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Anybody with elementary mathematical education will have seen improper fractions to refer to fractions where the numerator is greater than or equal to the denominator.

At a certain point in calculus studies, students are confronted with improper integrals, that is the limit of a definite integral as the bound(s) tend to $\infty$ or $-\infty$.

Improper prior distributions, in Bayesian statistics, are prior distributions that don't integrate to $1$.

Personally, as I learned these concepts, I found the use of the word improper to hint to a negative connotation.

A double question here.

$1)$ Are there any other objects in mathematical studies where this adjective is used?

$2)$ Were these mathematical objects named like this simply because they were extensions (one might also say "dares", such as taking the limit of a definite integral) of previously consolidated concepts or was this nomenclature used because of possible discord on the existence of such objects while they were being discovered?

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    $\begingroup$ There are also improper subsets and improper convex functions. The negative connotation is right, "improper" indicates "not behaving properly", i.e. so that handy technical properties hold. $\endgroup$ – Conifold Apr 9 at 19:25
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It is interesting to look at mathwords

IMPROPER FRACTION was used in English in 1542 by Robert Recorde in The ground of artes, teachyng the worke and practise of arithmetike: "An Improper Fraction...that is to saye, a fraction in forme, which in dede is greater than a Unit."

IMPROPER INTEGRAL occurs in "Concerning Harnack’s Theory of Improper Definite Integrals" by Eliakim Hastings Moore, Trans. Amer. Math. Soc., July 1901. The term may be much older.

PROPER FRACTION appears in English in 1674 in Samuel Jeake Arithmetic (1701): "Proper Fractions always have the Numerator less than the Denominator, for then the parts signified are less than a Unit or Integer" (OED2).

PROPER VALUE and VECTOR. See Eigenvalue.

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    $\begingroup$ What a website... how has its existence eluded me for so long? $\endgroup$ – Easymode44 Apr 9 at 14:32
  • $\begingroup$ Updated spelling for the first work: by Robert Recorde in The Ground of Arts, Teaching the Work and Practice of Arithmetic: "An improper fraction, that is to say, a fraction in form, which indeed is greater than a unit [one, unity]". $\endgroup$ – Robert Columbia Apr 10 at 3:10
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The usage of "improper" is too old to be usefully traced. It goes back to 1500s. Just checked the Unabridged Oxford English Dictionary. In such cases of relatively odd choices of words, I look up the same terminology in other languages. For instance, what do Germans use for improper fraction? A quick search shows it is unechter bruch and unechter means fake, false (fraction). It makes perfect sense from an purely arithmetical perspective. How can we have have more than 100 parts out of 100? It must have sounded fake and improper thing to do.

Here is what OED shows:

Not properly so called; improper fraction: a fraction whose numerator is greater than (or equal to) its denominator, and whose value is therefore greater than (or equal to) unity. (Formerly applied to analogous fractions in Algebra.) improper diphthong: see quot. 1795. 1552 R. Record Ground of Artes (rev. ed.) ii. sig. Q.vi An impropre fraction..that is to say, a fraction in form, which in deed is greater then an vnite. 1629 J. Fletcher Faithfull Shepheardesse (ed. 2) To Rdr. They [shepherds and shepherdesses in a Pastoral] are not to be adorn'd with any art, but such improper ones as nature is said to bestow, as singing and Poetry. a1690 S. Jeake Λογιστικη Λογια (1696) 44 Improper Fractions have alwayes the Numerator greater than the Denominator. 1795 L. Murray Eng. Gram. 4 An improper diphthong has but one of the vowels sounded; as, ea in eagle, oa in boat. 1806 C. Hutton Course Math. (ed. 5) I. 187 To Reduce an Improper Fraction to a Whole or Mixed Quantity.

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    $\begingroup$ Yes, I agree it must have felt really weird to experiment this way. Thanks for sharing the quote from the so-unaccessible OED! $\endgroup$ – Easymode44 Apr 9 at 14:36
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    $\begingroup$ It is an amazing resource for word origin lovers but a paid one of course (by the university). $\endgroup$ – M. Farooq Apr 9 at 15:55

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