# (k+1)th, (k+1)st, k-th+1, or k+1?

(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?

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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 2 '11 at 21:16
Community Wiki? –  JavaMan Aug 2 '11 at 21:20
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 2 '11 at 21:32
@kahen: to be honest, I can barely tell the difference between those two visually. –  Qiaochu Yuan Aug 2 '11 at 22:20
Now count further: kay, kay-plus-oneth, kay-plus-twoth, kay-plus-threeth, really? –  t.b. Aug 2 '11 at 22:45

If you want a whole lot of non-expert opinions, you can read the comments here.

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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

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Ah, Higham... :) +1. –  Guess who it is. Aug 4 '11 at 7:08
Also, the ordinal for 301 is pronounced "three hundred and first". –  Mitch Apr 10 '14 at 12:43