(k+1)th or (k+1)st?

Solution 1:

Major edit: while I still personally like (k+1)th, @Mitch has found confirmation for (k+1)st in a math handbook, to which I must concur. Therefore, (k+1)st appears to be the most correct to mathematicians. Barring any future pertinent revelations, I would suggest using (k+1)st.

Kay plus first seems worse to me because one visualizes k + 1st instead of (k+1)th. If only for this reason, I would say "(k+1)th". Adding "orthographic consistency" to the mix pretty much decides the issue for me.

I would say that the only reason why it might not sound natural is because we don't say "(k+1)th" in ordinary life, hence oneth is not a usual word. However, we often use the word first, and therefore it sounds more natural. This does not mean it is the best choice.

As to which is used more often, doing the googling ("(k+1)st" -grade -grades and "(k+1)th") turns up evidence for the inference that each is used about as much as the other. (Wikipedia uses (k+1)th, however, for what it's worth, while our friends at Math.SE seem mainly to support (k+1)st.)

As for spoken usage, any would be acceptable, but all are clunky. Try to avoid such terms.

As for usage difference based on context - there does not seem to be a usage/meaning difference between the two.

Either way, spoken or written, I recommend strongly that you reword your sentence so that facing the issue is unnecessary if possible.

Solution 2:

Thanks to everyone who answered this question. I'm very happy with the results! It looks like everyone has contributed to the ideal "best answer". Since I can't accept them all, I think it would be best to summarize all the answers as a community wiki post and accept that. (I hope this is not a breach of protocol.)

This question is mostly about usage. Some data we've collected:

• @PLL found a MathOverflow question with (n-1)th used by a non-mathematician and (n-1)st by a mathematician.
• @Neil Coffey broke out The Art of Computer Programming, Vol. 2. There's two instances (in Section 3.4.2, pages 136 and 137 in my copy) of Knuth using (t+1)st.
• @drm65 gives us at least one Wikipedia article using (k+1)th.
• @drm65 also tried to measure the usage of each on Google; although the page counts are unreliable because of how Google handles punctuation, it did find us some examples of (k+1)th in use.
• @Mitch turned this question over to our friends at Math.SE; as of now, three commenters report saying (k+1)th and three say (k+1)st
• Two blog posts (1) (2) about this same debate were uncovered by @Peter Shor here and David Speyer on Math.SE
• @Mitch uncovered a handbook for mathematical writing which advocates (k+1)st.
• @Peter Shor earns a commission on the eternal gratitude I retroactively owe Scott Morrison, who searched the arXiv and tallied 40 results for (k+1)th and 35 for (k+1)st.
• Finally, I checked Google Scholar to see if it gives better quality results than Google proper. Interestingly, most of the mathematical results on (k+1)st and (k+1)th appear to be legit. If we take Google Scholar at its word, the score is 19000 hits for (k+1)th versus 6120 for (k+1)st. The results for other possible variables (i, j, n) are similar. In comparison, kth gets 160000 hits.

So where does that leave us? I think we've conclusively shown that both variants are in (reasonably) common use. I am tempted to say, based on the last two results, that (k+1)th is somewhat more common in written mathematics. We have a ton of anecdotal data for spoken usage, although in the absence of a large enough spoken corpus of mathematics lectures it is hard to say anything for certain.

As for deciding what variant one should use, there seems to be a wide range of opinions. There are good arguments either way. You should follow your style guide if it has an opinion (at least one style guide advocates (k+1)st). However, the broadest consensus on this site seems to be that you should reword your sentence if you can. If you can't, pick one and stick with it; both appear to be in accepted use.

Solution 3:

The most definitive usage evidence on this I can find comes from Scott Morrison's comment on this blog post, where he does a "quick and dirty" search of the Math arXiv and finds a roughly 50-50 split (th's win, but not statistically significantly). And as far as I can tell, there's no grammatical reason for preferring one over the other. So my conclusion is that both are equally acceptable.

Solution 4:

It seems clear that -th is the "general" ordinal modifier in English. It's the one used with most numbers and the one we use in made-up numbers like "umpteenth", "somethingth" etc. So on those grounds, if what you have before the suffix isn't actually a number that fuses with that suffix, -th would seem to be the choice. I would posit that an intervening bracket is reason for the number not fusing. Or put another way, "(k+1)" isn't normally the component that is fused with "-st", and there's an argument for considering the whole constituent "(k+1)" rather than an individal component inside the brackets.

On the other hand, some mathematicians do write e.g. "(t+1)st". This example from Knuth, vol 2, section 3.4.2, bearing in mind that Donald Knuth is so obsessed with typography that he actually wrote a typography system to solve typographical issues he didn't like.

Or put another way: there's no consensus. Think about issues such as the above, then decide which you prefer. Either way, nobody will be able to say that you definitely "got it wrong".

Solution 5:

I think there is a disconnect between the need for parenthetical grouping in mathematics and what it does to the phonetic morphology of the "word". The variable k remains and we are adding one to that value. The parentheses have no bearing on the actual variable, only how the operations affect that variable. To that end I would say that the "correct" way to say it would be 'k-th plus one', unless you are specifically enumerating the grouping.