## Thursday, 25 March 2010

### (n+1)st? (n+1)th?

What is the ordinal numeral corresponding to the cardinal numeral (n+1)?
The ordinal numeral corresponding to 1 is first, so we write 1st.
The ordinal corresponding to 2 is second, so we write 2nd (or perhaps 2d if American).
The ordinal corresponding to 3 is third, so we write 3rd (or AmE 3d).
The ordinal numerals corresponding to 4 and upwards are formed with the suffix th, so we write 4th, 5th, 6th, 7th etc.
But after 20th come 21st twenty-first, 22nd twenty-second and 23rd twenty-third.
When we use algebraic expressions instead of ordinary numerals, the ordinal corresponding to n is nth, pronounced enθ. That corresponding to x is xth, pronounced eksθ. One of the teachers at my secondary school had the nickname kjuːθ, i.e. qth.
But it is not clear how to form ordinals for expressions such as x2, (x+1), and (x-2).
I was quite taken aback when, in the course of reading Seeing Further (ed. Bill Bryson, London: HarperPress for the Royal Society, 2010), I came across the sentence (p. 379)
[May, Oster and Yorke] identified simple features displayed by wide classes of difference equation relating the (n+1)st to the nth state of a system as it made the transition from order to chaos.

Do we really say en plʌs fɜːst? It feels wrong to me.
Imagine there is a queue of people waiting to enter a club. Your job is to pick which ones can go in. You might decide to take person number one (the first person), person number two (the second person), and person number three (the third person). More generally, you might decide to take person number n (the nth person), person number (n+1), (the (n+1)??? person), and person number (n+2) (the (n+2)??? person).
If pushed, I think I’d go for saying en plʌs wʌnθ, en plʌs tuːθ and writing (n+1)th, (n+2)th.
Any mathematicians have views on this?

1. I'm both a linguist and a mathematician, and it's always -th for me. I rationalize it, if necessary, by saying the bracket acts as a barrier for the combining rule.

2. I'm a mathematically inclined biochemist rather than a mathematician, but I've certainly had quite frequent occasion to pronounce (n + 1)th, and I've always said n plus wunth (sorry, I can't do IPA). I've had much less occasion to pronounce (n +2)th and I'm not entirely sure what I say, but it's probably n plus twoth.

3. "(n+1)st," "(n+2)nd," etc. seem to me justified by the analogy of "twenty-first," "twenty-second," etc. Nobody says "twenty-oneth," "twenty-twoth," etc.

4. In my former life as an undergraduate mathematician, it was definitely "(n+1)st" and "(n+2)nd"; if I ever heard someone saying "(n+1)th" it would sound extremely weird to me. The "n+" seems to me to be no different grammatically from the "one hundred and" in "one hundred and first".

(On the other hand, the final chapter of the novel The Hundred and One Dalmatians is "The Hundred and Oneth Dalmatian", so there may be some disagreement on that point as well.

5. I'd say that 2d for 2nd is archaic even in America.

I don't know who's reponsible for the following verse, but to my mind it settles the issue:

"The answer is," said Dr. North,
Who's worked on it a month,
"Not merely x raised to the fourth,
But x^(n+1)".

This also disposes of the claim that English lacks a rhyme for month.

Similarly, Edgar Allan Poe once wrote in a newspaper article: "Mr. Mathews once wrote some sonnets 'On Man,' and Mr. Channing some lines on 'A Tin Can,' or something of that kind — and if the former gentleman be not the very worst poet that ever existed on the face of the earth, it is only because he is not quite so bad as the latter. To speak algebraically:— Mr. M. is execrable, but Mr. C. is (x+1)-ecrable."

6. Nice! Did you think of that yourself? If so, in case I should quote it, may I change north to something that rhymes with fourth in my pronunciation?

7. I use (n+1)th.

And I also reacted to the poem by thinking that the other two lines didn't rhyme.

8. Depends on the accent. Now that we've been provided with a rhyme for month, does anyone have one for fourth? Can't think of any, unless you count archaic 3rd persons on -'th. Of course, for a name one could invent one, Dr. Boarth or something, or claim that family pronounces North with a dipththong/closed o, but that would feel like cheating.

9. As a mathematician, N+1st sounds right to me and N+1th sounds wrong. And N+2th sounds even wronger.

(And in my dialect, "fourth" and "north" rhyme.)

- Paul Clapham

10. I'm American and I've never seen "2d" or "3d" unless one is referring to computer graphics, in which case the "d" in both would be capitalized.

11. @Lipman: I'm not quite sure what you mean. For me, those words are /fɔːθ/ and /nɔːθ/. Which one are you pronouncing differently?

@JHJ: I agree. "One" rhymes with "gone", not "gun", for me too. But I do merge "non-", "none", and "nun". I wonder if we're playing with isoglosses here?

12. For those whose English has NOT undergone the FORCE-NORTH merger, I suspect there are no rhymes for "fourth".
fourth [foːrθ], north [nɔrθ] etc.
However the majority of English speakers DO have the merger, so the rhyme works for them.

