History of scientific nomenclature and mathematical symbols (2024)

Nomina si nescis, perit cognitio rerum.
If you ignore names, actual knowledge vanishes.
Carl von Linné (1707-1778)

(Jon of Visalia, CA.2000-10-18)
What is the origin of the word "algebra"?

"Algebra" comes from the arabic title of a book by the Persian mathematicianAbu Abdallah Muhammed bin Musaal Khwarizmi (c.783-fl.847)which is transliterated as: Kitab al-mukhasar fi hisab al-jabr wa'l muqabala(Overview of Calculation by Transposition and Reduction).The Latin version of that title included two neologisms:Algebra et Almucabala. The first one stuck.In fact, al-Khwarizmi describes three different techniques to reduceequations: al jabr (or transposition from one side of the equation to the other,mostly to obtain positive quantities), al muqabala (reduction, orcancellation of like terms on either side of the equation) and also al hatt(the division of both sides by the same number).All such techniques are used jointly and they becameknown collectively under the name originally given to the first of them...

The word "algorithm" comes from al-Khwarizmi's own name(also transliterated as "al-Khowarizmi") which became "Algorismus" in Latin.Technically, an algorithm is a procedure known to always terminate,with any input data (which is certainly not the case of all computer programs).

The rare term "algorism" is best reserved for elementary rules of computationusing decimal numeration and what we now call "Arabic numerals"(although they came from India)because of a seminal book of Al-Khwarizmi himself:The original Arabic text has been lost, but there's an English translation(entitled "Al-Khwarizmi on the Hindu Art of Reckoning")of a Latin edition (Algoritmi de numero Indorum)known to differ substantially from Al-Khwarizmi's original.

Al-Khwarizmi was named after his birthplace,a city whose modern name is Khiva, Khiwa, or Chiwa,located in modern Uzbekistan, south of the Aral Sea and north of the Caspian Sea(30km southwest of Urganch).Thewalled part of the city(Itchan Kala)is a UNESCO World Heritage Site, considered a shrine by the locals.Itsrecorded history goes back to the 7th century.Itwas once the capital city of Chorasmia(the country of Kharezm, Khwarazm, orKhorezm),also known as the Khanate of Khiva from 1511 to 1920(following conquest by nomadic Uzbeks) straddling modern Uzbekistan and Turkmenistan.Thecountry was conquered by Russia in 1873 and was known as theKhorezm Soviet People's Republic from 1920 to 1924...

Since little is known about al-Khwarizmi himself,we may wonder if he took the name of the capital cityinstead of a less prestigious birthplace in the vicinity.The only known biographical fact about al-Khwarizmi seems to be thathis parents had moved to a place south of Baghdad.[1]

There seems to be a widespread confusion with another astronomer who flourisheddecades later: Abu Jafar Muhammed bin Musa al Khwarizmi.

(2004-03-31) Avoirdupois System (1 av. lb = 1 lb = 0.45359237 kg)
What's the origin of the "avoirdupois" name for units of weight.

Any French-speaking person would immediately see this wordas a contraction of the sentence "avoir du poids",which means "tohave [a lot of] weight".Actually, this is a distortion of "avoir de poids", where "avoir"is a noun (not a verb), meaning "goods" or "assets". This is evidencedby the ancient expression "aver de poiz" which gave the alternatespelling "averdepois" (still used in an historical context).Thus, the thing simply means ponderous goods,as opposed to less bulky items like jewels orprecious metals for which theavoirdupois system was not intended.

The French word "poids" is spelledwith a silent "d" not found in the word avoirdupois.However, this letter was never dropped at all,since the French changed the spelling of their own word afterthe British had borrowed it.Furthermore, the French did so for a fallacious reason...

When French spelling was standardized, a few silent letters wereused so that some like-sounding words could be distinguished in writing.For example, the silent "g" in the word "doigt" (finger)was borrowed from its Latin etymology (digitus) to distinguish it fromthe word "doit" (a form of the verb devoir, which means "must").Thus, poids (weight) was differentiated frompois (pea) by a silent "d", ostensibly borrowedfrom the Latin word pondus (weight).The funny thing is that the correct etymology of poids isnot "pondus" but pensum (massive)from which the "s" in poi(d)s originates!

In a way, it's the French who made the spelling mistake, not the British.

(2003-11-03) Long Division
Cultural differences in writing the details of a division process.

If you're not a native speaker, you may need to be told that, in English, it'sthe same thing to "divide 145 by 5" or to "divide 5 into 145".People work out the division 145/5=29 very differentlyin different parts of the World:

	29		145		5
5	)145				29

The left layout is used in the US, the UK and Japan(thanks to Mitoko Sato-Chocat for pointing that out, 2018-10-12).In the UK, at least, another layout is used for short divisions(as discussed below).

The right layout is apparently dominant elsewhere: France, Brazil, etc.(Please tell ushow you were taught these or other layouts, where and when.)Most of the "action" takes place under the dividend (145 in this example).Either layout is thoroughly confusing to grown-ups who were taught the other way as kids!

On 2004-11-02, MaryNeerhout Borg (Oregon) asked:

[In the US layout]what is the "little house" over 145 called?

The order in the English layout (above left) is consistent with theidiom "5 goes into 145 [29 times]". Therefore, it's been suggested that the symbolconsisting of the top vinculum and the curly vertical part should becalled a guzinta (a tongue-in-cheek name meant to be pronounced like "goes into").

