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Mathematical Grammar II March 12, 2008

Posted by David Speyer in Uncategorized.

Which is better: (j+1)^{\textrm{st}} or (j+1)^{\textrm{th}}? What about (j-1)^{\textrm{st}} versus (j-1)^{\textrm{th}}?

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1. L. Zoel - March 12, 2008

if I was speaking, I would say “jay-plus-first” or “jay-minus-first” so I would tend to favor “st” in both cases

2. yo - March 12, 2008

The latter on both occasions. The “j plus one”-st sounds a bit strange I think.

3. D. Eppstein - March 12, 2008

“Jay plus first” or “Jay plus wunth”? Both sound wrong to me. Reword to avoid the awkward language.

4. Theo - March 12, 2008

It’s definitely the terms before and after the $j$th term in a sequence are definitely the $(j \pm 1)$th terms, when writing. The “th” is not to be pronounced — you should not read mathematics verbally, because that fails to capture the nonlinearity of mathematical writing — but rather it visually marks this as an ordinal rather than cardinal number.

In speaking, I think I say “Jay plus first”, but only if I believe that “two plus first” equals “third”.

5. complexzeta - March 12, 2008

I like (j+1)th and (j-1)th.

6. Steve - March 12, 2008

I agree with Theo. The “th” marks it as an ordinal and looks cleaner in writing, even if “jay plus first” sounds better when spoken.

7. Henry Cohn - March 12, 2008

Definitely not with “th”. I’d personally go with “st”, on the grounds that “j plus first” is what most people seem to say out loud, and it is a perfectly reasonable construction (given that we say “hundred [and] first”, etc.). The case of subtracting one sounds a bit weirder to me, probably because the analogy with English number formation is weaker, but the “th” still sounds weird to me. I don’t like constructions that can’t reasonably be read aloud.

Either way you choose, somebody’s not going to like it, so David’s right that it may be best to avoid the whole construction.

I disagree with Theo and Steve: I think the argument about what people say out loud should carry more weight. I also pass the Theo test of thinking “two plus first” is a way of saying “third”. :-) It’s certainly an eccentric and awkward way, but I’d be happy to say something like this in a talk. “We have proved that the j plus first element of the sequence is special. In this example, j equals two, so the two plus first, or third, element is the one we care about.”

8. Aaron - March 13, 2008

Strong preference for (j+1)st, weak preference for (j-1)th, for the reasons everyone else has given.

9. Charlie C - March 13, 2008

Ask the guys at Language Log.


-Charlie C

10. Todd Trimble - March 13, 2008

I’m with Henry all the way. People do frequently subvocalize or mentally articulate words while reading mathematics (and I don’t understand Theo when he says they “shouldn’t” — after all, any mathematics one reads is also mathematics one can talk about out loud with a colleague). The sound of language is an important consideration.

11. Kurt - March 13, 2008

(j+1) to the st power? What!? While it’s the grammatical convention to treat these things as ordinals and say first, second, etc. for the initial terms, I like to think of (j+1) as an index, not an ordinal. It’s the (j+1) term of the sequence, period. No need for any grammatical accouterments on that.

12. David Speyer - March 13, 2008

Huh. I’ve got to say, Kurt’s argument is only one so far which has no resonance to me at all. Integers can’t be used as adjectives in that way. When an integer is used as an adjective, it is a cardinality, not an index. You can say “The US has had 41 presidents.” but not “Abraham Lincoln was the 16 president.”

People’s responses are convincing me that these phrases should not be used at all. Before seeing the answers, my feeling was that “-th” was more logical but “-st” sounded better and I would probably go with the sound.

Why more logical? Order of operations — we substitute a value for j before we modify the integer into an adjective. So, by time the modifier -th hits, j+1 probably doesn’t end in 1 anymore.

Why do I ignore this argument? Because I think that the rule that -oneth changes to -first is fundamentally a phonetic rule, not a logical one. I think we can all agree that there are such rules; I think everyone says “an n-sphere”, not “a n-sphere$, even if we know that 2 \leq n \leq 7. (I’ve been trying to construct an example using ordinary English pronouns instead of variable names, but I haven’t found one yet.)

