User:JoergenB/Approximant

Approximant

• Approximant as a mathematical term. The article Approximant refers to the linguistic term, and IMHO should continue to do so; the term seems both central and in frequent usage in that field, while the usage in mathematics is more sporadic, and often older. (If it would turn out to be employed in any modern textbook in a not too specialised area of mathematics, I might reconsider that opinion.) I've added an alternative name redirect Approximant (continued fraction); and there is an article Padé approximant (where some other "approximants" are mentioned, too); the related Padé table shreds some light on the terminological connections, I think. Further use for should be sought, and then a fork page should be written, either Approximant (mathematics) or Approximant (other uses).

Other found usage includes Extrapolation (with not only the ubiquitous reference to Padé approximant, but also a red link to Rational approximant), History of the administrative divisions of China (in usage "approximative"?), Michael Fisher ("partial differential approximants", not linked), Square root of 5 (not linked, probably close to cont. frac. usage), IDN homograph attack ("approximants for...alphanumerics", sence approximately 'approximations'), Nth root ("Newton approximant", no link, but given explicitly), 0.999... (ref. to Tom Apostol, 1974), Percolation threshold (interesting ref. to article fr. 2005 whose title mentions "the Cubic Approximant Method"), Wikipedia:Reference desk/Archives/Mathematics/2010 February 7, Wikipedia:Reference desk/Archives/Mathematics/2007 January 20 (used without reference, in approx. the cont. fract. meaning, by User:KSmrq; might be an idea to ask it?), Wikipedia:Reference desk/Archives/Mathematics/2010 September 26 (by User:Count Iblis), Wikipedia:Reference desk/Archives/Mathematics/2008 December 4 (by User:Fredrik), Wikipedia:Reference desk/Archives/Mathematics/2009 March 5, Wikipedia:Administrators' noticeboard/IncidentArchive98#metric first or Imperial first (by User:CBDunkerson, for measures like "5000 cubits"), Wikipedia talk:German-English translation requests ("...a Kreis is a fairly good approximant of an English district), Talk:Golden canon of page construction, Talk:Vesica piscis (illuminative for "rational approximants"), Wikipedia talk:Disambiguation/Archive 27 ("approximantly" used as "approximately"; typo?), Wikipedia talk:WikiProject Mathematics/Archive 55, Wikipedia talk:WikiProject Mathematics/Archive 33 ("Riemann approximant sum"), Talk:Radial basis function, Talk:Yungas Road ("approximantly" = approximately), Talk:Golden rectangle, Talk:Chevrolet Volt, User talk:Misza13/Archives/2009/11 ("approximantly" = approximately), Talk:Optimal design, Talk:Rounding, Talk:Mathematical coincidence (mentioning planned section on "rational approximants"), Talk:Chronology of Jesus ("...the approximant date,..."), Talk:Hedge fund, Talk:Finite element method (interesting), Talk:Phonological history of English low back vowels/Archive 1 ("approximantly" = approximately - in a linguistic discussion!?), Talk:Golden ratio/Archive 1, Talk:Golden ratio/Archive 2, Talk:Ruđer Bošković/Archive 1 ("...a historical approximant" = approximation?), Talk:Philosophy/Archive 26, Talk:Continued fraction/Archive 1

OED on line gives two meanings to the noun; the first (older!!) mathematical

1. Math. An approximation to the solution of a given problem (whether a function, a series, etc.). (Should be copyright OED, thus ought not to be used directly.)

The second noun use (dating from 1964, seemingly) is the linguistic one. The adjective usage is considerably older (excerpt from 1641), but "obs. rare":

Approaching closely, resembling. (Should be copyright OED.)

Possibly, "Approximant" paired with "Approximate(ly)" and/or "Approximation" might be a candidate for the page List of commonly misused English words; but I do not know how "common" the "misuse" is. I'll anyhow put a question there.  Done, at Talk:List_of_commonly_misused_English_words#Approximant_versus_approximative.28ly.29_or_approximation, 14:22, 13 October 2010 (UTC); I'll wait a little, to see whether there is a response from some native English speaker...

Quote(s) Wall

Quote(s) from H. S. Wall, Analytic Theory of Continued Fractions, van Nostrand, 1948. In all probability, covered by "less than 70 years" copyright. Could not be employed directly; I'm making a quotation for the purpose of pinpointing Wall's meaning of the word, nothing else.

pp. 14-15:
We shall introduce some definitions with a view toward making these ideas more precise. The numbers a_p and b_p, called elements, may be any complex numbers; a_p/b_p is called the pth partial quotient, a_p is the pth partial numerator, and b_p the pth partial denominator. The quantity

$t_{0}t_{1}\cdots t_{n}(0)=b_{0}+{a_{1} \over b_{1}+{a_{2} \over b_{2}+\ldots +{a_{n} \over b_{n}}}}$

is called the nth approximant. (Fotnote: This is sometimes called the nth convergent.)