1}d_{H}d_{H+1}...$

Why do we use symbol ";" in this notation?

Thanks.

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  • $\begingroup$ If you mean something like $0.\bar 12$ , so infinite many ones followed by a $2$ this makes no sense , not even as a hyperreal number. $\endgroup$
    – Peter
    Commented Jun 1 at 11:55
  • $\begingroup$ You can use any symbol, $\&$, $\wp$, $\#$, or even a space. What exactly are you asking? The necessity of a symbol? Or why in particular the semicolon is chosen? $\endgroup$
    – Trebor
    Commented Jun 1 at 11:58
  • $\begingroup$ @Trebor The necessity of a symbol. $\endgroup$
    – Mike_bb
    Commented Jun 1 at 11:59
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    $\begingroup$ @Peter, it can be made sense of in the hyperreals (in a certain sense). Wikipedia's notation with $H$ is clearer than OP's notation with $\infty$, but basically if $H$ is a hypernatural number then "$0.$($H$-many ones followed by a two)" does make sense. The semicolon notation is more helpful as it reminds you that the choice of $H$ matters, and that the sequence of ones has a weird order-type (for example, just "$\Bbb N$-many ones" wouldn't make sense). $\endgroup$ Commented Jun 1 at 12:41
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    $\begingroup$ @Peter: You might be interested in the topic of ordinal numbers. For example, the ordered set of real numbers $\frac{1}{2},\frac{2}{3},\frac{3}{4},\ldots,\frac{n}{n+1},\ldots;1$ consists of an infinite sequence, followed by the single element $1$. This ordered set represents an ordinal number written in shorthand as $\omega+1$. $\endgroup$
    – Lee Mosher
    Commented Jun 1 at 13:11

1 Answer 1

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If $H$ is an infinite hyperinteger, in Lightstone's notation from

Lightstone, A. H. Infinitesimals. Amer. Math. Monthly 79 (1972), 242–251

the number $10^H$ would appear as $$0.000\ldots;\ldots01$$ where $1$ appears at rank $H$. The notation ";" separates between the digits at finite ranks from the digits at infinite ranks. Here a comma cannot be used because it has a different meaning in the context of decimals. The same goes for a period. One could use a slash "/" or perhaps a vertical bar, but ";" is better because it suggests a pause rather than a full stop, since the decimal digits continue beyond the standard ranks.

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