Oscar Cunningham on Nostr: The paper 'Infinitesimals' (Lightstone, 1972) shows that each hyperreal can be ...

The paper 'Infinitesimals' (Lightstone, 1972) shows that each hyperreal can be thought of as having a decimal expansion whose digits are indexed by the nonstandard integers. But in that system not every decimal expansion corresponds to a hyperreal number. In particular there's no hyperreal whose digits are 9 for the positive standard integers and 0 elsewhere. The decimal representation of the hyperreal (0, 0.9, 0.99, …) has 9s all the way up to the ∞th decimal place, where ∞ is the nonstandard natural number given by (0, 1, 2, …).

Of course the decimal expansion whose digits are 0 for every nonnegative nonstandard integer and 9 for every positive nonstandard integer does correspond to a hyperreal, namely 1.