Help from our readers :

[In a 2004-09-06 e-mail] JohnFannon tells us that, about 1950, young British pupils were instructedto use the above left layout[possibly with a straight vertical separation instead of a curly one] onlyfor long divisions...For short divisions (with a divisor of 12 or less)they used another layout, illustrated above,where the successive remainders appear as superscripts of the dividend's digits...

On 2008-03-10,Biniam Girma wrote:

This is how we learn to divide in Ethiopia.
The remainders are placed abovethe dividend.

Scientific usage is that general termsinclude special cases.
BigDawn(2002-02-01) Is a parallelogram a type of trapezoid?
wazupp (2002-02-26) Is a rhombus ever a square?

In a mathematical context, the answer to either question isdefinitely yes.

A trapezoid (British English: trapezium) is defined as a quadrilateralwith two parallel sides. If its other two sides happen to be also parallel, thetrapezoid (trapezium) happens to be also a parallelogram.Period.

Common usage may differ from the above becauselexicographers, dictionaries, and the general public often exclude from a generalcategory some common subcategories.Mathematicians, however, are much better off considering that(among many other similar examples):

• an equilateral triangle is an isosceles triangle,
• a circle is an ellipse,
• a square is a rectangle and a rhombus,
• a rectangle is a parallelogram,
• a rhombus is a parallelogram,
• a parallelogram is a trapezoid,
• a trapezoid is a quadrilateral,
• a quadrilateral is a polygon,
• a regular tetrahedron is a disphenoid,
• a plane is an helicoid,
• a total ordering is also a partial ordering,
• ... etc.

Although you may be able to get away with the opposing view at the most elementary level,it is poor mathematics to do so.

When "trapezoid" appears in actual mathematical discourse, it's universallyunderstood that any special "subtype" could occur.In the rare cases where it's essential to have a pair of nonparallel sides, it must be so stated.

Readon, if you're not convinced...

The lexicographers in charge of putting together general dictionaries oftenfail to consider the above facts.Either they copy each other's work, or are content withthe sole monitoring of common usage, ignoring actual mathematical usage.

Nonmathematical discourse is usually concerned with conveying the most informationin the fewest words about some specific instance of a concept,sothat the word with the narrowest meaning is used whenever possible.Ifyou're actually looking at some specific circular shape,you are describing it most accurately as a "circle".You would not use the term "ellipse" unless the shape failed to be circular...

Mathematical discourse, on the other hand, tries to issue general statements(theorems)applicable in the least particular set of circumstances:Ifsomething which is true of circles holds for other ellipses as well,then it's usually better to state it for all ellipses(practioners of projective geometry will oftenbe able to generalize such things further; to allconic sections).

See Also

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Mathematical terms are defined to maketheorems as simple and/or as general as possible.Nearly anything that is true of an ellipse is also true of a circle, and that iswhy mathematicians consider the circle to be a special type of ellipse.In the rare case when a theorem involving ellipses does not apply to circles,we must say so explicitely.For example, it's understood that the foci of an ellipse are not necessarilydistinct points...["Foci",plural of "focus" is pronounced "foe sigh".]

Occasionally, the mathematical definitions are in direct conflict withwhat general-purpose dictionaries state...For example, to a mathematician an ellipsoidis a special type of ovoid and aspheroid is a special type ofellipsoid (i.e, an oblate or prolate ellipsoid with [at least] oneaxis of symmetry). A sphere is a special type of spheroid.On the other hand, a general-purpose dictionary may [erroneously] define"spheroid" and "ovoid" as synonymous, so an ellipsoid would become a specialtype of "spheroid" (the Encarta dictionary makes that mistake).

Ultimate argument, for lexicographers:

The meaning of a word is ultimately revealed by its usage.Itstands to reason, then, that lexicographers should find out the meaning of amathematical word by analyzing its mathematical usage.Look at the word "in a sentence" so to speak, rather than put it on a pedestaland describe whatever prejudices you may have about its meaning.For the word "trapezoid", you may want to consider a description ofthe trapezoid method for approximating integrals:

The definite integral of a positive functionf,is the area bounded by the x-axis, the curve of cartesian equationy=f(x),and two "vertical" lines of equations x=aand x=b (a<b).For a smooth enough function f, this area is adequately approximated byusing the so-called trapezoid method:

Consider an increasing finite sequence (x_n) of pointsstarting at a and ending at b.An approximation of the integral of f from a to bis obtained as the sum (over the relevant range of n) of the areasof all the trapezoids with vertical bases x=x_nand x=x_n+1 whose vertices are either onthe x-axis or on the curve y=f(x).

Now, if f(x_n) = f(x_n+1), the correspondingtrapezoid happens to be a parallelogram (more precisely,a rectangle, possibly even a square).
Does this make the above description invalid?Do you suggest that we should even mention that thetrapezoid could in fact be a rectangle or a square?

Still not convinced about how pervasive inclusive concepts are in regularmathematical discourse?Look again at the meaning of other words in the above description of thetrapezoid method.We talked about an "approximation" to the integral,but we certainly did not mean to exclude the special case where thisapproximation happens to be the exact value, did we?The approximation is exact when f is linear, but thiscould happen in many other cases.Do you want to even mention such cases?Also, when f is linear, the "curve" of equationy=f(x)is actually a straight line.Does this bother you?If you (or someone you love) cannot come to terms with this,I humbly suggest staying away from anyscientific material whatsoever...

Are there exceptions to this rule?

Scientific concepts are as inclusive as they can be, unless a word is usedwhose etymology implies exclusion. For example, the term"pseudoprime" is normally understoodnot to apply to a prime number (although definitions and theoremswould be simpler if it did).We may then clarify things with locutions like"prime or pseudoprime" for the inclusive concept and "composite pseudoprime"for the exclusive one (it's better to be pleonastic than misunderstood).

