names of big numbers

In the English-speaking world two different systems have been used to name numbers larger than a million, one associated with the United States and the other with Great Britain. In the U.S., one billion is a thousand millions; in the U.K. it is, or was, a million millions. This difference has forced those writing for an international audience to such awkward phrases as “one thousand million,” to avoid the ambiguous “billion.”  In physics, the ambiguity helped lead to the demise of the term “Bev”–a billion electronvolts.

At present Britain is in a confusing period of transition. As early as 1926 H. W. Fowler, in his influential Modern English Usage, called for adopting the American system. Writers in the fields of economics and finance were early users of “billion” in the American sense, and in 1974 the British government announced that in its reports and statistics “billion” would henceforth mean 1,000,000,000.  The Times of London style guide now also uses the American definition. “Milliard” is rarely heard. Anecdotal reports suggest that at least among young people, everyday British usage is now the same as that of America.

In both systems the names of the numbers are constructed with Latin prefixes attached to “-illion” (see History). In the American system, which was taken from the French around 1800, the Latin prefix tells how many groups of 3 places are in the written number, not counting the first group of three (the one which represents numbers up to 999). In the other system, the Latin prefix tells what power of a million the number is. 

For example, “bi-” is from the Latin for “two”. To the Americans, that means adding 2 more “000” groups to “1,000,” so one billion = 1,000,000,000. In the other system, a billion is a million to the second power, that is, 1,000,0002, so a billion = 1,000,000 × 1,000,000 = 1,000,000,000,000. The discrepancy between the two systems grows as the numbers the prefixes stand for get bigger. The result is summarized in the table below.

# of zeros after 1
(power of 10)
# of “000”
groups
after 1,000
Name in U.S. (and others) Power of one million Name in
Germany
(& elsewhere)
6 1 million 1 million
9 2 billion milliard
12 3 trillion 2 billion
15 4 quadrillion   billiard
18 5 quintillion 3 trillion
21 6 sextillion   trilliard
24 7 septillion 4 quadrillion
27 8 octillion   quadrilliard
30 9 nonillion 5 quintillion
33 10 decillion   quintilliard
36 11 undecillion 6 sextillion
39 12 duodecillion   sextilliard
42 13 tredecillion 7 septillion
45 14 quattordecillion   septilliard
48 15 quindecillion 8 octillion
51 16 sexdecillion   octilliard
54 17 septendecillion 9 nonillion
57 18 octodecillion   nonilliard
60 19 novemdecillion 10 decillion
63 20 vigintillion    
66     11 undecillion
72     12 duodecillion
78     13 tredecillion
84     14 quattordecillion
90     15 quindecillion
96     16 sexdecillion
102     17 septendecillion
108     18 octodecillion
114     19 novemdecillion
120     20 vigintillion
303 100 centillion    
600     100 centillion

A cottage industry exists inventing names for even larger numbers, which serves no purpose but amusement. Beyond quadrillion, these words are only found in lists like ours, and not in dictionaries.  Anyone dealing with numbers beyond a quadrillion is likely to be a scientist who will use exponential notation, which is easier and clearer.  “1.234 × 1045” is not only shorter than “one quattordecillion, two hundred thirty-four tredecillions”; it is also unambiguous.

The second system is used by most of Europe, including the French, who decided in 1948 to convert to the system of their neighbors. Of course, the exact spellings of the number names differ from language to language: billiard is “Billiarde” in German, billion is “bilhão” in Portuguese, and so on.

Where does one billion = 1,000,000,000?

Where does one billion = 1,000,000,000,000?

History

Million

The word “million” originated with its present meaning in Old Italian, formed by adding to “mille,” thousand, the augmentative suffix “-one,” “great.” Thus “millione” is a “great thousand.” Another example of the same process is “padrone,” formed from “padre,” father, and meaning “great father,” hence patron, master, landlord1. “Millione” was in use in Italy by the early 14th century. In 1419 Charles VI of France wrote2

Deux millions qui sont vingt fois cent mille escus.

Two million, which is 20 times 100,000 escus.

By 1494 the Italian mathematician and inventor of double-entry bookkeeping, Luca Pacioli, wrote2

mille migliara che fa secondo el volgo el millione

a thousand thousands, which makes, according to the common people, a million

From Italy the word spread, appearing in French by the middle of the 14th century and in German by the early sixteenth century.  Interestingly, Tropfke3 remarks that in German the word first appeared only in the phrase “1 Million Gulden,” that is, “1 million guilders,” showing its close association with commerce.

