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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 :)

Phil Sweet
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Ross Churchley
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  • Let me know if the last bit is off topic or should be sput off into a separate question. I'll happily move it! – Ross Churchley Aug 02 '11 at 20:22
  • I think you can safely use (k+1)th ... The only ordinal changes involve numbers ending in 1, 2, or 3, and you wouldn't change your writing to (k+2)nd or (k+3)rd — no one would expect that from you — so all quantities not specifically involving literal values ending in 1, 2, or 3 may be expressed as you would with nth. Hey, zeroth is a word too! – Robusto Aug 02 '11 at 20:28
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    Because this is localized to mathematics, it might be worthwhile trying it on the math.SE site. – Mitch Aug 02 '11 at 20:37
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    Addition is commutative, so you can avoid the problem altogether and write (1+k)th. – John Bartholomew Aug 02 '11 at 20:41
  • I went ahead and posted this at math.SE. It's essentially the same answer but not necessarily definitive. Their's is a local pronunciation, and may not match what non-mathematicians naturally do. – Mitch Aug 02 '11 at 21:15
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    oneth is a rare rhyme for month – Henry Aug 02 '11 at 22:50
  • See this blog entry, as well. – Peter Shor Aug 03 '11 at 00:20
  • Isn't it funny that all the answers are downvoted once? – Daniel Aug 03 '11 at 00:37
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    This idea that "N plus oneth" doesn't sound good is potty! Even non-mathematicians talk about the "nth degree" etc. Who ever heard of the "nst degree"? I despair of people who think that because the last term within the brackets happened to be "1", they should adjust the ordinal indicator to reflect that particular element. – FumbleFingers Aug 03 '11 at 23:47
  • ...but I do like the fact that this Q has gotten a lot of interest! – FumbleFingers Aug 04 '11 at 00:15
  • @FumbleFingers I'm a little surprised myself at how much attention this question has gotten! It was even one of the StackExchange network's "Hot Questions" for a brief period yesterday. I'm happy to see that other people are as interested in this question as I am! – Ross Churchley Aug 04 '11 at 00:26
  • Self-gratification aside, the best information you can get is to be told things you didn't expect. I'm totally boggled there's even room for dispute about this one (I'm up for "-th", obviously), but that just goes to show I don't move in the right circles! :-( – FumbleFingers Aug 04 '11 at 00:39
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    @FumbleFingers Definitely. I knew about both variants, and could kind of guess the distribution of written usage (though I'm glad to finally have the data!), but I was surprised to see a style guide (1) have an opinion on it, and (2) come down for (k+1)st. – Ross Churchley Aug 04 '11 at 00:49

7 Answers7

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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.

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Daniel
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  • Wikipedia is not a reliable source for anything. But in that particular instance it does sound right. The search for good examples of either is difficult and part of the OP's question, how to deal with punctuation in on-line searches. – Mitch Aug 02 '11 at 21:11
  • Which is to say that if we find more evidence for (k+1)th it would finally be good evidence that there is a real extant word missing from the OED which would rhyme with month. I'm not convinced yet, but depends on evidence (and the answer one might get from this same question posed at math.SE). – Mitch Aug 02 '11 at 21:13
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    I'm satisfied that the "(k+1)th" Google search establishes this usage and the hit count is probably a good measure for its prevalence. However, your result for "(k+1)st" shows us that Google fails when the search term contains punctuation. It's finding pages which contain both K and 1st, where we'd expect early education sites to dwarf any sort of mathematical usage of (k+1)st – Ross Churchley Aug 02 '11 at 21:32
  • Check out the edit. Good call. – Daniel Aug 02 '11 at 22:33
  • Math.SE is also given this question. It seems as if (k+1)st is more popular there, but another strong opinion is that the construction should be avoided altogether if possible. I agree. – Daniel Aug 02 '11 at 23:39
  • (k+1)st has four times as many Google hits as (k+1)th. How on earth do you conclude that (k+1)th is preferable? – Peter Shor Aug 03 '11 at 00:15
  • Because, if you notice, most of the results for (k+1)st refer to Kindergarten to First grade. The mere number of Google hits are not a reliable way to demonstrate either way in this case. It's what's in the Google hits that matters. Anyway, I backed down from saying that (k+1)th is more often used, saying instead that they seem evenly matched. I only said that I prefer (k+1)th, and gave my reasons. – Daniel Aug 03 '11 at 00:16
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    See the formal answer that I found (hint: it's '(k+1)st' !), which I put in an answer to the math.SE question. But that doesn't -really- answer the question here as to real practice. Which is a combo of everyone's answer: (k+1)th and (k+1)st should be avoided, but both appear non-trivially often (the '-th' version more often by non-mathematicians), and (tangentially) this almost raises 'oneth' to legitimate word status. – Mitch Aug 03 '11 at 14:27
  • I edited my answer to make (k+1)th my opinion, but (k+1)st the usually correct choice. – Daniel Aug 03 '11 at 14:48
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    I think the definitive answer 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. – Peter Shor Aug 03 '11 at 16:19
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    @Mitch That's a great find! You're right that it doesn't really answer the question, but it definitely establishes the usage of (k+1)st in written mathematics. Thanks! – Ross Churchley Aug 03 '11 at 17:15
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    @Peter Shor could you make that comment an answer? I may end up accepting this one, but tracking down some hard data deserves at least one upvote and some portion of my eternal gratitude :) – Ross Churchley Aug 03 '11 at 17:16
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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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Ross Churchley
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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.

