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(Inspired by a question already at english.SE)

This is more of a terminological question than a purely mathematical one, but can possibly be justified mathematically or simply by just what common practice it. The question is:

When pronouncing ordinals that involve variables, how does one deal with 'one', is it pronounced 'one-th' or 'first'?

For example, how do you pronounce the ordinal corresponding to $k+1$?

There is no such term in mathematics 'infinityeth' (one uses $\omega$, with no affix), but if there were, the successor would be pronounced 'infinity plus oneth'. Which is also 'not a word'.

So then how does one pronounce '$\omega + 1$' which is an ordinal? I think it is simply 'omega plus one' (no suffix, and not 'omega plus oneth' nor 'omega plus first'.

So how ist pronounced, the ordinal corresponding to $k+1$?

  • 'kay plus oneth'
  • 'kay plus first'
  • 'kay-th plus one'
  • 'kay plus one'

or something else?

Community
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Mitch
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    I use $(k+1)^{st}$ but I can't speak for anyone else. It's easier for me to pronounce, although I guess it is a bad idea if you have variables named $s$ and $t$... – Qiaochu Yuan Aug 02 '11 at 21:16
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    Community Wiki? – JavaMan Aug 02 '11 at 21:20
  • omega and first ? – Andrea Mori Aug 02 '11 at 21:20
  • I say "kay plus first" though I try to avoid such terms in writing. – JavaMan Aug 02 '11 at 21:21
  • Rejoinder to the @drm65's answer: So, do you say "one-hundred-and-oneth" instead of "one-hundred-and-first"? – t.b. Aug 02 '11 at 21:23
  • @Theo: It could be pronounced differently, the ordinal for 101 and that for k+1. – Mitch Aug 02 '11 at 21:31
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    I say ‘kay plus first’ and ‘omega plus first’ if I’m using then as ordinal numbers (adjectives); as cardinal numbers (nouns) they are of course ‘kay plus one’ and ‘omega plus one’. – Brian M. Scott Aug 02 '11 at 21:32
  • Maybe but for me $k+1$ has no semantic difference to $101$ with $k=100$. Hence $(k+1)^{st}$ and $101^{st}$. It was meant to invalidate the bracketing argument. – t.b. Aug 02 '11 at 21:33
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    As the OP on English.SE, allow me to further ask whether there's a difference between spoken and written math. I've caught a prof in lecture writing (k+1)th at the same time as he said (k+1)st out loud! – Ross Churchley Aug 02 '11 at 21:42
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    @Qiaochu: How about $(k+1)^{st}$ versus $(k+1)^{\text{st}}$ then? – kahen Aug 02 '11 at 22:19
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    @kahen: to be honest, I can barely tell the difference between those two visually. – Qiaochu Yuan Aug 02 '11 at 22:20
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    Now count further: kay, kay-plus-oneth, kay-plus-twoth, kay-plus-threeth, really? – t.b. Aug 02 '11 at 22:45
  • kay plus oneth is the only rhyme I know for month – Henry Aug 02 '11 at 22:48
  • @Qiaochu, I think it is commonly said that it is $n$ plus oneth (I have not come across $n$ plus first). So, perhaps the common extension is to have $k$ plus oneth. – picakhu Aug 03 '11 at 01:05
  • @kahen: I -can't- tell the difference. What is it? – Mitch Aug 03 '11 at 13:44
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    @Theo: Yes. Weirdly enough, in fact that's what I often say, though I usually avoid the issue with "the number at position k+1" or similar. (I don't really like the idea that numbers that are 1, 2, or 3 mod 10 should have a special linguistic status. :P) – ShreevatsaR Aug 03 '11 at 14:26
  • @Mitch: The first example has ‘st’ typeset in italic, and the second example has it typeset in roman. The latter is considered preferable as it avoids confusion with exponentiating by $st$... but I think it's still a bad idea so I write $(k+1)$-th. – Zhen Lin Aug 03 '11 at 14:27
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    @ShreevatsaR: Don't forget "kay-plus-fifth" vs. "kay-plus-fiveth"; the difference (in speaking) isn't only for 1,2, or 3. Also, we're already stuck with "twenty-first" rather than "twenty-oneth", etc. – Jonas Meyer Aug 04 '11 at 07:29
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    The naive solution: Let $j=k+1$. :-) – Asaf Karagila Aug 06 '11 at 10:56

2 Answers2

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From the Handbook of Writing for the Mathematical Sciences section 5.5 p. 63:

Here are examples of how to describe the position of a term in a sequence relative to a variable k:

kth, (k+1)st, (k+2)nd, (k+3)rd, (k+4)th, … (zeroth, first, second, third, fourth, …)

Generally, to describe the term in position k±i for a constant i, you append to (k±i) the ending of the ordinal number for position i (th, st, or nd), which can be found in a dictionary or book of grammar."

So the formal answer is that it should be:

(k+1)st

J. M. ain't a mathematician
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Mitch
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If you want a whole lot of non-expert opinions, you can read the comments here.

David E Speyer
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