Whenever there's only one prominent or "maximal"special case, the qualifier "proper" may be usedto exclude it.

In particular, the qualifier proper turns areflexive relation into an anti-reflexive one.That's a fancy way of saying that if a relation with respect to someobject always apply to the object itself, then the matchingproper relation never does. The qualifierproper may exclude something else only if we didn'thave any reflexivity to begin with...

Examples, including a few tricky ones:

• A proper multiple of an integer isn't equal to that integer.
• A proper ellipse isn't a circle.
• A proper class isn't a set.
• A proper subset isn't the whole set
  (the empty set is a proper subset of any set but itself).
• A proper subgroup isn't the whole group
  (a one-element group is not a proper subgroup of itself).
• A proper ideal of a ring isn't the whole ring.
• A proper (positive) divisor of an integern > 1 can't be n (it can be 1).
• A proper multiperfect numberis not a perfect number.
• A proper filter of a poset isn't the whole set.
• A proper saddlepoint isn't an extremum.
• A proper asymptotic series isn't convergent.
• A proper distributive algebra isn't associative.
• A proper theorem applies to infinitely many things.
• A proper rhombus isn't a square.
• Last (and possibly least) a proper trapezoid isn't a parallelogram!

Confusion arises when the qualifier proper is dropped fromsuch examples.This may happen even in reputable textbooks, especially when the vocabularyis introduced incidentally.When the concepts are actually manipulated extensively,almost all authors will use their inclusive definititons(it would be utterly inconvenient to systematically disallow the use of a general termunless special cases have been specifically excluded).

(L. T. of Austin, TX.2000-03-30)
What are the names of the polygons 10 sides and up?
(Jason of Canajoharie, NY.2000-12-06)What is an 11-sided polygon called?[What are] the names for polygons with sides numbering 12-20?
(M. Q. of New Port Richey, FL.2000-11-10)
What are the names of polygons with 11 ,12, 13, 14, ... sides?
(H. I. of Martinsville, VA.2000-11-30)
What do you call an 11-sided polygon?
(Fred F. of Beverly Hills.2001-02-10)
What do you call a 13-sided polygon?
(M. K. of Uzbekistan.2001-02-10)What is a 32-sided shape called?

An 11-sided polygon is an hendecagon.

Terms like "undecagon" and "duodecagon" have sometimes appeared to denote polygonswith 11 or 12 sides. These are macaronic terms (namely, terms built from a mixtureof different languages, like Greek and Latin) and they should be avoided.Unfortunately, the term "undecagon" seems to be used almost as often as "hendecagon"to describe an 11-sided polygon (the very questionable spelling "endecagon" is, mercifully,a very distant third).

The systematicnaming of polygons ispurely based on Greek roots (we do not call polygons "multigons").The classification below starts with "polygons" with one or two sides,which are legitimate topological objects.Such sides may not be straight lines in euclidean geometry,but they can be "straight" in noneuclidean geometries:On the surface of a sphere, the equivalent of a straight line is a great circleand a monogon or henagon consists of a single vertex andany great circle going through it,whereas a digon consists of two vertices and both of the great arcs joining them(for two antipodal vertices, many digons may be constructed whose edges aregreat half-circles).

There's no specific name for the empty polygon, 0-gon (0 vertices, 0 edges),sothe entire sequence is as follows (it's also acceptable to call an "n-gon"any polygon with n edges):

henagon or monogon (1; almost unused), digon (2; almost unused),triangle or trigon (3; adjective is "trigonal" or "triangular"),quadrilateral, quadrangle or tetragon (4; adjective is "tetragonal" or "quadrangular"),pentagon (5), hexagon (6), heptagon (7; avoid "septagon"), octagon (8),enneagon (9; avoid "nonagon"),

decagon (10),hendecagon (11; avoid "undecagon"),dodecagon (12; avoid "duodecagon"),triskaidecagon or tridecagon (13),tetrakaidecagon or tetradecagon (14; avoid "quadridecagon"),pentakaidecagon or pentadecagon (15; avoid "quindecagon"),hexakaidecagon or hexadecagon (16),heptakaidecagon or heptadecagon (17; avoid "septadecagon"),octakaidecagon or octadecagon (18),enneakaidecagon or enneadecagon (19; avoid "nonadecagon"),

icosagon (20),icosikaihenagon or henicosagon (21),icosikaidigon or docosagon (22),icosikaitrigon or tricosagon (23),icosikaitetragon or tetracosagon (24),icosikaipentagon or pentacosagon (25),icosikaihexagon or hexacosagon (26),icosikaiheptagon or heptacosagon (27),icosikaioctagon or octacosagon (28),icosikaienneagon or enneacosagon (29; avoid "nonacosagon"),

triacontagon (30), triacontakaihenagon or henitriacontagon (31),triacontakaidigon or dotriacontagon (32)...tetracontagon (40)... pentacontagon (50)... hexacontagon (60)... heptacontagon (70)...octacontagon (80)... enneacontagon (90; avoid "nonacontagon")... hectagon (100), ...chiliagon (1000), ... myriagon (10000).

For some obscure reason,the corruption of the prefix hexaconta- into hexeconta- became acceptableor dominant when applied to families of objects other than polygons.For example "hexecontahedron" is used more often than"hexacontahedron" to denote a polyhedronwith 60 faces.Iadvise against this alternate spelling in spite of its apparent popularity,because the trailing vowel isn't irrelevant in other cases;hexane and hexene are different,so are hexanoic(caproic) and hexenoic.The former term refers to a saturated chain of carbon atoms,the latter doesn't, as discussed next.

