# Why do we use symbol ";" in extended decimal notation for hyperreal numbers?

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.

• 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. Commented Jun 1 at 11:55
• 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? Commented Jun 1 at 11:58
• @Trebor The necessity of a symbol. Commented Jun 1 at 11:59
• @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). Commented Jun 1 at 12:41
• @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 $\omega+1$. Commented Jun 1 at 13:11

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