Peter Shor
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  • I've upvoted, but with misgivings. You of all people should have a definite position. I personally think "st" is ridiculously indefensible, on the basis of logic (at most it could apply to one possibility, compared to an infinite number of contexts where it was incorrect), but whilst I wouldn't presume to tell others what they should say, I think they should at least know what they do say. – FumbleFingers Aug 03 '11 at 23:40
  • I've upvoted, with many thanks. This answers my question the best of any so far (excepting the Frankenstein's monster I wrote summarizing everybody's answers) because I really wanted some data on actual usage. – Ross Churchley Aug 03 '11 at 23:51
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    Personally, I use "st". Both constructions are perfectly comprehensible, and both seem to be used roughly equally often. I don't see any reason to prescribe either one. Based on the answers here, I suspect that the percentage of mathematicians that use "st" may be higher than the percentage of non-mathematicians that use it. – Peter Shor Aug 03 '11 at 23:53
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    @FumbleFingers I don't think a definite position is necessary. There are strong arguments in both directions, and people with strong opinions on each side. I don't think it's unreasonable to not take sides if one believes the opposing arguments are equally strong. In any case, reasons for preferring one over the other are only tangentially related to this question, which is about usage. – Ross Churchley Aug 03 '11 at 23:57
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    @Peter Shor I went around our department asking people what they thought. (k+1)st was favoured by a majority of faculty, while (k+1)th won out among grad students. – Ross Churchley Aug 03 '11 at 23:59
  • Okay, I didn't want to stir up a hornets nest! I still think st sounds daft, but what do I know? If that's what people say, that's what they say. I just thought @Peter was being a bit weasely not taking sides. I won't deny that I personally prefer th because I read whatever came before it as (some unspecified number) "n", regardless of the particular form of the expression, but I accept that I don't do maths (read "math" if non-Brit), so I really shouldn't comment. – FumbleFingers Aug 04 '11 at 00:09
  • @FumbleFingers: It comes from how you pronounce it in your head--- you say "kay plus one-th" or "kay plus first", using the rule one-th -> first. One-th is just annoying to pronounce. If you have "k+1" as a concept which has no voicing, logic dictates "th", because k is unlikely to be 0. But in pronunciation, "st" is more natural. – Ron Maimon Mar 14 '12 at 06:45
  • @Ron Maimon: I suspect your "One-th is just annoying to pronounce" position is more US than UK. From comments on this question about "oner" I get the impression Americans are more leery of generating "regularised" forms from the word "one" - they tend to either avoid them completely, or opt for explicit oddities such as "one-er". I don't think it's to do with anything being "more natural"; it's to do with US usage being more biased to favour the relatively high proportion of non-native English speakers in the country. – FumbleFingers Mar 14 '12 at 15:16
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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".