(2020-09-05) Latin Numerical Prefixes
They are less often used than the above Greek ones.

The most prominent use is for bases of numeration.

1. Unary: Just one digit (arguably 0): "", "0", "00", "000", "0000"...
2. Binary: Base two.
3. Ternary: Base three.
4. Quaternary: Base four.
5. Quinary: Base five.
6. Senary: Base six.
7. Septenary: Base seven.
8. Octonary, octonal or octal: Base eight.
9. Denary or decimal: Base ten.
10. Nonary; Base nine.
11. Undecimal: Base eleven.
12. Duodecimal, dozenal or uncial: Base twelve.

List_of_numeral_systemss

(2001-06-24) Chemical Nomenclature:

The Greek numerical prefixes are not limited to thenaming of polygons; they are thebasis of the systematic naming of other families of scientific objects which depend onsome primary count.One important example is the(extended)official nomenclature for organic molecules,based on the number of carbon atoms in the backbone of the molecule,as established in 1957 by the IUPAC(International Union of Pure and Applied Chemistry).As is the case with simple polygons, simple organic molecules may have a common namewhich was used in various languages before systematic naming was introduced.Much more so, in fact...

Note that systematic classifications may or may not be extended to start with zero:Some early chemists did classify water as the simplest(carbon-free) alcohol.(Water is about as good a polar solvent as other alcohols.)However,it's much less useful to view hydrogen as the "simplest alkane".

It would be a counterproductive, misguided and dubious endeavor[you've been warned!] to introduce degenerate cycles of1or2 carbon atoms into chemical nomenclature,but let's have fun:"Cycloethane" is ethylene(C₂H₄ or H₂C=CH₂)"cyclomethane" is methylene (CH₂)."Cycloethanol" would be yet another name forethenol (H₂C=CHOH), also known as vinyl alcohol or hydroxyethylene.Finally, "cyclomethanol" would be the elusive hydroxymethylene(HCOH).

n	Polygon	Alkane H(CH2)nH cyclo: (CH2)n	p-monoalcohol CnHn+2O H(CH2)p-1CHOH(CH2)n-pH Cycloalcohol: (CH2)n-1CHOH
0		hydrogen	water
1	henagon monogon	methane	methanol (or carbinol)
2	digon	ethane	ethanol
3	trigon triangle	propane cyclopropane	1-propanol, 2-propanol cyclopropanol
4	tetragon quadrangle quadrilateral	butane cyclobutane	1-butanol, 2-butanol cyclobutanol
5	pentagon	pentane cyclopentane	1-pentanol, 2-pentanol, 3-pentanol cyclopentanol
6	hexagon	hexane cyclohexane	1-hexanol, 2-hexanol, 3-hexanol cyclohexanol
7	heptagon	heptane cycloheptane	1-heptanol, 2-heptanol, 3-heptanol 4-heptanol, cycloheptanol
8	octagon	octane cyclooctane	1-octanol, 2-octanol, 3-octanol 4-octanol, cyclooctanol
9	enneagon	nonane [sic] cyclononane	1-nonanol [sic], 2-nonanol, 3-nonanol 4-nonanol, 5-nonanol, cyclononanol
... etc.

In 1957, the IUPAC (carelessly) endorsed the use ofthe Latin prefix "nona-" for "9" in the names of organic and other chemicals,and we must now use "nonane" or "nonanol" and refrain from any witty remarks to theeffect that "enneane" or "enneanol" would have been more correct...Also, "eicosa-" (rather than "icosa-") is the recommended form of the prefix for "20"in a chemical context.

Many alternate names exist for a large number of important organic chemicals.For example, the simplest carboxylic acid (methanoic acid, HCOOH) was originally calledformic acid, because it was first distilled (!) from ants(Latin: formicae, French: fourmis).Formaldehyde (CH₂O, CAS 50-00-00) is thus the common nameof what is more properly called methanal. The French commonly call formaldehydeformol, with an unfortunate use of a suffix normally reserved for alcohols(the name formal [sic] has been proposed, which would feature the proper suffixfor an aldehyde, but it never caught on).Just to take a cheap shot at the practical lack of standardization in some chemicalnames, here are some of the published names used for O=CH₂,in alphabetical order:BFV, CH₂O, FA, Fannoform, Floguard1015, FM282, Formaldehyde,Formalin, Formalin40, Formalith, Formic aldehyde, FYDE,H₂CO, Hoch, Ivalon, Karsan, Lysoform,Methaldehyde, Methanal, Methyl aldehyde, Methylene glycol, Methylene oxide,Morbicid, Oxomethane, Oxymethylene, Paraform, Superlysoform,NCI-C02799, RCRA waste numberU122, UN1198, UN2209.Other foreign designations occasionally surface in Englishtexts, including: "Aldéhyde formique" or "Formol" (French),"Aldeide formica" or "Formalina" (Italian),"Aldehyd mravenci" (Czech), "Formaldehyd" (Polish),"Formaline" (German), or "Oplossingen" (Dutch)...

The numerical designations n:0, shown above, are commonly used by chemists for asaturated chain of n carbons (no double bonds). An unsaturated chain of n carbons withp double bonds would be designated n:p. For example:

Myristoleic is 14:1,palmitoleic is 16:1,oleic is 18:1, linoleic is 18:2, linolenic is 18:3,moroctic is 18:4,gadoleic is 20:1, arichidonic is 20:4, timnodonic is 20:5,erucic is 22:1, clupanodonic is 22:5,selacholeic or nervonic is 24:1,...

