Testwiki:Reference desk/Archives/Mathematics/2022 February 18

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February 18

Name for natural numbers plus their reciprocals?

Is there an established term for the union of the natural numbers and the unit fractions? Or maybe, more precisely, of the natural numbers greater than one, the unit fractions smaller than one, and one?

Programming languages typically have native data types neither for natural numbers only, nor for fractions. When I need a scaling variable that is known to be integral and positive, but the scaling can be either up or down, my preferred solution, instead of using a float, is to wrap an integer with a class and have it map negative values to unit fractions, so z is read as 1/(-z) when z < 0. Or -n^1 is read as n^-1, if you will, in which case the mapping conceptually reduces to moving the sign to a different place.

Never quite know what to call that class, though. :P

- 2A02:560:42A8:BF00:5878:2D04:1599:115 (talk) 16:51, 18 February 2022 (UTC)

There's no name I know of for that exact set, but see Egyptian fraction for a related notion. --Trovatore (talk) 17:58, 18 February 2022 (UTC)

It's an unusual mapping. I suppose it would be the set $\{\exp (\pm \log (n))|n\in \mathbb{N}\}$. For clarity, I'd probably call it something like NaturalsAndReciprocals. --{{u|Mark viking}} {Talk} 19:52, 18 February 2022 (UTC)

This is a little off-topic, but it might be worth pointing out that "natural numbers" often include zero these days, and many programming languages do have a type for that, or more precisely for natural numbers up to a certain limit, say 32 or 64 bits. --Trovatore (talk) 20:56, 18 February 2022 (UTC)