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Sometimes when writing I find myself looking for a word to describe something in the “³⁄₂th” place — exactly between first and second. I would like to ask, does there exist an easy expression for this? If so, does it generalise to other “in-between” places as well?

(I mostly encounter this problem when writing mathematics, where I wish to write the “³⁄₂” order derivative, and normally have to endure clumsy or imprecise expressions (“Taking the fractional derivative. . . .”).)

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Can you provide more context? Maybe it's just me, but I mean, between the first and the second... what? :D Between two objects? Between 1st and 2nd competitor/athlete? –  Alenanno May 23 '11 at 9:53
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Ordinal numbers are used for non-negative integers, I doubt if there is a word like that for "3/2st". –  user8568 May 23 '11 at 10:20
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@Boob No, that isn't correct. One would like to have such words for the (much much more) general case of topological spaces with a well-defined ordering, but this was too much to hope for out of English. I shot rather low with this question. There is motivation (and validity) to much more than just these 'extra' words. –  Glen Wheeler May 23 '11 at 12:27
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@Boob Topology is contained in mathematics. –  Glen Wheeler May 23 '11 at 15:37
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@Boob Again, that is just not true. An example of a topological space with an ordering is the natural numbers, which are the positive integers. Topology and geometry are related subfields of mathematics but one is not contained in the other. –  Glen Wheeler May 24 '11 at 7:47
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6 Answers

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The word you are looking for is the sesquialter, sesquialteral, sesquialterate, sesquialteral, or sesquialterous element.

You can also use sesquialter as an ordinal noun, for the midpoint between first and second:

I’ve skipped the zeroth, but then jumped straight from the first to second, so I’m am still missing the sesquialter I expected midway between those two.

That’s because sesquialter means “one-and-a-halfth”,1 but is substantially easier to say.2 It is one of those sesquipedalian terms surpassing both in erudition and utility alike. Per the OED, it means:

1. Of a proportion: That is as 1½ is to 1. Of an object: Proportionate to another object as 1½ is to 1; that is such a multiple of.

Here are a few of its more recent citations:

  • 1698 Phil. Trans. XX. 81 ― We assign to a Fifth··the Sesquialter Proportion (or that of 3 to 2).
  • 1711 H. Needler in J. Duncombe Lett. (1773) I. 90, ― 6 is only sesquialter of 4.
  • 1715 Cheyne Philos. Princ. i. 222 ― In all the Revolutions of the Planets about the Sun,··the periodical Times is [sic] in a Sesquialter Proportion to the middle Distances.
  • 1784 J. Keeble Harmonics 29 ― The sesquialter chromatic.
  • 1846 Penny Cycl. Suppl. II. 369/2 ― The following ratios are super‐particular: 15 to 10, which is sesquialter.

And its etymology is:

Etymology: L., f. sesqui- (see prec.) + alter second. For the formation cf. ONor. hálfr annarr, OE. óþer healf, G. anderthalb.

The sequi- prefix is today most familiar in terms like sesquicentennial, for the 150-year anniversary of some event. It gives rise to a delightful multitude of derived terms.

1. a. With designations of measure or amount, denoting one-and-a-half times the unit; as sesquihōra an hour and a half; sesquipēs a foot and a half (see sesquipedalian); so † sesquiˈhoral a., lasting an hour and a half; ˌsesquioˈcellus Ent. (see sesquialterous); † ˈsesquitone Mus., an interval consisting of a tone and a semitone, a minor third; also used loosely in † sesquiˈdecuman a., consisting of fifteen; † sesquiˈdecury, a set of fifteen.

The OED provides not only senses 1b through 1d, with which we need not here concern ourselves, but also the operative sense 2a:

2. a. With an ordinal numeral adjective, denoting the proportion 1 + 1n:1, i.e. n + 1:n, where n is the corresponding cardinal number, as sesquioctāvus, bearing the ratio 1⅛:1, i.e. 9:8; so sesquialter, -altera, etc., sesquitertia, etc.; † ˌsesquibiˈtertial, involving a proportion of 5:3; † ˌsesquiˈdecimal, of 11:10; sesquiˈnonal, of 10:9; ˌsesquiocˈtaval, -ˈoctave, of 9:8; ˌsesquiˈquartal, -quartan, of 5:4; ˌsesquiˈseptimal, of 8:7.