These traditional adjectives for unsaturated carbon chainsusually apply to only one particular position of thedouble bond(s) and/or one particular cis/trans configuration.For other unsaturated carbon chains with the same numerical designations,it's better to use numerical adjectives based on the two numbers involved(except if the second one is 1).The ending to use is "-enoic", as a reminder that an alkene series is involved.Examples include docosenoic or docosaenoic (22:1,instead of erucic),docosadienoic (22:2), docosatrienoic (22:3), docosatetraenoic (22:4)and docosapentaenoic (22:5, instead of clupanodonic).Thepopular (overmarketed) compound DHA does not have a competing traditionalname, it's "simply" called docosahexaenoic acid (22:6).

In the above table for saturated chains,the official systematic adjectives are given first in each list.Notice many competing semi-regular formations with a few exceptional cases of their own:Butyric is used instead of butylic (which is unused in English)because of the etymological influence of butyrum (butter); the French do usebutylique.Pentyl and amyl are used as names, but do notserve as the bases for adjectives.

Inanumber of cases, systematic adjectives are far less popular than thetraditional ones whichappears in boldface toward the end of some lists.Suchtraditional adjectives are usually derived from the names of plantscontaining the corresponding unsaturatedfatty acids(alkanic carboxylic monoacid).Inatleast two cases, the etymology is the name of an animal;a tiny one (the ant) for formic, as discussed above,and the largest one (the whale) for cetylic:Theadjective cetylic does come fromcetus, the Latin name of the whale (the corresponding alternate nameof hexadecane is cetane, but cetanoic is unused).Thecorresponding fatty acid was originally obtained from spermacetiextracted from the head of the sperm whale or other cetaceans(spermaceti products were used to make candles, cosmetics, and ointments).Cetylic acid is better known as palmitic acid (rather than hexadecanoic acid),as a reminder that it is one of the primary components of palm oiland coconut oil.Let's summarize the etymologies of some of the words tabulated above:

Methylic (Greek methu wine and hyle wood; wood alcohol),formic (Latin formica ant),ethylic (Latin aether upper air, volatile spirit),acetic (Latin acetum vinegar),propionic (Greek pro- first and pion fat; first fatty acid),butyric (Latin butyrum butter),amyl (amylum starch, French amidon),caproic, capronic, capryilic, caprinic, capric(Latin caper goat; because of the associated smell),valeric and valerianic (the fatty acid occurs in the root of the valerian plant),pelargonic (Latin pelargonium, genus name of the geraniums),lauric (Latin laurus laurel),myric and myristic (Greek muron perfume, and muristikos fragrant),cetylic and cetane (Latin cetus whale),palmic and palmitic (palmite, pith of the palm tree, and palmitin palm oil),margaric and margarinic (Greek margaron pearl; pearly white aspect of margarin),stearic (Greek stear tallow, hard fat),arachic and arachidic (New Latin arachis, genus name of peanuts and groundnuts,from the Greek arakis legume),behenic,lignoceric (wax from beech; Latin lignum wood and cera wax).

Help from our readers:

In an early version of this article,we had wondered about the etymology of behenic(wrongly guessing a relation with behemoth).More than a year after being posted here, ourplea for helpwas answered by Valerio Parisi, whom we thank for the following comment about thekelor tree(also known asmoringa,horseradish tree,dangap in Somalia,etc.)In 1848, behenic acid was found as a constituent (up to 8.6%) ofthe Moringa oleifera seed oil and was then named afterthat oil's common designation:ben oil or behenoil.

On 2002-12-19, Valerio Parisiwrote: [edited text + hyperlinks]

Dear Dr. Michon,

Here is a loose translation of what my Italian dictionary has to say aboutthe etymology of "behenic"(Zingarelli: vocabolario della lingua italiana- Zanichelli editore; decima edizione, 1970).Note the spurious coincidence linking the eleventh month of the yearand the eleven pairs of carbon atoms in the behenic chain:

Ben-oil tree (Moringaoleifera)Tree of the Moringaceae family, bearing white flowers.Behen oilis extracted.from its seeds. [Italian bèen.From the Persian bahman: eleventh month of thePersian year,corresponding to the sign of Aquarius,when the roots of this tree were traditionally harvested and consumed.]

Sincerely yours,
Valerio Parisi
Tor Vergata University, Rome, Italy

You may notice that chains with an odd number of carbon atoms have fewertraditional designations.The reason is that plants synthesizefatty acids with an even number of carbons.In the original version of this article,we asked any "biologist or organic chemist" for anexplanation of this fact.About 9months later, Bruce Blackwell answered the call:

On 2002-04-01, Bruce Blackwellwrote:

I must say I have enjoyed your Numericana web site thoroughly.Iam a physicist by training but I am self-educated in many sciences,including organic and biochemistry.

The reason most biological fatty acids have an even number of carbon atoms isthat the predominant mechanism of synthesis in the cellis the repetitive condensation of acetate(CH3COO^-)to the growing chain; hence 2 carbons at a time.The carboxyl end of the growing chain (which begins with a single acetate)is condensed to the alpha carbon of a new acetate in a reaction similar to a synthesisin the laboratory known as the Claisen condensation.Inthe living cell, the reaction is mediated by enzymes and sulphur atoms.In the laboratory,Grignard reagents are used.

It's been a pleasure working through the examples on your site.

Bruce Blackwell
Oracle Corporation, Nashua, NH

Thanks for the explanation, Bruce. Your kind words were also appreciated...

