Like the crest of a peacock, like
the gem on the head of a snake, so is
mathematics at the head of all knowledge. Vedanga Jyotisa
by Lagadha
(c. 500 BC)^{ } Oldest extant text of
Indian astronomy

[...]
No subject loses more than mathematics
by any attempt to dissociate it from its history. J.W.L.
Glaisher^{ } (1848-1928)

(Marc of Philippines.
2000-11-28)
When did mathematics originate?

Foremost among the earliest mathematicians on record is
Thales of Miletus
(c.625 - c.550 BC).
However, in a poem about Thales, the famous librarian
Callimachus/Callimaque
(c.300-240 BC) seems to credits an earlier Phrygian mathematician,
by the name of Euphorbus, with the straightedge-and-compass construction
of the circle circumbscribed to a scalene triangle.

There is some debate about this...
Pythagoras (c. 569-475 BC)
claimed to be a reincarnation of the
legendary Euphorbus who died
in the Trojan war
(c.1200 BC) where Phrygia
was an ally of Troy.
Ancient poets may have used the name
Euphorbus to refer to Pythagoras himself
(they were taking reincarnation at face value).

Callimaque himself was also the founder of Library Science.
He was responsible for the 120 volumes of the Pinakes, the annoted catalog
of the Library of Alexandria, created when the Great Library had been in existence
for about 20 years, around 265 BC.

A few Indian authors are remembered for the Indian Sulbasutras (geometrical rules appended to the Vedas for the construction
of religious altars): Baudhayana (fl. around 800 BC) and Manava (fl. around 750 BC) are the oldest, whereas Apastamba
(fl. around 600 BC) was probably a contemporary of Thales.

Some form of mathematics predated the first mathematicians on record:

The oldest known artifact linked to the basic mathematical activity of counting
is the Lebombo
Bone, which was found in the Maputaland
Border Cave,
an important prehistorical shelter first excavated in 1934 and located high in
the Lebombo (Lubombo) escarpment which separates Swaziland
from Maputaland
(the northern part of the South African province of
KwaZulu-Natal, the Zulu Kingdom).
The Lebombo bone is a piece of a baboon fibula which
has been dated to 35 000 BC.
It's carved with 29 notches and resembles calendar sticks
still used by modern Namibian tribes.
The bone may have helped count the days in a lunar month,
or a woman may have kept track of her menstrual cycle with it.

The so-called Rhind
Papyrus is an Egyptian scroll (about 18 feet by 13 inches)
dating from the same period (around 1650 BC) which contains mathematical tables and problems
copied from an older document (probably dating from about 2000 BC) according to the
comments left on the papyrus by the scribe Ahmes
(or A'h-mose).
Among its 87 problems, the Papyrus includes a "recreational" puzzle,
which can be translated as follows:

A man has seven houses,
Each house contains seven cats,
Each cat has killed seven mice,
Each mouse had eaten seven ears of spelt,
Each ear had seven grains on it.
What is the total of all of these?

The answer is 19607 (not counting the man himself).
This puzzle is unlikely to have been handed down directly across
several cultures and many generations,
but the fascination about geometric progressions of common ratio 7
seems to have been strong among early mathematicians.
In 1202, a similar riddle appeared,
in Fibonacci's Liber Abaci (the answer is 137256):

Seven old women are travelling to Rome.
Each has seven mules.
On each mule are seven sacks.
In each sack are seven loaves of bread.
In each loaf are seven knives.
Each knife has seven sheaths.
What is the total of all of these?

However, the most popular incarnation of this ancient puzzle is
probably the following anonymous English riddle:

As I was going to St. Ives,
I met a man with seven wives.
Every wife had seven sacks,
Every sack had seven cats,
Every cat had seven kits.
Kits, cats, sacks and wives,
How many were going to St. Ives?

Rather unfortunately, this version of the riddle is often considered
an exercise in lateral thinking,
rather than arithmetic: Since the narrator met the wives
on the way to [the medieval fair of] St. Ives,
he/she was going there, but they (probably) were not...
So the answer could be zero or one (instead of 2800 or 2801),
depending on whether the narrator happens to be a wife or not!
Also, we can't possibly guess how many are going to the fair
that we're not even told about...
The "modern" discussion about this nice little poem seems endless and pointless.

(2002-11-24)
How did the [ positional ] decimal system appear in India?

Ancient Indians held the art of reckoning (ganita) in the highest esteem.
They used symbols for marks or divisions (ankas) which are the
ancestors of modern digits (1, 2, 3, 4, 5, 6, 7, 8, 9).
The introduction of a tenth symbol for
zero (0) paved
the way to our positional system of decimal numeration...

