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


Mathematicians commonly have to form ordinals from variables: you might look at the kth element of a sequence, for example. When the variable is a single letter, the ordinal is always formed with the suffix -th. Thanks to the unwritten conventions of variable naming, this usually results in one of the easily pronounceable words ith "eyeth" jth "jayth" kth "kayth" nth "enth" or mth "emth".


Things get a little bit trickier when you look at the elements near the kth one. Two opposing forces drag you in different directions when you try to form the ordinal for (k+1). On the one hand, orthographic consistency favours (k+1)th; on the other hand, "kay plus oneth" sounds less natural out loud than "kay plus first". In fact, I've caught one of my professors in a lecture saying (k+1)st at the exact same time as he was writing it on the board — as (k+1)th.


I think both variants are in use, and each has a defensible argument behind it, so I don't think one or the other is "correct". But I wonder if one gets used more than another. Ideally, I'd like to know



Between (k+1)th and (k+1)st, is one used more often than the other?



and better yet



Does usage differ based on context?



(I'm mainly curious about written vs spoken contexts, but I imagine it could be different based on discipline. Mathematicians can't be the only ones with this problem...)


Those are the questions I'd like to answer, anyways. I fear it might not be definitively answerable (although anecdotal evidence is still welcome!). My usual method — count the Google hits for each variant — is useless here because of the way Google handles punctuation. So, as a side note, I'd also love to hear answers to the more general question: How do I determine how widely used a term is when it consists mostly of punctuation?



EDIT (x2)


I thought I should clarify that the motivation for this question doesn't come from a particular instance I had in mind: I typically say (k+1)st out loud but might lean towards (k+1)th if I ever had to write it. I'm mostly curious about how they're used by others. I appreciate all the data points in your answers and the parallel Math.SE question!


It seems that some of you have shared my Google troubles when trying to measure the relative usage of (k+1)st and (k+1)th. I realized that the underlying issue is applicable to more common usage problems, so I've spun the last part off as a separate question: see Compare usage between punctuation variants. Whoops, looks like that was a bit too meta as a separate question. So just keep in mind that Google page counts seem to be unreliable for this particular question unless you find a workaround to make it search for punctuation properly.


Anyone who can get some sort of tool to find hard data on this question gets an invisible second upvote from me and my eternal gratitude :)



Answer



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.


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