13. Scott Carter - March 13, 2008

“Integers can’t be used as adjectives” Huh?
I was about to go eat two cookies before I read that. Real numbers, despite their fictitious nature, are precisely adjectives. The cookie jar is 3 \sqrt{2} meters across the room.

14. Kenny - March 13, 2008

“can’t be used as adjectives in that way” – they’re used as adjectives for cardinalities (and related concepts like measures) but not as ordinals. Interestingly, it seems possible to use bare integers for ordinal purposes if you put them after the noun instead of before – “item 16 in the list” vs. “the 16th item in the list”. But maybe this isn’t quite right – “President 16 was Abraham Lincoln” sounds a little weird, and Wayne Gretzy certainly wasn’t the 99th player on the Edmonton Oilers, even though he was player 99.

15. John Palmieri - March 13, 2008

How about (1+j)th and (-1+j)th, to finesse the whole issue? I mean, if either of your two choices (j+1)th and (j+1)st is going to distract a large number of readers, try rewording it in a way that is less likely to do so.

16. Ben Webster - March 13, 2008


That would be so much more distracting…

17. Odd Man Out - March 13, 2008

I’ve always used the ‘st’ for first (1st) and ‘th’ for the others (except 2nd and 3rd). IMO, in general, it’d be best to use ‘th’ for a general term since the first 3 number are rather exceptions when considering the number line as a whole.

So, my vote is ‘th’ in both cases.

18. Robert Samal - March 13, 2008

I’m surprised nobody suggested the dash sign, what I would use is
$(j+1)$-st (or $j$-th, $(j+5)$-th, …).

I’m not sure which is more proper English, but it definitely seems to me,
that it is better
1) not to write th (or st) as superscript (to avoid confusion with powers)
2) add a dash – just for more pleasant reading

19. Isabel Lugo - March 13, 2008

I don’t think I can add to what people have said about written mathematics. But when speaking I think I’d refer to “the jay plus one element” or “the jay minus one element”.

Then again, questions like this are always a bit hard to answer by pure introspection; what one really wants is a corpus of mathematical speech. (Or mathematical writing — but that wouldn’t be too hard.)

20. Scott Carnahan - March 13, 2008

Google says:
“n plus oneth” – 65 hits on the web, 3 in Google scholar
“n plus first” – 834 on the web, 29 in scholar
“n plus 1th” – 37 on the web, 0 in scholar
“n plus 1st” – 7 on the web, 0 in scholar

I had some parsing difficulties with e.g., “(j+1)th”, since most punctuation gets ignored or turned into a space, and it becomes some kind of biblical reference.

21. Scott Morrison - March 14, 2008

I knew that having a local copy of the entire arxiv would come in handy one day!

A quick and dirty grep on every maths papers from 2007:

for file in `find .`; do zcat $file | grep -a “+1[^ ]th”; done | wc

says that it finds 40 results; the corresponding grep for st returns only 35. The “th”s have it!

(You too can have a local copy of the arxiv, if you have 35gb free, and some bandwidth; no guarantees, but try your luck with this torrent I prepared.)

22. John Armstrong - March 14, 2008

it seems possible to use bare integers for ordinal purposes if you put them after the noun instead of before – “item 16 in the list” vs. “the 16th item in the list”

I tend to think of that as using an alternate referent, combined with a certain shorthand. Think of the objects as data records. Gretzky has a name and a number, among other properties. We can refer to them by name (“Wayne Gretzky”) or by another field (“[the] player [whose number is] 99″). In practice we drop “one” a lot of the time.

The confusion is that in many cases such indexing numbers are assigned chronologically, and in sequence, so we conflate indices with orderings.

23. Kevin - March 15, 2008

While we’re on this topic, how should “nth” be written in TeX?


24. Andy P. - March 15, 2008

Kevin — IMHO, the best way is $\text{n}^{\text{th}}$, or for numbers $5^{\text{th}}$. Of course, this is best accomplished with a macro.

25. James - March 15, 2008

I use “$n$-th”. I think it is less likely to be misinterpreted as “$n-th$” than “$n^{\text{th}}$” is for “$n^{th}$” or “$n$th” is for “$nth$”. But if if I were completely certain there would be no confusion with any of these, I would use “$n^{\text{th}}$.