The Sanskrit name for zero is shoonya
(void, "nothingness" or emptiness)
but other words with related meanings (including "sky" or "endless")
have been used for this "new" concept:
kha, gagana, akasha, nabha, ananta...

The need for zero had been bypassed by ancient Indians, who used different
Sanskrit words
for all the successive powers of ten,
as shown in the last column of the table below.
(The current usage,
reflected by the numerical punctuation in India and Sri Lanka, is to
name only odd powers of ten beyond 1000, so that a combination
with dasa [= 10 times] is needed for even powers).

The original scheme called for first naming the largest possible power of ten which could go
into a given integer, along with a nonzero anka (from 1 to 9)
stating how many times it could do so
(the rest of the integer, if any, being named according to the same recursive scheme).

About 2000 years ago, it occurred to some bright anonymous Indian soul(s)
that the powers of ten need not be mentioned at all,
if the proper ankas are always given in decreasing order of importance,
provided a symbol is explicitely given for each power of ten that's not present.
This needed zero symbol took the form of a small circle (with a center dot,
which has been dropped in the modern "0"). The modern decimal system was born.

The oldest surviving use of the modern decimal system seems to be a sacred text
called Agni Purana,
revealed to its author (Vasistha) around AD 100.

The decimal system had not reached
Roman Syria [modern Jordan] when Nicomachus
of Gerasa (c.AD 60-120)
wrote his famous Arithmetike Eisagoge ("Introduction to Arithmetic")
which established arithmetic as a field of study separate from geometry and
remained a standard textbook, in spite of its many flaws,
for more than 1000 years.
Arithmetike Eisagoge
featured multiplication tables using the Greek system of numeration.

kdomenick (2001-04-02)
What are the Roman numerals for 18 034?

There are several correct answers, including the ones listed below.
For historical details and more information,
see our unabridged answer.

MMMMMMMMMMMMMMMMMMXXXIV (correct but awkward)

MMMXXXIV
which may be typed as
((I)) I)) MMMXXXIV

XVIII

XXXIV (a vinculum is used to multiply by 1000)

XVIII^{M}XXXIV (preferred over XVIIIMXXXIV )

(A. D. of Australia.
2000-12-01)
How did John Napier originally work out values for his logarithm tables?
(Daniel of Marietta, GA.
2000-11-06)
Who came up with "ln" (natural logarithm)?
(Jerry of Winter Springs, FL.
2001-02-07)
Why did Leonhard Euler come up with "e" ?

The man most often credited with the invention of logarithms is the Scottish mathematician
John Napier (1550-1617).

Yet, a Swiss watchmaker by the name of
Jobst Bürgi
(1552-1632) had the same idea in 1588
(six years before Napier) but did not publish until 1620 (six years after Napier).
[His first name is variously spelled: Jobst, Jost, Joost, Joose, or Joobst.]
Prior to that, Bürgi was one of the foremost pioneers of the so-called
prosthaphaeresis
method which reduces a multiplication to the computation of
a sum, a difference and a half-sum with 4 lookups of available trigonometric tables.
In its purest form, prosthaphaeresis
requires only a table of cosines and yields a product as the average of the
cosines of a sum and a difference by applying the following formula
(which Bürgi brought to the attention of
Tycho Brahe who used it very extensively for many years):

A B =
cos a cos b =
½ [
cos ( a + b ) + cos ( a - b ) ]

In systematic computations, prosthaphaeresis, is considerably faster
than long
multiplication and much simpler to tabulate
(a multiplication fits in a single written line).
The invention of logarithms would further reduce the work to only one addition and two lookups
per multiplication.

Napier, who had also been using prosthaphaeresis,
published the first printed table of logarithms in 1614, under the title
Mirifici Logarithmorum Canonis Descriptio,
after spending twenty years on the problem (from 1594 to 1614)
and eventually computing
x = 10000000(0.9999999)^{L}
for successive integral values of L.
He shunned decimal fractions, which had only recently been
introduced in Europe (by
François Viète in 1579 and
Simon Stevin in 1583).
John Napier himself coined the Latin term logarithmus for L,
from the Greek logos arithmos ("proportion number").
In modern terms, the original
"Napierian logarithm" L(x) was thus the following
decreasing function of x:

L(x) = (7 - log(x)) / (7 - log(9999999))

» 161180150.6088 - 9999950.00025 ln(x)

Modern logarithms are such that
the logarithm of a product is the sum of the logarithms of its factors.
A modern reader may be surprised to discover that Napier's original function
does not possess that crucial property!

