[im3w1l]

So something I was thinking of: A number in decimal notation can be seen as a function from the integers to {0,1,2,3,4,5,6,7,8,9} (where negative numbers map to digits left of the decimal point and non-negative to digits right of the decimal point) such that only finitely many negative numbers map to non-zero.

Could you generalize this to include the hyperreals by lifting the restrictions on finitely many, and also adding in some transfinite ordinals to the domain of the function?

[bubblyworld]

I suspect yes - no need to introduce transfinite ordinals, you simply map from the set Z*, which is the integers but including the nonstandard ones. In fact you don't even need to remove the finiteness hypothesis, the transfer principle should guarantee that every hyperreal has such a representation since you can prove that every real does for the standard version.

(if the finiteness thing seems confusing, remember that there are infinitely large nonstandard integers in the hyperreals, and you can't tell them apart from the others "from the inside")