Billions, trillions, etc.

At least as early as the middle of the 15th century the “-illion” part of “million” had been extracted and was used as a root to which Latin prefixes were added to name other large numbers.

The first known recorded use of the words billion and trillion are in a work of Jehan Adam (1475)4:

item noctes que le premier greton dembas vault ung, le second vault [...here some words seem to be omitted...] cent, le quart vult mille, le Ve vault dix M, le VIe vault cent M, le VIIe vault Milion, Le VIIIe vault dix Million, Le IXe vault cent Millions, Le Xe vault Mill Millions, Le XIe vault dix mill Millions, Le XIIe vault Cent mil Millions, Le XIIIe vault bymillion, Le XIIIIe vault dix bymillions, Le XVe vault mil bymillions, Le XVIe vault mil bymillions, Le XVIIe vault dix Mil bymillions, Le XVIIIe vault cent mil bymillions, Le XIXe vault trimillion, Le XXe vault dix trimillions.

Also note that the first counter from the bottom stands for one, the 2nd stands for [...] one hundred, the 4th stands for one thousand, the 5th stands for ten thousand, the 6th stands for one hundred thousand, the 7th stands for a million, the 8th stands for ten millions, the 9th stands for one hundred millions, the 10th stands for one thousand millions, the 11th stands for ten thousand millions, the 12th stands for one hundred thousand million, the 13th stands for a billion, the 14th stands for ten billions, the 15th stands for one hundred [the "mil" in the original is an obvious error] billions, the 16th stands for one thousand billions, the 17th for ten thousand billions, the 18th stands for hundred thousand billions, the 19th stands for a trimillion, the 20th stands for ten trimillions. 

Nine years later, in 1484, a French mathematician, Nicolas Chuquet (1445 – 1488), completed a treatise entitled Le Triparty en la science des nombres. Florian Cajori has written: “Chuquet elaborates the exponential notation to a completeness apparently never before dreamed of. On this subject Chuquet was about one hundred and fifty years ahead of his time.”5  In the very first pages of the Triparty Chuquet wrote:

Et pour plus facilement nõbrer ung grant nombre lon peult diuiser les figures de six en six en commancant tousiours a dextre. et sus la premiere figure dune chascune sixne la premiere exceptee lon peult mettre ung petit point.  Et doit on sauoir que toutes les figures depuis le premier point jusques au secõd se tant on y a sont tous. millions et du second au tiers sont millions de millions et du tiers au quart sont millions de millions det millions Et ainsi des ault's pointz en proferant ce vocable million autant de foiz comme il y aura de pointz Ou lon peult mettre 1. ou lieu du premier point et 2. ou lieu du second et 3. ou lieu du tiers et. 4. ou lieu du quart qui auront semble signification comme les pointz ¶ Ou qui veult le premier point peult signiffier million Le second point byllion Le tiers poit tryllion Le quart quadrillion Le cinqe quyllion Le sixe sixlion Le sept.e septyllion Le huyte ottyllion Le neufe nonyllion et ainsi des ault's se plus oultre on vouloit preceder ¶ Item lon doit sauoir que ung million vault mille milliers de unitez. et ung byllion vault mille milliers de millions. et tryllion vault mille milliers de byllions. et ung quadrillion vault mille milliers de tryillions et ainsi des ault's. Et de ce en est pose ung exemple nombre diuise et punctoye ainsi que deuant est dit. tout le quel nombre monte 745324. tryllions 804300 byllions. 700023 millions 654321 [endnote 3]

     .      .     .
7453248043000700023654321 [sic]

And for more easily enumerating a large number, one can divide the figures into groups of six, always from the right.  And above the first figure of each group of six, except the first, one can put a small mark. And one should know that all the figures, as many as there are, from the first mark to the second are millions, and from the second to the third are millions of millions, and from the third to the fourth are millions of millions of millions, and so on for the other marks, saying the word "million" as many times as there are marks. Or one can put a 1 in place of the first mark and a 2 in place of the second, and 3 in place of the third, and 4 in place of the fourth, which will have the same meaning as the marks.  Or if you prefer the first mark can signify million, the second mark byllion, the third mark tryllion, the fourth quadrillion, the fifth quyillion, the sixth sixlion, the seventh septyllion, the eighth ottyllion, the ninth nonyllion and so on with others as far as you wish to go.  Item: one should know that a million is worth a thousand thousand units, and a byllion is worth a thousand thousand millions, and tryillion is worth a thousand thousand byllions, and a quadrillion is worth a thousand thousand tryllions, and so on for the others. And an example of this follows, a number divided up and punctuated as previously described, the whole number being seven hundred forty-five thousand three hundred and twenty-four tryllions, 804300 byllions 700023 millions 654321.