Napier's breakthrough was "merely" a device to reduce any geometric
progression to an arithmetic progression,
thereby allowing easy numerical solutions to previously delicate questions.
Also, the pioneering work of Napier paved the way for the convenience,
which would soon follow from two critical modifications...

In the above, log(x) denotes what is now known as the common logarithm
or decimal logarithm of x (its logarithm in base ten).
Common logarithms were developped by Henry Briggs (1561-1681)
after he had met with Napier.
Both men agreed that it would definitely be more convenient to:

Let a zero logarithm correspond to 1 (instead of 10 000 000).

Facilitate decimal computations by letting the logarithm of 10 be 1.

Today, the logarithm of unity is always zero, which is a necessary condition
for the relation log(ab) = log(a)+log(b) to hold.
However, several choices are commonly made for the base
(i.e., the number whose logarithm is chosen to be unity).
The most notable non-decimal logarithmic bases are e (2.718281828459...) and 2,
for natural and binary logarithms
[respectively denoted, unambiguously, by
ln(x) and lg(x) ].
Other popular logarithmic scales feature other bases implicitly
(e.g., 10^{ 0.1} = 1.2589254... for
decibels, 10^{ 0.4} = 2.51188643...
for star magnitudes,
2 for grain size).

Logarithms in different bases are proportional. To convert between different
logarithmic systems,
it's useful to know that log(x) / log(y) doesn't depend on the
chosen "log" base.

Napier felt too old to undertake the task of building new tables,
so Briggs did it...
Under the title "Arithmetica Logarithmica" Briggs published his first
(incomplete)
table in 1624 giving the decimal logarithms of all integers from 1 to 20 000
and from 90 000 to 100 000 at an accuracy of 14 decimal places.
The gap was closed by the Dutch publisher Adriaan Vlacq (1600-1667),
whose results
were included in the second (1628) edition of Arithmetica Logarithmica,
the first complete table of Briggsian logarithms.

Natural Logarithms :

In modern terms, the natural logarithm of a,
denoted ln(a) or Log(a),
is defined as the integral
of 1/x from 1 to a.

The fact that "the area under the hyperbola" has the logarithmic property
[ L(ab) = L(a)+L(b) ] was made clear
in 1649 by Alfonso Antonio de Sarasa (1618-1667).
He derived this much from the following property of the hyperbola
[of equation y = 1/x]
which had been discovered around 1622 by his teacher (at the court of
Philip IV
of Spain) the Belgian Jesuit Grégoire de Saint-Vincent
(1584-1667, a.k.a. Gregory of St. Vincent)
who made the following remark in
Opus
Geometricum (1647).

If b/a = d/c, then the area under the hyperbola
above the segment [a,b] is equal to the area above [c,d].

This is so because the shapes "above" two such segments can be obtained from each other
using two successive stretches (horizontal and vertical)
of reciprocal ratios—one stretch is really a "squeeze".
The combination of these two transforms leaves areas unchanged.

De Sarasa's crucial observation is easily derived from Saint-Vincent's property:
If we let L(x) be "the area above the segment [1,x]", the above statement
with a = 1 applies to d = bc and
does translate into L(b) = L(bc)-L(c).
Although this holds for any rectangular hyperbola
[y = k/x] the unit hyperbola [y = 1/x]
is a more convenient basis to define what are now called natural logarithms...

The Swiss mathematician
Leonhard Euler (1707-1783) started to popularize
natural logarithms in 1728, under the name of hyperbolic logarithms.
They would become the one kind of logarithms most favored by scientists.

Euler had used the symbole for
2.718281828459...
as early as 1727.
In a 1731 letter, he finally describes e as
"the number whose hyperbolic logarithm is one".
Euler himself never explained why he chose the letter e.
Therefore, we may only guess what he had in mind, by observing
that this is the first letter of "exponential" and of German words like
Eins
(one) or Einheit (unity).
Also, e was just the first letter of the alphabet available
after the letters a,b,c,d, which are commonly used for various parameters...
It's highly unlikely that Euler used e because it happened to be
the initial of his own name, since he was a modest man.
(See, in particular, page 156 of "e: The Story of a Number" by Eli Maor,
Princeton University Press.)

The Slovanian mathematician Jurij Vega (1754-1802)
published his first table of logarithms in 1783.
He endeavored to build reliable 7-figure log tables,
based on the original Vlacq tables.
These were checked by soldiers who received a gold ducat for every mistake they found...
Vega's first edition of Thesaurus Logarithmorum Completus appeared
in 1794; the 90th edition [!] was published in 1924...