26. Henry Cohn - March 16, 2008

The confusion is that in many cases such indexing numbers are assigned chronologically, and in sequence, so we conflate indices with orderings.

In my dialect of mathematese, this is precisely the distinction between sequences and functions defined on the natural numbers. The two notions are identical from an abstract point of view, but I consider the ordering to be crucial for a sequence and not for a function. In particular, it is reasonable (and even mandatory) to speak of the first, second, etc. terms of a sequence, while it would sound weird to call f(5) the fifth value of a function f from {1,2,3,…} to the real numbers. So from my perspective, you can’t avoid this issue by treating everything as an index. Ignoring the implicit ordering in a sequence sounds bad too.

As for writing the “th”, I definitely prefer $n$-th, since the hyphen helps with readability, although I don’t think I could give a principled defense of it. I don’t like superscripts in any case, even when $n$ is replaced with an actual number. I tend to use superscripts when writing by hand, but I view them as a defense against bad handwriting, to keep the superscript from being misread as part of the number. Similarly, when I write checks, I often write the number of cents as a superscript, not because I think it ought to be typed that way but rather because I don’t want it to be misread by the bank. Of course, typed text can be tricky to read as well (which is why I prefer $n$-th to $n$th), but I’m more worried about handwriting so I take more precautions there.

Of course, this gets into the whole issue of whether typing ought to imitate handwriting. In general, I think the answer is no, but I do like blackboard bold symbols in type, despite the argument that they were originally just an attempt to write bold symbols by hand.

27. Allen Knutson - March 16, 2008

Absolutely $(j+1)$st. And similarly, $n$th. Why should $n$th get an exponent, when fourth and fifth aren’t four^th and fif^th?

I think it’s very important that $n$th isn’t nth. Part of the question of “how to write this” involves what fonts are used. And once the font has changed from math to prose, the exponent construction is unnecessary.

(Similarly, I always speak of the $0$s and $1$s in a string of bits, rather than the $0$’s and $1$’s — what if I want to use the possessive at some point?)

28. James - March 16, 2008

But $4^{\text{th}}$ does get an exponent, and you could specialize to $n=4$ but not $n=\text{four}$.

29. Todd Trimble - March 16, 2008

Re this whole business of writing the “th” as a superscript: there’s nothing logical about it of course; it’s just based on an old, old manner of abbreviating. For example, Mister or Master was once M^r (often with the “r” underscored), and similarly there was D^r, M^mme, etc. This would also apply to names (e.g. Tho^s for “Thomas”).

I think many manuals of style now recommend against the superscript notation, but old habits die hard.

(An irrelevant and off-topic factoid about another old shorthand: all of us have seen quaint names like “Ye Olde Tea Shoppe”. What is not so well known is that this “ye” or “y^e” is really just a way of writing “the”: the “y” here is not really a “y”, but a way of scribing the old runic character þ (pronounced “thorn”), which functioned as a “th”. In particular, it was not originally pronounced as “yee”.)

30. joe shmoe - March 19, 2008

According to several math style guides, the correct form is to include the (j-1) in math mode and the “th” in text mode, not superscripted, directly after. This also requires ensuring that the two appear together and are now broken by a linebreak.

31. Ben Webster - March 19, 2008

You know, if there are several of them, it shouldn’t be hard to provide examples.

32. Kevin - March 19, 2008

joe shmoe: what math style guides are you referring to?

33. Kurt - March 19, 2008

I have to admit, until I read joe shmoe’s comment, it didn’t occur to me to actually look in a math style guide. Here’s the relevant section from “Handbook of Writing for the Mathematical Sciences, 2nd Ed.” by Nicholas Higham:

5.5. Ordinal Numbers

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, firsst, 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.

I still like my earlier answer, though. :)

34. Request: Modular forms « Secret Blogging Seminar - July 15, 2008

[...] have Fourier expansion , where B denotes Bernoulli numbers (which are rational), is the sum of the (2k-1)st powers of all divisors of n, and is a coordinate on the unit [...]

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