Neil Coffey
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  • +1 because you're right in exactly the same way as @Peter Shor, but again I wish you'd nail you personal colours to the mast. So far as I know, Donald Knuth is a man I should respect - but more in matters of mathematics than linguistics, I feel. I will back off if anyone wants to interpose "semantics" there! :) – FumbleFingers Aug 04 '11 at 00:59
  • Well as I say, the reason for mentioning Donald Knuth specifically is that (a) he's a mathematician (or at least, computer scientist, but his maths expertise is pretty jolly good by most people's standards), and (b) he does actually care about issues of typography so may have thought about this more than your average mathematician. But no, that doesn't mean you intrinsically have to respect his or anyone's linguistic decisions. (For what it's worth, I actually disagree with him here and would write -th for the reasons I've stated.) – Neil Coffey Aug 04 '11 at 03:41
  • Fair enough. Being primarily a descriptivist I think an accurate summary of usage prevalence matters more than personal preference, rules, or logic. But it's just nice to know where people are coming from. – FumbleFingers Aug 04 '11 at 12:27
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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.

Mark T
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    But no one I've ever seen uses (kth+1). I can't say I'd be satisfied to use this method. – Daniel Aug 02 '11 at 20:55
  • I was referring to the spoken usage, but I agree it is not conventional. – Mark T Aug 02 '11 at 20:59
  • For all I know, it's never happened to me, but if someone spoke of the "nth degree" and I wanted to go one higher, I'd probably say the "nth plus one". At the outside, maybe "(n plus one)th". I just can't imagine trying to outdo the "nth" with the "n plus first", but apparently people will say (or at least think) that. – FumbleFingers Aug 04 '11 at 00:48
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Kay plus one-th occurs often informally, it is a legitimate generalization (even though it clashes with 'first' and 'one-th' is not a generally recognized in dictionaries), and is not so often used by mathematicians. (I personally don't use this, except possibly humorously). That it is used often enough should make 'one-th' a de facto neologism that should be recognized.

Kay plus first also occurs often informally (the distribution of these two forms is unclear), it avoids the questionable word 'one-th', is acceptable and used more often by mathematicians, and is stipulated by a math style guide. (I would use this one in speech if at all).

More commonly in mathematical writing, one tries to avoid using 'k+1' as an adjective needing the ordinal ending, and tries to say the 'successor to k' or the 'item k+1'.

Mitch
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    Please give the source of your knowledge that (k+1)th is not used by mathematicians. – Daniel Aug 02 '11 at 20:42
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    @drm65: I looked at the links in your search and found the one by a student for (k+1)th, and the rest were (k_1)th or unrelated (and so irrelevant) tokenizations were different punctuation is removed. My source is purely anecdotal, having worked with ordinals mathematically. But my usage could be idiosyncratic. – Mitch Aug 02 '11 at 20:50
  • Doesn't (k-1)th count? 5 out of the first 10 results contain (k+1)th or (k-1)th, in a mathematical context. – Daniel Aug 02 '11 at 20:52
  • @drm65: an unrepresentative blog entry shows that it goes both ways (it looks like 1/2 and 1/2). – Mitch Aug 02 '11 at 20:53
  • I see your math.SE question, and its results, but Wikipedia is still on my side. :) – Daniel Aug 02 '11 at 22:34
  • I'll remove my downvote if you say (k+1)th is not used by most mathematicians, and if you make the "sounding terrible" more subjective (e.g. to me). But I won't upvote until you say (k+1)th is acceptable, too. IMHO, it is. – Daniel Aug 03 '11 at 14:52
  • @drm65: Done (I think to your liking). Since I see now that people use 'oneth' it must be acceptable. But my personal feeling is that k+oneth sounds to much like a wrongly generalized past tense verb 'seed' for 'saw'. Since 'oneth' is acceptable, I think that means it should be recognized formally. – Mitch Aug 03 '11 at 15:56
  • @drm65: oh and I never down-voted anybody. I did up-vote you after some edit. – Mitch Aug 03 '11 at 15:58
  • And here's my end of the bargain... (and the downvoter just removed their downvote from my answer too, so I guess that was because of the edit as well) – Daniel Aug 03 '11 at 16:21
  • (k+1)-th/st is such a useful expression in mathematics that it's stupid to try to avoid using it. Just choose one. Saying "the successor to k" instead is like saying "that North American marsupial that plays dead" because you don't know whether to spell it possum, 'possum, or opossum. – Peter Shor May 11 '19 at 12:05
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As a mathematician, I’ve definitely heard both used; I think I use (n+1)st myself, but of thinking self-consciously about it, it’s hard to be sure.

I can’t find many examples online, but this mathoverflow question shows both usages: (n–1)th from the questioner (who seems to be a non-mathematician), and (n–1)st in comments (from a mathematician).

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PLL
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