The three standard logarithms: log, ln (or Log) and lg

Special notations are used for logarithms to the three most commonly used bases, namely
2, e and 10, otherwise the base is indicated as a subscript.
The ratio of two logarithms doesn't depend on the base:

Log_{ a } x

=

Log_{ b } x

Log_{ a } y

Log_{ b } y

Thus, the base may be omitted when we give ratios of logaritthms.
You may use this with y = a or b to properly convert
a logarithm in one base to a logarithm in the other
(knowing only that the logarithm of the base is always 1).
A good way to avoid mistakes, however, is to forget all this and
just memorize the following formula (which does imply the above):

^{ }Log_{ a } x

=

Log x

Log a

In modern practice, the capitalized symbol without subscript (Log)
is used for logarithms when the base doesn't matter or is understood from the context
(by default, natural logarithms are understood in a scientific context).
The other standard notations are:

ln x = Log x / Log e (Natural logarithms.)

log x = Log x / Log 10 (Decimal logarithms.)

lg x = Log x / Log 2 (Binary logarithms; notation due to
Knuth.)

(V. F. of Waynesville, NC.
2001-01-24) What
was the world's first [mechanical] calculator?

The French philosopher Blaise Pascal (1623-1662)
built a famous mechanical calculator (the Pascaline) around 1642, when he was 19.
More than 50 prototypes were made; and one of them was formally presented to Chancellor
Pierre Séguier in 1645.

A similar contraption had been built by the German scientist
Wilhelm
Shickard (1592-1635) for Kepler,
in 1623 (the year Pascal was born). Shickard's
Rechenuhr
was destroyed in 1624 and all but forgotten until a working replica was built at Tübingen by
Baron Bruno von Freytag-Löringhoff (1912-1996) in 1957.

JoBarra
(2001-02-14)
What is significant about 100 degrees Fahrenheit?

In 1714, when G.D. Fahrenheit devised the temperature scale now named after him,
he meant 100° to be the normal temperature of the human body.

At first, he had defined 0°
as the lowest temperature he had achieved by using salt to melt ice.
(It's possible to go a few degrees lower under ideal conditions.)
That temperature was also a record low for Danzig (in 1709).

This fuzzy basis was later improved by
making the ice point exactly 32°F and the steam point 212°F.
In this redefined Fahrenheit scale,
the normal temperature of the human body is around 98.6°F
(= 37°C).

At what temperature do the Fahrenheit and Celsius scales coincide?

The Italian physician
Santorio Santorio (1561-1636)
had a passion for describing all natural phenomena in terms of numbers.
This led him to invent several types of measuring instruments.
He is credited with the invention of the thermometer, because he was apparently the first
to put a numerical scale on what's known as an
air
thermoscope.
In 1599, Santorio Santorio set up a medical practice in Venice and
became part of a circle of learned men which included Galileo Galilei (1564-1642)
who may have devised a rudimentary water thermometer in 1593.

The alcohol thermometer was introduced in 1709 by Gabriel Daniel
Fahrenheit (1686-1736), who also invented the modern mercury thermometer in 1714
and introduced the temperature scale now named after him.
The French scientist René Antoine Ferchault de Réaumur
(1683-1757) devised the competing Réaumur scale
(with a 0° ice point and an 80° steam point) in 1731.
In 1948, the centigrade scale
(featuring a 0° ice point and a 100° steam point)
was officially named after the Swedish astronomer Anders Celsius (1701-1744)
who had devised a backward centigrade scale in 1741
(on that obsolete scale, water freezes at 100° and boils at 0°).

The modern orientation of the centigrade scale was apparently first used on 1743-05-19
by the Frenchman Jean-Pierre
Christin (1683-1755) in Lyon.
Back in Sweden, it was the famous naturalist
Carl Linné (1707-1778) who first specified
and used that modern scale sometime before 1745, after ordering a thermometer
so graduated from the instrument maker
Daniel Ekström (1711-1755) at Uppsala University.

Although this makes a difference only when the utmost in precision is called for,
the Fahrenheit and the Celsius scales are no
longer defined in terms of actual measurements of the ice point or the steam point.
Instead, both scales now refer to the Kelvin
thermodynamic temperature scale, in which 273.16 K is the exact temperature
of the triple point of water (the only temperature at which
all three states of water can be observed at once).

This is pronounced and spelled "273.16 kelvins", foregoing the
word "degree" or capitalization, as with any proper unit
(the symbol, however, is capitalized because that unit is named after a
person).

