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The main question here is whether using -(e)th to create ordinals out of cardinals1 is still considered a productive suffix in English. Is it?

If so, then does it matter whether we are in a formal or informal register? Is it accepted in some contexts but not in others? Is it blocked by certain circumstances?

Consider these sorts of formations, arranged by least to most shocking; are they legal, illegal, or suspect?

  • the nth or nth item
  • the zeroth element
  • googolth for which item is number googol
  • epsilonth and ¹⁄epsilonᵗʰ (¹⁄epsilonth)
  • alphath or perhaps alphath
  • calculating the (1−alpha)th percentile
  • three-halfth for ³⁄₂ᵗʰ (³⁄₂th)
  • two-thirdth for ²⁄₃ᵗʰ (²⁄₃th

Footnotes:

  1. Such as we do with four > fourth, fifty > fiftieth, and billion > billionth
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  • 2
    And another hat-seeking missile is launched.
    – Robusto
    Commented Dec 22, 2012 at 14:55
  • Very suspect. It will work in informal situations only. The meaning of the suffix is untenable unless it is attached to a count. (ie '3/2th' doesn't make sense).
    – Mitch
    Commented Dec 22, 2012 at 16:05
  • @Mitch But why does is (1/epsilon)th accepted then?
    – tchrist ♦
    Commented Dec 22, 2012 at 16:10
  • Accepted by who? Do you have an instance where it makes sense?
    – Mitch
    Commented Dec 22, 2012 at 18:04
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    If it works, then so do the other ordinalizing allomorphs -- -st, -nd, and -d.
    – John Lawler
    Commented Dec 22, 2012 at 19:16

2 Answers 2

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The short answer: yes, it is productive, because you can create words using this suffix that have never been heard before, such as the two-trillion-and-sixteenth coin in Scrooge McDuck's Money Bin. From Wikipedia:

A productive grammatical process defines an open class, one which admits new words or forms. Non-productive grammatical processes may be seen as operative within closed classes, but only previously formed and learned structures show the results of those processes. (Italics mine.)

The point of productivity is not that you can add a suffix to any word you please, but that it can be added to some words to create new words. Only if it cannot be added to any words in order to create new words is it said to be unproductive.

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    +1, but I am sure I have seen that example before.
    – Robusto
    Commented Dec 22, 2012 at 15:26
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    Yes, but there are only so many of those words, right? True, we are in the countably infinite Cantorian realm of ℵ₀, but it is still countable.
    – tchrist ♦
    Commented Dec 22, 2012 at 15:34
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    @tchrist: This counts as infinite. The crux is that a speaker can and will form a word he hasn't heard before using this suffix. The theoretical number of such possibilities doesn't matter in the end.
    – Cerberus - Reinstate Monica
    Commented Dec 22, 2012 at 15:48
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    I am suspecting you cannot add -th to a word that already has an irregular ordinal: first, second, third. So no two-thirds-th for (2/3)th — maybe. But that probably doesn’t count for blocking productiveness. I almost wonder whether half was once an ordinal.
    – tchrist ♦
    Commented Dec 22, 2012 at 16:28
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    I think this is right, not only because I personally can coin the (previously unknown, I'm sure) seven-trillion-and-nineteenth possibility, but because thousands of people have referred to the googolth power since this became possible (around 1940?). Interestingly, Google itself gets a bit confused there - it insists on showing me the googleth power first, even though apparently that's a (slightly) less common form.
    – FumbleFingers
    Commented Dec 22, 2012 at 22:38
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A formal register tends to eschew neologisms, so using -th to produce a word like coolth1 would tend to restrict it to informal use. It does mean that it is a productive suffix, though.

1 Coolth meaning fashionability, fashionableness

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  • I was thinking of the ‑th suffix used in zeroth and nth (or nᵗʰ), not the ‑th suffix used in length, width, breadth — and historically, heighth [sic]. Nor the inflection ‑th suffix used in doth and thinketh, either, lest the matter should proceed so far. :)
    – tchrist ♦
    Commented Dec 22, 2012 at 15:37
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    'coolth' ... That's not the ordinal of a cardinal.
    – Mitch
    Commented Dec 22, 2012 at 16:02
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    And oughtn't it really to be chilth (or something like that), anyway?
    – Janus Bahs Jacquet
    Commented Dec 20, 2013 at 3:27
  • @JanusBahsJacquet: umlaut before -th hasn't been productive for a while. Compare growth and sloth (the umlauted form "sleuth" inherited from Old English is now obsolete).
    – herisson
    Commented Nov 19, 2016 at 7:26
  • As Mitch mentioned, isn't this addressing a separate suffix from the one that the question is about?
    – herisson
    Commented Nov 19, 2016 at 7:27

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