Lynda Brown(Canada, 2001-08-21; e-mail.) [ ... about "sesqui" ... ]
I wondered if you happen to know why phosphorus sesquisulfide isP₄S₃where neither element is the "sesqui" of the other. Thanks.

The usual meaning of the "sesqui" prefix is "one and a half".Asesquicentury is 150 years anda sesquicentennial marks the passing ofthat many years.

The only fractional prefixes in common scientific usage are hemi(1/2) and,indeed, sesqui(3/2).However, it is acceptable to combine hemi with any prefix representing awhole number (almost always an odd one).The most "common" example is hemipenta(5/2).Although we've never actually encountered prefixes like hemihepta(7/2),or hemiennea(9/2), these would be allowed toform [new] scientific names with unambiguous meanings...

P4S3is now called tetraphosphorus trisulphide.This compound was once an important discovery, with huge socialimplications (see below)around the turn of the 20th century.

The curious name ofphosphorus sesquisulphide was apparently given to it at the time by itsFrench inventors, the chemists Henri Savène and Emile David Cahen(and/or their entourage).It may have been just a catchy name, whose use was considered somewhat acceptablebecause of the scarcity of fractional prefixes outlined above.The only other accurate names for the chemical, besides tetraphosphorus trisulphide,would be diphosphorus sesquisulphide or phosphorus hemisesquisulphide,which would look even weirder to any modern chemist.

It may well be the case that Savène and Cahen originally useda "proper" name which was shortened later because there is really nopossible risk of confusion!You were right to observe that all this is not very satisfying and/or logical.Others found it unsatisfying as well, and this is why themore precise name of tetraphosphorus trisulphide is now used.

Matches,Phosphorus, and P₄S₃

An household match is such a common item nowadaysthat it may be hard to imagine what a revolutionarymarvel it once was! Vigorously rubbing two pieces of wood togethermay generate enough heat to allow some dry material to smolder and then ignite in thepresence of a spark from metal or silex.This can be very hard work, as many boy scouts will tell you!The idea of the match is to carefully choose the materials involved, so thatfriction will cause enough heat locally as well as some sparking to trigger ignition offlammable material...

Surprisingly, the first recognizable friction matches didn't useany phosphorus at all:They were made in 1826 froma fifty-fifty mixture of potassium chlorateand antimony trisulphide,together with some gum arabic, sugar and starch.The inventor was an English pharmacist from Stockton-on-Tees[at 59 High Street] namedJohn Walker (1781-1857).Walker didn't bother to patent his creation that he called friction lights.He sold the very first batch on April7, 1827(to a local solicitor named Hixon).Those bulky three-inch splints of wood, were still expensive, unreliable andsomewhat tricky to use.

All this would change 4 years later (in 1830 or 1831)when a young chemistry student from the French village ofSaint-Lothain(Jura) had the idea to substitute white phosphorusfor the antimony sulfide in Walker's recipe. This is howCharlesSauria (1812-1895) managed to make the first modern strike-anywhere matches.The idea was first applied industrially in 1832 by Jakob Friedrich Kammerer.Sauria himself did not profit from his invention anddied a pauper.

Thosewhite phosphorus "strike-anywhere" matchesbecame known as Lucifers, which wasthetrademark coined by Samuel Jones in 1829 or 1830for the previous generation of matches.The name may have been far more appropriate than it was meant to be:For one thing, Lucifers could ignite accidentally rather easily(pure white phosphorus can ignite spontaneously in the air above 34°C).It was also soon discovered that white phosphorus is highly toxic:Continuous exposure among factory workers(impoverished "match girls") would cause a dreaded and often fatalbone disease known as phossy-jaw.Whitephosphorus became a public health issue on the international scene...

The French goverment sponsored research to find a suitable replacementfor white phosphorus.The outcome, the work of H. Savène and E.D.Cahen,was based on tetraphosphorus trisulphide, a yellow solid melting at 172°C(then called phosphorus sesquisulphide, as mentioned above).Apaste including 13% of that chemical and 28% potassium chlorate worked very well(the rest of the recipe included powdered glass, glue and fillers such as zinc oxideand iron oxide).It was not spontaneously flammable, not toxic and didn't cause phossy-jaw!Aperfect product with a lousy name.

In 1906, an international treaty (the so-called Berne Convention)was signed in Switzerland, obligating the signing countries to ban white phosphorusfrom the manufacture of matches.TheUS did not sign that treaty (on the grounds that the requiredban would not have been constitutional)but the US Congress created punitive taxes which had the same effect, in 1913.All this did not prevent toxic Lucifers from being manufactured in China,as late as 1950...

In the U.S., the patent for P4S3matches was secured in 1910 by the Diamond Match Company.However, the public health issue was such that President Taftpublicly urged the company to voluntarily surrenderits patent into the public domain, despite its enormous moneymaking potential.The Diamond Match Company did so on January 28, 1911.

At this writing,the nontoxic P4S3strike-anywhere matches are still quite popular in the US.In many other countries, however, they have been all but replaced bythe so-called safety matches,which you can only strike on a special patch(normally located outside each package)coated with some red phosphorus, which is essential to ignition.This type of safety match was inventedin 1855, byJohan Edvard Lundstromof Sweden.

That invention was only made possible by the prior discovery ofred phosphorus,the nontoxic form of phosphorus obtained by heating ordinarywhite phosphorus between 230°C and 300°C, in the absence of oxygen...

For other aspects of the whole story, see the wonderful book of John Emsley,The13th Element: The Sordid Tale of Murder, Fire and Phosphorus(JohnWiley & Sons, New York, 2000. ISBN 0-471-39455-6).