The temperature of 273.15 K (exactly 0.01 K below the triple point of water)
is now equal, by definition, to both 0°C and 32°F, whereas 373.15 K
is equal to both 100°C and 212°F.

Considering the early technological advances of China,
it may be surprising that China was eventually overtaken
by the West.
This problem was most prominently raised by
Joseph Needham
(1900-1995) and has become known as Needham's Question
(or "Needham's Grand Question").
Needham himself has speculated that Taoism would not encourage
the type of nonconformist driven individuals who have been leading
scientific progress in the West, especially since the advent of the
modern Scientific Method
(something which the Chinese did not invent).

Nomenclature & Etymology

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 mathematician
Abu Abdallah Muhammed bin Musa
al 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 reduce
equations: al jabr (or transposition from one side of the equation to the other,
mostly to obtain positive quantities), al muqabala (reduction, or
cancellation 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 became
known 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 computation
using 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
(30 km southwest of Urganch).
The walled part of the city
(Itchan Kala)
is a UNESCO World Heritage Site,
considered a shrine by the locals.
Its recorded history goes back to the 7th century.
It was once the capital city of Chorasmia
(the country of Kharezm, Khwarazm, or Khorezm),
also known as the Khanate of Khiva from 1511 to 1920
(following conquest by nomadic Uzbeks) straddling modern Uzbekistan and Turkmenistan.
The country was conquered by Russia in 1873 and was known as the
Khorezm 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 city
instead of a less prestigious birthplace in the vicinity.
The only known biographical fact about al-Khwarizmi seems to be that
his parents had moved to a place south of Baghdad.
[ 1 ]

There seems to be a widespread confusion with another astronomer who flourished
decades 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 word
as a contraction of the sentence "avoir du poids",
which means "to have [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 evidenced
by the ancient expression "aver de poiz" which gave the alternate
spelling "averdepois" (still used in an historical context).
Thus, the thing simply means ponderous goods,
as opposed to less bulky items like jewels or
precious metals for which the
avoirdupois system was not intended.

The French word "poids" is spelled
with 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 after
the British had borrowed it.
Furthermore, the French did so for a fallacious reason...

When French spelling was standardized, a few silent letters were
used 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 from
the word "doit" (a form of the verb devoir, which means "must").
Thus, poids (weight) was differentiated from
pois (pea) by a silent "d", ostensibly borrowed
from the Latin word pondus (weight).
The funny thing is that the correct etymology of poids is
not "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's
the same thing to "divide 145 by 5" or to "divide 5 into 145".
People work out the division 145/5 = 29 very differently
in 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 us
how 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!

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

The order in the English layout (above left) is consistent with the
idiom "5 goes into 145 [29 times]". Therefore, it's been suggested that the symbol
consisting of the top vinculum and the curly vertical part should be
called 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] John
Fannon tells us that, about 1950, young British pupils were instructed
to use the above left layout
[possibly with a straight vertical separation instead of a curly one] only
for 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...

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

Scientific usage is that general terms_{ }include 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 is
definitely yes.

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

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

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 universally
understood 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.

Read on, if you're not convinced...

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

Nonmathematical discourse is usually concerned with conveying the most information
in the fewest words about some specific instance of a concept,
so that the word with the narrowest meaning is used whenever possible.
If you'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:
If something 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 often
be able to generalize such things further; to all
conic sections).

Mathematical terms are defined to make
theorems as simple and/or as general as possible.
Nearly anything that is true of an ellipse is also true of a circle, and that is
why 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 necessarily
distinct points...
["Foci", plural of "focus" is pronounced "foe sigh".]

Occasionally, the mathematical definitions are in direct conflict with
what general-purpose dictionaries state...
For example, to a mathematician an ellipsoid
is a special type of ovoid and a
spheroid is a special type of
ellipsoid (i.e, an oblate or prolate ellipsoid with [at least] one
axis 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 special
type of "spheroid" (the Encarta dictionary makes that mistake).

Ultimate argument, for lexicographers:

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

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

Consider an increasing finite sequence (x_{n}) of points
starting at a and ending at b.
An approximation of the integral of f from a to b
is obtained as the sum (over the relevant range of n) of the areas
of all the trapezoids with vertical bases x = x_{n}
and x = x_{n+1} whose vertices are either on
the x-axis or on the curve y = f(x).

Now, if f(x_{n}) = f(x_{n+1}), the corresponding
trapezoid 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 the
trapezoid could in fact be a rectangle or a square?