13. @James D: no, I do rhyme "month" and "(x+1)th" (although I think that given where I'm from I probably "shouldn't"). My non-rhyme is the same as Lipman's, for me "north" [nɔːθ] vs. "fourth" [foəθ].

14. n plus first and n plus second, definitely. I've never heard any different. I find them as natural as "twenty-first" and "twenty-second", and I'd find "n plus one-th" and "n plus two-th" as unnatural as "twenty-one-th" and "twenty-two-th".

15. Wikipedia's List of English words without rhymes offers "(n+1)th" and "hundred-and-oneth" for month. Also Thackeray's Major Bangles of the Onety-oneth Hussars.

The same page lists "eighth" as rhymeless; with my Irish dental plosive THs, it rhymes with "faith". Some of my countrymen have alveolar plosives for TH, giving plenty of rhymes for "month", not all printable in a family blog.

"Forth" rhymes with "fourth", as does "Twenty-fourth". And you could pronounce "(n+1)th" as "(en and one more)th".

How does one pronounce 0th? "noughth" sounds too like "nought" so I think it has to be "zeroth". You could try "noneth" if you need a second rhyme for "month".

I think "2d" and "3d" for "2nd" and "3rd" is still used in the US for legal citations, but not so much by the military .

16. As I said, I do not know who wrote the rhyme: it certainly was not me. There is another version in which the third line ends in x^n and doesn't rhyme with the first line. Shoulda just cited that one in the first place, ~~ scroan ~~.

Let me add this quatrain by Lewis Carroll:

But what are all such gaieties to me
Whose thoughts are full of indices and surds?
x^2 + 7x + 53
= 11/3

Lipman: Where do you come from that you don't have the NORTH=FORCE merger?

Mollymooly: Quite right about the legal citations, where 2d means 'second series' and so for 3d — this is probably about brevity as well as archaism. I don't know what happens when we get to the fourth series, though.

17. I'm not from Ireland, and I rhyme eighth with faith; eighth is just /eɪθ/.

18. How does one pronounce 0th? "noughth" sounds too like "nought"

"Noughth" definitely exists, for example in noughth week, the week before the start of term in some universities. Though logically it should be noughtth.

19. 0th is "zeroth" in maths, as in e.g. "zeroth-order equation", "zeroth-order approximation". What it is in music in the English language I don't know, but Bruckner's symphony number 0 is designated "Nullte" in German.

20. @AJD:
But in Ireland they would both have a dental or less commonly alveolar plosive at the end of course. The fact that there is a plosive at the end of "eighth" is what causes the preceding plosive not to be pronounced I believe, because it would be difficult to have two plosives in a row. I think Wells mentions this in one of his books. But Dictionary.com and American Heritage both list /eɪθ/ as a possible pronunciation, which I never realized until now.

21. I've always seen (n + 1)th. I don't think it's said out loud very often.

What I've always wanted to know is how to say words such as 1/22, 1/32, 1/42, etc. I remember a teacher at school saying 1/32 as ðɜːtɪ tuːθ and then laughing at how childlike it sounds. Is there any rule for pronouncing these fractions?

22. @AJD: the same would I apply to me. Also from that list, I would rhyme "twelfth" with "health", "stealth", "wealth", etc. (I think most people do) and "wolf" with "Gulf" (northern half of England).

23. 1/32 - that's much stranger to me than (n+1)th. Of course, you could avoid the dilemma by saying "a two-and-thirtieth". But indeed, "a thirty-twoth" and "a thirty-second" both sound wrong (the latter's better for me), let alone constructions involving "half", which may even be misleading.

24. 1/32 for me is "one thirty-second" or "one over thirty-two", unless we are talking about music, in which case it is "a thirty-second-note".

25. I'm an American who studied math in France before returning to the U.S. years later.

So I developed a personal way of reading some words, not being exposed to mathematical English on a daily basis, and was surprised by some things when I was eventually confronted with the way people actually said them.

I used to anglicize the name "Euler" as "Yooler", only to discover on my return that it was actually pronounced the German way.

I also read the Greek letter as "fie", and was surprised most people said "fee". I still say "fie".

I had always said "n plus oneth", but eventually changed to "n plus first" when I realized most people seemed to say it that way. I have to say, I find (n+1)th more logical. But (n+1)st is at least analogous to "hundred and first" if "plus" is considered to be the same as "and".

By the way, in French, there's no ambiguity whatsoever. It's "n plus unième", just like "vingt-et-unième".

26. Americans normally say phi, xi, pi, and psi with the PRICE vowel, but anomalously pronounce beta, eta, and zeta with the FACE vowel.

27. If you visit an American math or physics department, you'll see that usage is divided on phi, xi, and psi (though not pi).

28. Do you say (n-1)st or (n-1)th ?

29. The last three comments, by Joan Brown, Inversiones en oro, and Ben, are spam.

I came across this because I'm reviewing a paper that uses (n+1)th, and I wanted to find a definitive answer...

My 2 cents: It is (n+1)st and (n+2)nd, and wherever possible. It is important to have the right sound when you read math aloud, even in your own head.