The title of that book comes from the fact that phosphoruswas discovered around 1669 in Hamburg, by the alchemist Hannig Brandt,at a time when only 12 other chemical elements were known:Gold, Silver, Mercury, Copper, Iron, Zinc, Tin, Lead, Antimony, Arsenic, Carbon,and Sulphur.

On 2001-08-21, Lynda Brown(Canada) wrote:

Thank you very much for your prompt and detailed answer.Ihadnot guessed that "hemi" might have been involved,but that makes perfect sense.

Ironically, I had just finished reading John Emsley's book on phosphorus (I am now readingMolecules at an Exhibition).

Regards, Lynda Brown

References:

• Phosphorus:from urine to fire, by Peter E. Childs (Univ. of Limerick).
• TheHistory of Matches, by Mary Bellis.

(M. P. of Saint Petersburg, FL.2000-11-04)
If you have a million, billion, and a trillion,what are the next 5 large numbers that come after that?

Here's the sequence: million (n=1), billion (n=2), trillion (n=3), quadrillion (n=4),quintillion (n=5), sextillion (n=6), septillion (n=7), octillion (n=8),nonillion (n=9), decillion (n=10), ... vigintillion (n=20), ... centillion (n=100).[See table below.]The n-th word in this sequence may be referred to as the n-th zillion.

This word pattern was devised around 1484 by Nicolas Chuquet (1445-1488),who authored the first treatise of algebra ever written by a Frenchman.Chuquet (a self-described "algorist")used the word for the n-th zillion to denote a million to the n-th power,namely 10⁶ⁿ, where n is as listed above [or tabulated below].However, things did not remain so simple with the passage of time...

In the 17th century, a few influential French mathematicians decided to usethe same names to denote the successive powers of a thousand instead,namely 10³ⁿ⁺³, where n is as listed above [or tabulated below].This was described as a "corruption of the Chuquet system"but was considered more "practical".That's the system used in the US today (where a billion is indeed 1000000000)and increasingly in English texts of any origin.In 1974, British Prime Minister Harold Wilson even informed the House of Commonsthat the word "billion" in statistics from the British government wouldthenceforth mean 10⁹, in conformity with American usage...However, since the original Chuquet system is still used in the UK,it's probably best to avoid such names in international communications,if there is any risk of ambiguity whatsoever.Astronomers, in particular, routinely speak of a "thousand million"(legal in the Chuquet system, weird but unambiguous in the American one)or a "million million" (not legal in either system, but unambiguous in both).

After using the "American system" for quite a while, France reverted back tothe original Chuquet system in 1948 and declared any other system illegal in 1961.Also in 1948, the 9th CGPM approved the original Chuquet system for internationaluse in scientific fields.

The trend seems to be that the Chuquet system is used in all languages but English,where the American system is increasingly dominant (especially in a financial context).A "billion" in English almost always means 1000000000,the corresponding British term "milliard", which would be unambiguous,is apparently rarely used nowadays. (The term "milliard" itself was apparentlycoined around 1550 and is credited to Jacques Pelletier.)Nevertheless, a "zilliard" sequence is being used to denote 10³⁺⁶ⁿ.These are numbers 1000 times largerthan the corresponding "zillions" of the original Chuquet system, whose gapscompared with the American system are thus filled:milliard(10⁹),billiard[sic!](10¹⁵),trilliard(10²¹),quadrilliard(10²⁷),etc.

n	nth zillion	US	World
1	million	106	106
	milliard		109
2	billion	109	1012
3	trillion	1012	1018
4	quadrillion	1015	1024
5	quintillion	1018	1030
6	sextillion	1021	1036
7	septillion	1024	1042
8	octillion	1027	1048
9	nonillion	1030	1054
10	decillion	1033	1060
11	undecillion	1036	1066
12	dodecillion duodecillion	1039	1072
13	tredecillion	1042	1078
14	quattuordecillion	1045	1084
15	quindecillion	1048	1090
16	sexdecillion	1051	1096
17	septendecillion	1054	10102
18	octodecillion	1057	10108
19	novemdecillion	1060	10114
20	vigintillion	1063	10120

n	nth zillion	US	World
21	unvigintillion	1066	10126
22	dovigintillion duovigintillion	1069	10132
23	trevigintillion	1072	10138
24	quattuorvigintillion	1075	10144
25	quinvigintillion	1078	10150
26	sexvigintillion	1081	10156
27	septenvigintillion	1084	10162
28	octovigintillion	1087	10168
29	novemvigintillion	1090	10174
30	trigintillion	1093	10180
31	untrigintillion	1096	10186
32	dotrigintillion duotrigintillion	1099	10192
33	tretrigintillion	10102	10198
40	quadragintillion	10123	10240
50	quinquagintillion	10153	10300
60	sexagintillion	10183	10360
70	septuagintillion	10213	10420
80	octogintillion	10243	10480
90	nonagintillion	10273	10540
100	centillion	10303	10600

In her 1975 review of numeration systems(Histoire comparée des numérations écrite)the French mathematicianGeneviève Guitel(1895-1982) found it useful to introduce the terms "short scale"(échelle courte) for the American system and"long scale" (échelle longue) for the original Chuquet system.

The American system (Guitel's "short scale")is also used in Russian, except that "milliard" is used instead of"billion" (which is apparently a rarely used synonym).Officially at least, other languages use Chuquet's original "long scale" to namelarge numbers.Unconfirmed exceptionswe've gleaned so far include Turkish, Greek and Romanian, as well asSpanish in Puerto Rico (or in the U.S.), and Portuguese in Brazil.If you are absolutely certain about any other language and/or country in whichthe American system is used, pleaselet us know.