Still not convinced about how pervasive inclusive concepts are in regular
mathematical discourse?
Look again at the meaning of other words in the above description of the
trapezoid method.
We talked about an "approximation" to the integral,
but we certainly did not mean to exclude the special case where this
approximation happens to be the exact value, did we?
The approximation is exact when f is linear, but this
could happen in many other cases.
Do you want to even mention such cases?
Also, when f is linear, the "curve" of equation
y = 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 any
scientific material whatsoever...

Are there exceptions to this rule?

Scientific concepts are as inclusive as they can be, unless a word is used
whose etymology implies exclusion. For example, the term
"pseudoprime" is normally understood
not to apply to a prime number (although definitions and theorems
would 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 used
to exclude it.

In particular, the qualifier proper turns a
reflexive relation into an anti-reflexive one.
That's a fancy way of saying that if a relation with respect to some
object always apply to the object itself, then the matching
proper relation never does. The qualifier
proper may exclude something else only if we didn't
have any reflexivity to begin with...

Examples, including a few tricky ones:

A proper multiple of an integer isn't equal to that integer.

Last (and possibly least) a proper trapezoid isn't a parallelogram !

Confusion arises when the qualifier proper is dropped from
such examples.
This may happen even in reputable textbooks, especially when the vocabulary
is 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 term
unless 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 polygons
with 11 or 12 sides. These are macaronic terms (namely, terms built from a mixture
of 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 systematic
naming of polygons is
purely 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 circle
and a monogon or henagon consists of a single vertex and
any 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 are
great half-circles).

There's no specific name for the empty polygon, 0-gon (0 vertices, 0 edges),
so the 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"),

For some obscure reason,
the corruption of the prefix hexaconta- into hexeconta- became acceptable
or dominant when applied to families of objects other than polygons.
For example "hexecontahedron" is used more often than
"hexacontahedron" to denote a polyhedron
with 60 faces.
I advise 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.

(2001-06-24)
Chemical Nomenclature:

The Greek numerical prefixes are not limited to the
naming of polygons; they are the
basis of the systematic naming of other families of scientific objects which depend on
some 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 name
which 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 of
1 or 2 carbon atoms into chemical nomenclature,
but let's have fun :
"Cycloethane" is ethylene (C_{2}H_{4} or H_{2}C=CH_{2})
"cyclomethane" is methylene (CH_{2 }).
"Cycloethanol" would be yet another name for
ethenol (H_{2}C=CHOH), also known as vinyl alcohol or hydroxyethylene.
Finally, "cyclomethanol" would be the elusive hydroxymethylene
(HCOH).

In 1957, the IUPAC (carelessly) endorsed the use of
the 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 the
effect 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 called
formic acid, because it was first distilled (!) from ants
(Latin: formicae, French: fourmis).
Formaldehyde (CH_{2}O, CAS 50-00-00) is thus the common name
of what is more properly called methanal. The French commonly call formaldehyde
formol, with an unfortunate use of a suffix normally reserved for alcohols
(the name formal [sic] has been proposed, which would feature the proper suffix
for an aldehyde, but it never caught on).
Just to take a cheap shot at the practical lack of standardization in some chemical
names, here are some of the published names used for O=CH_{2},
in alphabetical order:
BFV, CH_{2}O, FA, Fannoform, Floguard 1015, FM 282, Formaldehyde,
Formalin, Formalin 40, Formalith, Formic aldehyde, FYDE,
H_{2}CO, Hoch, Ivalon, Karsan, Lysoform,
Methaldehyde, Methanal, Methyl aldehyde, Methylene glycol, Methylene oxide,
Morbicid, Oxomethane, Oxymethylene, Paraform, Superlysoform,
NCI-C02799, RCRA waste number U122, UN 1198, UN 2209.
Other foreign designations occasionally surface in English
texts, including: "Aldéhyde formique" or "Formol" (French),
"Aldeide formica" or "Formalina" (Italian),
"Aldehyd mravenci" (Czech), "Formaldehyd" (Polish),
"Formaline" (German), or "Oplossingen" (Dutch)...

Nomenclature of Saturated Carbon Chains

n:0

Chemical adjectives commonly used for straight counting :

The numerical designations n:0, shown above, are commonly used by chemists for a
saturated chain of n carbons (no double bonds). An unsaturated chain of n carbons with
p 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 chains
usually apply to only one particular position of the
double 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).
The popular (overmarketed) compound DHA does not have a competing traditional
name, 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 use
butylique.
Pentyl and amyl are used as names, but do not
serve as the bases for adjectives.

