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People sometimes use the term "finite" to mean "non-zero" or "non-infinitesimal". For example, physicists often say "finite temperature" to emphasize that the temperature under consideration is not zero. Similarly, when speaking of symmetry groups one might say "finite translation" to emphasize that a translation is not infinitesimal. This is not strictly a correct usage of the term finite, which technically encompasses zero and infinitesimal values just as well as the rest. So my question is: Is there a better word to use in these cases, to refer to non-zero, non-infinitesimal, but still finite values?

Sesquipedal
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  • How about a non-zero constant? – Studentmath Dec 06 '14 at 23:34
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    A nonzero real number? By the fact that it is a real number it is finite, and by the fact that it is nonzero, there are many numbers smaller than it (in magnitude). – JMoravitz Dec 06 '14 at 23:34
  • The difference between "finite transformations" and "infinitesimal transformations" has little to do with numbers. Physicists refer to an element of the Lie algebra of a Lie group as an infinitesimal transformation. They aren't actually elements of the group, though they do map to elements of the group via the exponential map. – Matt Samuel Dec 06 '14 at 23:37
  • Yes, I'm not looking for a term peculiar to real numbers--notions of infinity, zero, and finitude are much more general. In many cases it is useful to distinguish between finite objects which are also non-zero, and it would be useful to have an adjective for this. – Sesquipedal Dec 06 '14 at 23:41
  • Here's another example: What do you call a hyperreal number that is neither infinitesimal nor infinite? – Sesquipedal Dec 06 '14 at 23:42
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    "Finite nonzero" works for me. – Qiaochu Yuan Dec 07 '14 at 02:02
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    @Sesquipedal: Goldblatt (Lectures on the Hyperreals, p. 50) calls such numbers "appreciable" (= "limited but not infinitesimal"). – Hans Lundmark Dec 07 '14 at 15:09
  • @HansLundmark: That's precisely the sort of term I'm looking for! Feel free to submit it as an answer if you like, and I'll mark it as accepted. – Sesquipedal Dec 07 '14 at 20:14
  • All right, I've done that. – Hans Lundmark Dec 08 '14 at 07:10

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On p. 50 in Goldblatt's book Lectures on the Hyperreals, hyperreal numbers $b$ which are "limited but not infinitesimal" ($r<|b|<s$ for some $r,s \in \mathbb{R}^+$) are called appreciable.

Hans Lundmark
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