 

Chuquet's treatise was not published in his lifetime, but a man who may have been his pupil, Éstienne de la Roche (1470–1530), used it as the first part of a textbook he published in 15204.  (A scholar, Aristide Marre, discovered Chuquet's manuscript in the Bibliothèque Nationale in the late 1870s–with notes apparently in de la Roche's handwriting.)

Neither Adam nor Chuquet claim they invented these names. After the first section, Chuquet never mentions them again. That he defines these terms without making any use of them suggests that the passage is an obligatory presentation of a notation numerically literate persons were expected to know, rather than an innovation by Chuquet.

Milliards, billiards, etc.

A few decades later we find the second series of still extant words for big numbers recorded in an arithmetic text of 1549 by Jacques Peletier, who is often credited with inventing the term “milliard” (originally “milliart”). This distinction he disclaims:

Les Francois ont deux motz numeraux significatifz, l'un au septiesme lieu, qui est Milliõ, et l'autre au treziemse, qui est Milliart, c'est a dire Million de Millions. Et pource ont ilz autre maniere de Nombrer que n'ont les Latins, qui n'ont point de mot simple parsus Milini que les Grecz, qui n'en ont point parsus Miriade, qui vaut dix Mil. Je toucherai seulement icy la mode Francoise, laissant la mode Latin et Grec que a cause de brieueie

...

Je n'eusse point usurpè ce mot de Milliart, n'eust etè l'autoritè de Budè au Traittè de la Livre et de ses parties et me fusse contentè de demeuer aux Millions. Mais avec ce que la difficultène m'en semble point plus grande, la maniere de Nombrer sen trouve plus riche.8

The French have two words of numeric meaning, one for the seventh place, which is “million,” and the other for the thirteenth, which is “milliart,” that is to say, a million millions. And therefore, they have a different way of numbering than the Latins do, who do not have a simple word for numbers above a million, like the Greeks, who have no word beyond “myriad,” which is ten thousand. I will write here in only the French method, ignoring the Latin and Greek method for the sake of conciseness.

...

I would not have usurped [Ed: that is, “taken the liberty of introducing” or “presumed to use”] this word “milliart,” if not for the authority of Budè in “De asse et partibus ejus”9, and I would have contented myself with “millions.” But this doesn't seem to me to raise any great difficulty, and enriches our method of numeration.

“Milliart” was admitted to the dictionary of the Academie Francaise in 1740.

A pause for reflection

Chuquet's practice of dividing a number into groups of six is almost unique, even in its own time, though it is used by John Locke (1690):

And I doubt not but we ourselves might distinctly number in words a great deal further than we usually do, would we find out but some fit denominations to signify them by; whereas, in the way we take now to name them, by millions of millions of millions, &c., it is hard to go beyond eighteen, or at most, four and twenty, decimal progressions, without confusion. But to show how much distinct names conduce to our well reckoning, or having useful ideas of numbers, let us see all these following figures in one continued line, as the marks of one number: v. g.

Nonillions Octillions Septillions Sextillions Quintrillions Quartrillions Trillions Billions Millions Units
857324 162486 345896 437918 423147 248106 235421 261734 368149 623137

The ordinary way of naming this number in English, will be the often repeating of millions, of millions, of millions, of millions, of millions, of millions, of millions, of millions, (which is the denomination of the second six figures). In which way, it will be very hard to have any distinguishing notions of this number. But whether, by giving every six figures a new and orderly denomination, these, and perhaps a great many more figures in progression, might not easily be counted distinctly, and ideas of them both got more easily to ourselves, and more plainly signified to others, I leave it to be considered. This I mention only to show how necessary distinct names are to numbering, without pretending to introduce new ones of my invention.10

It was far more common to divide numbers into groups of three, as Peletier does, using marks like Chuquet’s. Even today we use thousands separators, whether a space (as is recommended with SI), a point or a comma. In contrast, conventional millions separators are unknown. Similarly the modern metric prefixes differ in leaps of a thousand, not a million. The “-ard” words appeared because the use of the powers of a million notation created a need for them. Unfortunately the words that were coined create pairs that sound confusingly similar.