In a number of cases, systematic adjectives are far less popular than the
traditional ones which
appears in bold face toward the end of some lists.
Such traditional adjectives are usually derived from the names of plants
containing the corresponding unsaturated
fatty acids
(alkanic carboxylic monoacid).
In at least 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 :
The adjective cetylic does come from
cetus, the Latin name of the whale (the corresponding alternate name
of hexadecane is cetane, but cetanoic is unused).
The corresponding fatty acid was originally obtained from spermaceti
extracted 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 oil
and 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, our
plea for help
was answered by Valerio Parisi, whom we thank for the following comment about the
kelor tree
(also known as
moringa,
horseradish tree,
dangap in Somalia,
etc.)
In 1848, behenic acid was found as a constituent (up to 8.6%) of
the Moringa oleifera seed oil and was then named after
that oil's common designation:
ben oil or behen oil.

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

Dear Dr. Michon,

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

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

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

I must say I have enjoyed your Numericana web site thoroughly.
I am 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 is
that the predominant mechanism of synthesis in the cell
is 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 synthesis
in the laboratory known as the Claisen condensation.
In the 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 is
P_{4}S_{3}
where neither element is the "sesqui" of the other. Thanks.

The usual meaning of the "sesqui" prefix is "one and a half ".
A sesquicentury is 150 years and
a sesquicentennial marks the passing of that 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 a
whole 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 to
form [new] scientific names with unambiguous meanings...

P4S3
is now called tetraphosphorus trisulphide.
This compound was once an important discovery, with huge social
implications (see below)
around the turn of the 20th century.

The curious name of
phosphorus sesquisulphide was apparently given to it at the time by its
French 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 acceptable
because 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 used
a "proper" name which was shortened later because there is really no
possible 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 the
more precise name of tetraphosphorus trisulphide is now used.

Matches,
Phosphorus, and P_{4}S_{3}

An household match is such a common item nowadays
that it may be hard to imagine what a revolutionary
marvel it once was! Vigorously rubbing two pieces of wood together
may generate enough heat to allow some dry material to smolder and then ignite in the
presence 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 that
friction will cause enough heat locally as well as some sparking to trigger ignition of
flammable material...

Surprisingly, the first recognizable friction matches didn't use
any phosphorus at all:
They were made in 1826 from
a fifty-fifty mixture of potassium chlorate
and antimony trisulphide,
together with some gum arabic, sugar and starch.
The inventor was an English pharmacist from Stockton-on-Tees
[at 59 High Street] named
John Walker (1781-1857).
Walker didn't bother to patent his creation that
he called friction lights.
He sold the very first batch on April 7, 1827
(to a local solicitor named Hixon ).
Those bulky three-inch splints of wood, were still expensive, unreliable and
somewhat tricky to use.

All this would change 4 years later (in 1830 or 1831)
when a young chemistry student from the French village of
Saint-Lothain
(Jura) had the idea to substitute white phosphorus
for the antimony sulfide in Walker's recipe. This is how
Charles
Sauria (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 and
died a pauper.

Those white phosphorus "strike-anywhere" matches
became known as Lucifers, which was
the trademark coined by Samuel Jones in 1829 or 1830
for 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 fatal
bone disease known as phossy-jaw.
White phosphorus became a public health issue on the international scene...

The French goverment sponsored research to find a suitable replacement
for 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).
A paste 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 oxide
and iron oxide ).
It was not spontaneously flammable, not toxic and didn't cause phossy-jaw!
A perfect 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 phosphorus
from the manufacture of matches.
The US did not sign that treaty (on the grounds that the required
ban 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 P4S3
matches was secured in 1910 by the Diamond Match Company.
However, the public health issue was such that President Taft
publicly urged the company to voluntarily surrender
its 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 P4S3
strike-anywhere matches are still quite popular in the US.
In many other countries, however, they have been all but replaced by
the 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 invented
in 1855, by
Johan Edvard Lundstrom
of Sweden.

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

The title of that book comes from the fact that phosphorus
was 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.

(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 10^{6n}, 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 use
the same names to denote the successive powers of a thousand instead,
namely 10^{3n+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 1000 000 000)
and increasingly in English texts of any origin.
In 1974, British Prime Minister Harold Wilson even informed the House of Commons
that the word "billion" in statistics from the British government would
thenceforth mean 10^{9}, 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 to
the 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 international
use 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 1000 000 000,
the corresponding British term "milliard", which would be unambiguous,
is apparently rarely used nowadays. (The term "milliard" itself was apparently
coined around 1550 and is credited to Jacques Pelletier.)
Nevertheless, a "zilliard" sequence is being used to denote 10^{3+6n}.
These are numbers 1000 times larger
than the corresponding "zillions" of the original Chuquet system, whose gaps
compared with the American system are thus filled:
milliard (10^{9}),
billiard [sic!] (10^{15}),
trilliard (10^{21}),
quadrilliard (10^{27}), etc.