The above naming scheme is unused in China, India, Japan and Korea...

The Conway-Wechsler System (1995)

On page 14-15 of the Book of Numbers(by John H. Conway and Richard Guy, 1995)Conway puts forth the unlimited naming system he devisedwith Alan Wechsler.That linguistic proposal is based on a positional numeration system in base 1000(one thousand)applied to the exponents of the "zillions" (a "zillion" is a power of 1000).The scheme specifies a prefix of the form Xilli- to denote any integer from 0 (nilli-) to999 (novenonagintanongentilli-). You may append an unlimited number of such prefixesand obtain a name for a zillion by adding the syllable"on" at the end (so the word ends in "illion").Thus, something like XilliYilliZillion denotes 1000 to the power of 1000000X+1000Y+Z+1.Nice.

Conway and Wechsler suggest to keep the traditional names for the first zillions(thousand, million, billion, trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion)and to use their system only after that point (that's wise).This works nicely, as describedby Robert Munafo,as soon as the basic scheme for the first 1000 prefixes is flawless.This was achieved by "going through dozens of iterations and ironing out difficulties"(in the words of Allan Wechsler himself on 2000-03-01,as quoted by Nicolas Graner).

Robert Munafohas pointed out that a previous complaint of mine was technically unfounded, at least in writing,since the spelling details of the published Conway-Wechsler system do indeed specify thata trescentillion denotes 1000 to the power of 103 while atrecentillion is 1000 to the power of 300.Arguably, thisnarrowly avoided ambiguity remains a flaw when the names are merelypronounced (arguably, the extra "s" is virtually silent) andI maintain my previous proposal that the Conway-Wechsler system should be amendedby using a different prefix for 300 like "tercenti-" or "tricenti-"(I like the latter, but the former is more robust in languages other than English).

(Mark of Edmond, OK.2000-11-06)
What is the name of the number represented by 100000 to the power of 100000(a 1 with 1/2 million zeros)?

Following Rudy Rucker (quoted by John Conway and Richard Guy), we may usethe suffix plex at the end of a number to denote 10 to the power ofthat number: zeroplex is 1, oneplex is 10, twoplex is 100, threeplex is 1000, etc.The celebrated googol is a hundredplex (which exceeds by far the total number ofelementary particles in the observable Universe).

The neologism "googol" was coined in 1938 by the American mathematicianEdward Kasner(1878-1955) who had been looking for a cute term to stand for 10 to the 100th.Kasner had previously asked for the opinion of his nine-year-old nephew(Milton Sirotta, b. 1911) who came up with "googol". The name stuck.

The standard names for this number would be "ten dotrigintillion" or "ten duotrigintillion"in the American system of numeration (where a billion is a thousand million)and "ten thousand sexdecillion" in the original Chuquet system of numeration(still used by a few British subjects, and by virtually the entirenon-English-speaking world, who considers that a billion is a million million).

A googolplex is 10¹⁰¹⁰⁰, a number which isimpossible to write down with ordinary numeration, since this would entail the digit "1"followed by a googol of zeroes. (The suffix plex is a contraction of"plus exponent". The suffix minex has been proposed byTadashi Tokieda as a contractionof "minus exponent" to denote small numbers: zerominex is 1, oneminex is 1/10,twominex is 1/100, etc.)

One answer to your question would therefore be that 100000¹⁰⁰⁰⁰⁰may be called "500000-plex".This is allowed, but you may argue that the use of numerals does notmake this more of a "name" than100000¹⁰⁰⁰⁰⁰ or 10⁵⁰⁰⁰⁰⁰.

The problem is that the plex suffix leads to ambiguity when used with number namesthat consist of several words:Does "five hundred thousand-plex" mean 500000-plex or 500 times a 1000-plex?

To solve the problem almost unambiguously in this particular case,we need a single word to represent 500000...There happens to be a legitimate one, using the standard prefix hemi- for "one half":500000 is a hemimillion and your number could therefore be called ahemimillionplex.

That is a correct answer unless you make a different parsing and consider that a"hemimillionplex" is a "hemi[millionplex]" instead of a "[hemimillion]plex" as intended.I argue that this should not be done on the basis that the former parsing is less"useful" than the latter, since it would put large numbers with short names"too close" to each other and leave more severe gaps in-between.In other words, the hemi- prefix (like the other standard numerical prefixessesqui-, di-, tri-, quadra-, penta-, etc.)should have a stronger parsing priority than the suffix plex.An hyphen (hemimillion-plex) would probably makethe whole thing less ambiguous, but this breaks the pattern established for lesser numbers.

I don'tknow what a professional linguist would have to say about all this.If you happen to be one, please let me know.

Did I really say "useful" ?

On 2003-03-22, BetsyMcCall wrote: [edited summary]

I really like your site, and I decided to weigh in on your linguistic problem.I study mathematical linguistics (the mathematics of language,not the linguistics of mathematics, but close enough).

[...]You are basically correct.You've obviously considered this problem carefully,and have tried to follow the normal numerical usage of the prefixes.[...]In the end, only usage can determine the "correct" parsing,but I don't really envision hemimillionplexbecoming popular enough to set a trend.Another alternative would be to cast 500000 [purely] in Greek terms[to obtain] pentacontamyriaplex.The Greek prefixes will naturally parse together.[...]I seriously doubt that the Oxford English Dictionarywill be quoting any of these words any time soon.

Betsy McCall

University of Pittsburgh

Betsy is right: pentacontamyriaplex is better thanhemimillionplex.