Sense 2b also gives several nice words for harmonic situations:

b. in Music, after sesquialtera and sesquitertia; sesquiquarta, -quinta, -sexta, -octava (-octave), -nona, applied

  • (i) to harmonic intervals producible by sounding four-fifths, five-sixths, etc. of a given string;
  • (ii) rhythmic combinations of four notes against five, five against six, etc.

As you correctly perceive, having a word that means ³⁄₂th is exceedingly convenient at times, which no doubt why are ancestors invented sesquialter way back in the 16th century. This is the first citation given for the term in the OED:

  • 1570 Dee Math. Pref. c j b, ― A Cylinder, whose heith, and Diameter of his base, is æquall to the Diameter of the Sphære, is Sesquialter to the same Sphære.

Footnotes:

  1. Regarding the productivity of the -th suffix for creating ordinals from cardinals in English, see this question.

  2. Insofar as words that have end in -fth or -xth are often considered difficult to pronounce by non-native speakers, like fifth, sixth, twelfth, and if they find twelfth tough, they seem likely to find halfth tough as well.

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Wow! Thanks, +1 –  Glen Wheeler Feb 1 '13 at 2:20
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If you google halfth, you get 114,000 hits, the first of which is "One-and-a-halfth-order logic." And there's the famous cartoon Duck Dodgers in the 24½th Century (the official pronunciation of this is twenty-fourth and a half).

UPDATE:

Googling, I get two hits for "one-and-a halfth derivative" and three for "first-and-a-half derivative". So these are both possibilities.

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+1, you were faster, sir –  Unreason May 23 '11 at 11:38
    
Indeed, the pronounciation is a big problem. When one gives a talk, and wishes to say "Computing the (3/2)-th derivative gives us..." it becomes cumbersome and sometimes embarrassing. With yours and Unreason's answers, I shall use "X-and-a-halfth". The question now becomes, how do I accept both answers? –  Glen Wheeler May 23 '11 at 12:25
    
It's not correct. There is no place for 3/2, when we count members of a queue, we cannot say "1st one, one-and-a-halfth one, 2nd one, ..". there's no order for non integer numbers. –  user8568 May 23 '11 at 15:22
    
@Boob You are missing the point. The situation is that we have a set of objects, they have a well-defined ordering, one wishes to refer to those in the "3/2th" place. That they have a well-defined ordering is not in question, and that seems to me to be what you are struggling with. –  Glen Wheeler May 24 '11 at 7:49
    
@Boob: There's a first derivative, and a second derivative, and the OP wants to talk about a mathematical object halfway between them. You could say the "3/2 order derivative" or the "derivative of order 3/2." But I think "one-and-a-halfth derivative" or "first-and-a-half derivative" are more comprehensible, albeit less grammatical. If the OP needs to start talking about the "1.743th derivative", then "derivative of order 1.743" starts sounding better. –  Peter Shor May 24 '11 at 11:24
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This is a hard one, and maybe you would have better luck at mathoverflow (but do link to here, so that they don't send you away).

In mathematical papers it is not uncommon to find constructs which bring out situations that common language would not encounter (or that would be so rare that short of a poem you would have trouble find it).

Back to your issue - it is really not uncommon to find and see as completely normal linguistic monstrosities such as for example

... of the (α+1)-th order ...

which seem natural when reading the paper, but if you stop to consider how you would pronounce it if you were presenting the paper you are soon to realize that you would probably rephrase it on the spot to something else.

So, strictly mathematically speaking, if the above notation is acceptable to you then, mathematically, since α+1 can be anything then for α = 1/2 it is equal to 3/2, so for that case you would write

... of the (3/2)-th order ...

But now the problem of pronunciation is even more proclaimed.

I did find an article called one-and-a-halfth-order logic in oxford journals.

Though I did not find dictionary definitions of halfth I did find quite a few results for it in books.

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I did think of asking on mathoverflow or math.stackexchange, but normally matters of language are asked here as a first stop. The suggestion of "halfth" is exactly what I was looking for. Is it really correct? –  Glen Wheeler May 23 '11 at 12:22
    
@Glen, no it is not really correct - the word "halfth" is not listed in dictionaries, so it is relatively newly coined word. But if oxford journal accepts it you have a precedent that you can refer to. Logical confusion remains - ordinal numbers are normally reserved for positive integers. The situation is similar to merriam-webster.com/dictionary/zeroth, where word was coined to expand the domain to which the construct applies to for reasons similar to yours (with the difference that zeroth already made it to the dictionaries). –  Unreason May 23 '11 at 12:50
    