n

n^{th} zillion

US

World

1

million

10^{6}

10^{6}

milliard

10^{9}

2

billion

10^{9}

10^{12}

3

trillion

10^{12}

10^{18}

4

quadrillion

10^{15}

10^{24}

5

quintillion

10^{18}

10^{30}

6

sextillion

10^{21}

10^{36}

7

septillion

10^{24}

10^{42}

8

octillion

10^{27}

10^{48}

9

nonillion

10^{30}

10^{54}

10

decillion

10^{33}

10^{60}

11

undecillion

10^{36}

10^{66}

12

dodecillion
duodecillion

10^{39}

10^{72}

13

tredecillion

10^{42}

10^{78}

14

quattuordecillion

10^{45}

10^{84}

15

quindecillion

10^{48}

10^{90}

16

sexdecillion

10^{51}

10^{96}

17

septendecillion

10^{54}

10^{102}

18

octodecillion

10^{57}

10^{108}

19

novemdecillion

10^{60}

10^{114}

20

vigintillion

10^{63}

10^{120}

n

n^{th} zillion

US

World

21

unvigintillion

10^{66}

10^{126}

22

dovigintillion
duovigintillion

10^{69}

10^{132}

23

trevigintillion

10^{72}

10^{138}

24

quattuorvigintillion

10^{75}

10^{144}

25

quinvigintillion

10^{78}

10^{150}

26

sexvigintillion

10^{81}

10^{156}

27

septenvigintillion

10^{84}

10^{162}

28

octovigintillion

10^{87}

10^{168}

29

novemvigintillion

10^{90}

10^{174}

30

trigintillion

10^{93}

10^{180}

31

untrigintillion

10^{96}

10^{186}

32

dotrigintillion
duotrigintillion

10^{99}

10^{192}

33

tretrigintillion

10^{102}

10^{198}

40

quadragintillion

10^{123}

10^{240}

50

quinquagintillion

10^{153}

10^{300}

60

sexagintillion

10^{183}

10^{360}

70

septuagintillion

10^{213}

10^{420}

80

octogintillion

10^{243}

10^{480}

90

nonagintillion

10^{273}

10^{540}

100

centillion

10^{303}

10^{600}

In her 1975 review of numeration systems
(Histoire comparée des numérations écrite)
the French mathematician
Geneviève 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 name
large numbers.
Unconfirmed exceptions
we've gleaned so far include Turkish, Greek and Romanian, as well as
Spanish 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 which
the American system is used, please
let 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 devised
with 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-) to
999 (novenonagintanongentilli-). You may append an unlimited number of such prefixes
and 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 described
by 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 Munafo
has 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 that
a trescentillion denotes 1000 to the power of 103 while a
trecentillion is 1000 to the power of 300.
Arguably, this
narrowly avoided ambiguity remains a flaw when the names are merely
pronounced (arguably, the extra "s" is virtually silent) and
I maintain my previous proposal that the Conway-Wechsler system should be amended
by 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 100 000 to the power of 100 000
(a 1 with 1/2 million zeros)?

Following Rudy Rucker (quoted by John Conway and Richard Guy), we may use
the suffix plex at the end of a number to denote 10 to the power of
that 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 of
elementary particles in the observable Universe).

The neologism "googol" was coined in 1938 by the American mathematician
Edward 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 entire
non-English-speaking world, who considers that a billion is a million million).

A googolplex is 10^{10100}, a number which is
impossible 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 by
Tadashi Tokieda as a contraction
of "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 100000^{100000}
may be called "500000-plex".
This is allowed, but you may argue that the use of numerals does not
make this more of a "name" than
100000^{100000} or 10^{500000}.

The problem is that the plex suffix leads to ambiguity when used with number names
that 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 a
hemimillionplex .

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 prefixes
sesqui-, di-, tri-, quadra-, penta-, etc.)
should have a stronger parsing priority than the suffix plex.
An hyphen (hemimillion-plex) would probably make
the whole thing less ambiguous, but this breaks the pattern established for lesser numbers.

I don't know 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, Betsy
McCall 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 hemimillionplex
becoming popular enough to set a trend.
Another alternative would be to cast 500 000 [purely] in Greek terms
[to obtain] pentacontamyriaplex.
The Greek prefixes will naturally parse together.
[...]
I seriously doubt that the Oxford English Dictionary
will be quoting any of these words any time soon.