Numeration then lay dormant until the late 1700's. At least as early as 1834 American schoolbooks described the current American system. In 1948 the 9th CGPM (Resolution 7) endorsed separation in groups of three, not six:  

In numbers, the comma (French practice) or the dot (British practice) is used only to separate the integral part of numbers from the decimal part. Numbers may be divided in groups of three in order to facilitate reading; neither dots nor commas are ever inserted in the spaces between groups.

The conversion of the French in 1961 and the position of the British has already been described.

1. We owe this example to:
Graham Flegg, Cynthia Hay and Barbara Moss.
Nicolas Chuquet, Renaissance Mathematician.
Dordrecht: D. Reidel Publishing Company, 1985.

Menninger gives the same example, and adds “sala,” room; “salone,” sitting room, lounge, saloon.

Karl Menninger.
Paul Broneer, translator.
Number Words and Number Symbols. A Cultural History of Numbers.
Cambridge, MA: MIT Press, 1969.
Reprinted in facsimile by Dover Publications, 1992[BACK]

2. Jean Juvénal Des Ursins.
Histoire de Charles VI.
Paris: impr. royale, 1653.

Luca Pacioli.
Summa de Arithmetica Geometria Proportioni et Proportionalita.
Venice: Paganino de Paganini, 1494.
Page 19 verso

3. Johannes Tropfke.
Geschichte der Elementar-Mathematik in Systmatischer Darstellung mit besonderer Berücksichtigung der Fachwörter. Erster Band, Rechnen. 2nd enlarged edition.
Berlin, Leipzig: Vereinigung Wissenschaftlicher Verleger; Walter de Gruyter & Co., 1921.

4. Lynn Thorndike.
The Arithmetic of Jehan Adam, 1475 A. D.
The American Mathematical Monthly, volume 33, number 1, pages 24-28 (January 1926).

The quotation and its translation are excerpted from Thorndike.

5. Florian Cajori.
A History of Mathematical Notations. Volume I.
La Salle, IL: Open Court Press, 1928.

Page 100.  Reprinted in facsimile, two volumes in one, by Dover Publications in 1993.

6. Aristide Marre, ed.
Le Triparty en la science des nombres.
Bullettino di Bibliografia e di Storia delle scienze matematiche et fisiche, volume 13, pages 555-659, 693-814; volume 14, pages 413-460. (1880)
Reprinted in facsimile in 1964 by Johnson Reprint Corporation, #10 in the Sources of Science series.

Page 593.  We have written out "premier" and “preceder,” to avoid having to try to depict Chuquet’s handwritten abbreviation for “pre-”.

7. L'arismethique nouvellement composee.

8. Jacque Peletier.
L'Arithmetique de Jacque Peletier du Mans, departie en quatre Livres, a Theodore Debesze.
Poitiers: at the sign of the Pelican, 1549.
Reprinted in facsimile by Slatkine Reprints, Geneva, 1969.

Selections from chapter 1, sections 6 and 10.

9. Guillaume Budè was a renowned scholar and humanist, secretary to Louis XII and later Francis I. "De asse et partibus ejus" (Venice, 1522). There is a wonderful portrait of Bude at www.metmuseum.org/toah/ho/08/euwf/ho_46.68.htm

10. John Locke.
Essay on Human Understanding.
1690.

Chapter 16, section 6.

Chinese names of big numbers

In China, names for big numbers appeared before the Qin dynasty (before 211 bce): yi (ten thousand); zhao (million); jing (ten million); yi (again, one hundred million); zhao (again, a thousand million). During the Qin the ambiguity of zhao was resolved when the term for million was changed to baiwan. In modern times, ten thousand became shiwan and ten million qianwan; otherwise the old terms remain in use.

Resources

Steve Trussel has an interesting page comparing American and Japanese names for big numbers: www.trussel/com/jnumbers.htm

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