Other difference is that in similar situation, for example of exponentiation, ero did make it to be accepted, but 1/2, 2/3, 3.14, etc did not take on the -th principle and you will not say "to (2/3)-th power", but "to power of 2/3". This suggest that such approach should be more appropriate in this case, too. Especially if your orders are arbitrary rationals; if you mostly talk about derivations of order 1/2, 1 1/2, etc... you might take "halfth" (though pronunciation suffers again). –  Unreason May 23 '11 at 13:02
    
Pronounciation is one major motivation for this question, although another is aesthetics. When writing a large computation on a chalkboard, and grouping terms with different orders (derivatives or powers or whatever), one likes to write on the side "first order terms", "second order terms", and so on. It looks very clumsy and frankly unsatisfying to have to write "terms of order 3/2" in between the lines "terms of first order" and "terms of second order". I'd rather write "3/2th order terms" than change all to be "terms of ...th order". –  Glen Wheeler May 24 '11 at 7:52
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Found this one sesquialteral. Although I must admit, this word was suggested by google translate as a translation for not quite used - not in the sense OP intended - word in my native language

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Not indicative of ordinal place –  trideceth12 May 23 '11 at 10:40
    
@92MGRXvmoFfCkCd1JH4p9jpZjFQRKp OP wanted a word for a fractional derivative - here's one. I wouldn't use either sesquialteral or it's original counterpart to count objects - in normal world there can be no object with the count "3/2" (if you're not talking about platform 9 3/4 from well known book) –  Philoto May 23 '11 at 10:46
    
OP's comment "For positive integral orders, we have words in English (first, second, third...). I just want to know if there is a word for "3/2 st"." :p –  trideceth12 May 23 '11 at 10:47
    
@92MGRXvmoFfCkCd1JH4p9jpZjFQRKp The OP's mentioning (in brackets) of fractional derivatives made me post my answer. As I said, I wouldn't find an object with number 3/2 in real world. –  Philoto May 23 '11 at 10:55
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Ordinal numbers are used for non-negative integers, I doubt if there is a word like that for "3/2st".

Decimal and rational numbers and fractions cannot be written as ordinal.

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You should not refer to 3/2 as a decimal number, it is a simple fraction which is a rational number. Decimal numbers can represent certain fractions, but not all. –  Unreason May 23 '11 at 11:04
    
@Unreason: It's kind of rational number which can be written as a decimal fraction because of prime factor"2". anyway, Answer improved. –  user8568 May 23 '11 at 11:21
    
@Boob, indeed 3/2 can be written as decimal number 1.5, however it is wrong to say "because of prime factor 2". Rational numbers do not have prime factors, only integers do (unless you were referring to the fact that 2 is a prime factor of 10?). –  Unreason May 23 '11 at 11:54
    
@Unreason: What? Rational numbers which have prime factors "2" and "5" can be written as fraction decimal. –  user8568 May 23 '11 at 11:55
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@Boob, it is related. Decimal is short for decimal fraction (denominator is a power of ten). Therefore, to represent other fractions as decimal, you should be able to make their denominator a power of ten. This is possible if that denominator is a prime factor of some power of ten. In this case 2 is a prime factor of 10, so it is possible. Rational numbers still don't need (nor have) prime factors (unless they are also integers). If you allow repeating decimals in decimal numbers you can represent all fractions. Still you would refer to 3/2 as fraction, and to 1.5 as decimal. –  Unreason May 23 '11 at 12:05
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  • 'halfth' is not (currently) a 'word' in English.
  • mathematically, there is no such thing as an ordinal between one and two. If you are discussing one and two as relative orders of things, and then want to refer to something between those, then you have to re-order and rename, -or- you're not talking about an order any more. For example, with derivatives, if you want to talk about the derivative $f^{(3/2)}$ which is somehow between $f'$ and $f''$, then you're not talking anymore about repeating the derivative operator an integral number of times, you're applying the continuous derivative operator with the parameter 3/2. And so you should use a locution that reflects that (i.e. what you think is clumsy actually sounds right to me).
  • mathematically pedantic, actually one can have total orders (I use those terms technically now) on things other than the naturals (here the expected '<' order on the rationals or reals). But now we're pretty far away from any general English rules and we'd follow the jargon standards of the mathematical community. And there you would avoid entirely any informal neologisms like 'three/half-s-th' or 'third-halfs'.
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