Name of the great numbers

The names of the great numbers (higher than the trillion) are practically never used, at least in a context of normal communication. Many systems were proposed to name great numbers, but none seems to have had of practical utility.

; Note: This article uses the scientific Notation: 10⁶ corresponds to one 1 follow-up of six 0, that is to say 1.000.000; in the same way, 10⁹ is worth 1.000.000 000, and so on.

Use of the great numbers

A few great numbers have really a direction for the man, and are of a use relatively running. Thus, at the beginning of 2007, a research in the topicality (Google) gives:

7.365 pages found for the word million.
9.988 for billion.
22 for billion.
36 for trillion.
Nothing for quadrillion.

Beyond that, the names of great numbers have hardly than an artificial, mathematical existence: pages containing the word " sextillion" are nothing any more but mathematical pages of definitions of these great numbers.

In the everyday usage, these great numbers are expressed with the scientific Notation. With this notation, which exists since the years 1800, the great numbers are expressed by one ten and one number while exposing. One will say for example: " The emission in x-rays of this radio-galaxy is of 1.3· 10⁴⁵ erg". The 10⁴⁵ number is read simply " ten forty-five power " : it is easy to read, easy to understand, and much more speaking that " quattuordécillion" (which presents moreover the disadvantage of meaning two different things, according to whether convention used is the long or short scale).

When it is a physical quantity which must be indicated, in fact the prefixes of the international system are preferentially used. Everyone will include/understand what is a " femtoseconde" , whereas a " billiardième of seconde" will be difficult to include/understand.

It is thus not for their practical utility that the great numbers are named, but they from time immemorial fascinated those which are leaning on them while trying to apprehend what " large nombre" could mean well.

System of Archimedes

One of the first known examples is the calculation which Archimedes made number of grains of sand that the universe could contain, in Arénaire (Ψάμμιτης). For that, it generalized the Greek numbering system, whose highest term was called the myriad (10⁴), which thus made it possible to the Greeks to count up to 99.999.999 (either 10⁸-1, the myriad of myriad not having a name).

Archimedes called these nommables numbers in Greek of the " numbers of first ordre" ; and the myriad of myriad called, that is to say 10⁸, the basic unit of the " numbers of second ordre". By taking this number like new unit, Archimèdes was able to name 99.999.999 " numbers; of second ordre" , until 10⁸· 10⁸=10¹⁶. This number is in its turn taken as the unit of the " numbers of third ordre" , and so on.

Archimedes continued its logical construction for all the " ordres" who could be named in Greek, i.e. until the number of order a myriad of myriad, is $(10^8) ^ {(10^8)}=10^ {8 \ cdot 10^8}$ , natural end of this series of designation.

Archimedes prolonged this construction by taking this number like basic unit again, which enabled him to extend the system of denomination until $\ left ((10^8) ^ {(10^8)}\ right) ^ {(10^8)}=10^ {8 \ cdot 10^ {16}}.$

This point, Archimedes made use of this designation to estimate the number of grains of sand which the universe could contain, because " innumerable like the grains of sable" for the Greeks the archetypal example of something represented which could not be counted. It found like order of magnitude " thousand myriads of the eighth ordre" (either 10⁶³).

Family of - lions

System of Nicolas Chuquet

Nicolas Chuquet wrote a book, Triparty in the science of the numbers, where one finds first exposed use modern to group the great numbers per packages of six digits, that it separated by points (it will be noticed that the names employed by Chuquet are not completely the modern names).

Or which wants the first point peult to mean million the second point byllion the third poit tryllion the quarter quadrillion the cinq^e quyllion the six^e sixlion the sept.^e septyllion the huyt^e ottyllion the neuf^e nonyllion and thus of the ault'^s more oultre one vouloit to precede itself. Item lon must know that ung million vault thousand thousands of unitez, and ung byllion vault thousand thousands of million, and tryllion vault thousand thousands of byllions, and ung quadrillion vault thousand thousands of tryllions and thus aultres: And of it in is divided installation ung number example and punctoye as in front of is said, very which number assembles 7 squared 453248 tryllions 043000 byllions 700023 million 654321. Example: 7 ' 453248 ' 043000 ' 700023 ' 654321.

However, the work of Chuquet was not published of alive sound. A good part was copied by it by Estienne of the Rock in a work which it published in 1520, arismetic the .

This description is that which corresponds to the system known as of the long scale, where the prefixes correspond to the powers of the million. The bymillion of Adam ( byllion for Chuquet) thus corresponds to 10¹², and the trimillion / tryllion is worth 10¹⁸.

It is in Chuquet that one allots the invention of the system, but the first terms existed before him. The words bymillion and trimillion appear in 1475 in a manuscript of.

the term Million existed before Adam and Chuquet. It is a word of probably Italian origin, millione , intensified form of the word thousand : a million is étymologiquement a a large thousand, pointing out the units of second order of Archimèdes.
the way in which Adam and Chuquet present these terms suggests that they describe a preexistent use, rather than a personal invention. It is probable that terms like billion and trillion were already known at that time, but that Chuquet (expert in art to handle the exhibitors) generalized the system of it, inventing the names corresponding to the higher powers.

Chunquet specified only the first ten prefixes; the extension of its system to the higher numbers always caused alternatives in the solutions adopted to adapt the Latin names to the suffix - lion.

Formation of the names in - lion

The system of Nicolas Chuquet consists in making follow the prefixes bi-, sorting,… of the suffix - lion, to form the successive names of unit. In the original system, which corresponds to the long scale, each unit is worth 10⁶ time the preceding unit. One thus has, in a regular way:

These ten units make it possible to reach 10⁶⁰, which is enough largely with the normal physical uses. It is the system recommended in 1948 by the ninth general conference of the weights and measures (and made legal to France by decree 61-501 of May 3rd, 1961). This regular system is that known as of the long scale . The Anglo-Saxon countries tend to use an irregular system, the short scale , where one billion is worth 10⁹ and a trillion 10¹² (other units being without practical applications).

Standardization suggested by Conway and Wechsler

Proposed by John Horton Conway and Allan Wechsler, this system regularizes and prolongs that of Nicolas Chuquet. The first stage of its system consists in standardizing the writing of the Latin prefixes, from 1 to 999 (in the table which follows, the indents are intended only to facilitate the reading, and do not form part of the name of number).

The radicals of the units can take or lose linking consonants:

tre becomes very in front of the words preceded by a S : thus, 303= tre' trecenti .
is become its in front of the words preceded by a S : thus, 306= se' trecenti .
is become sex in front of the words preceded by a X : thus, 106= se' x' centi , while 600 = se' centi .
septe becomes septem in front of the words preceded by a m , and septen in front of the words preceded by a N : thus, 107= septe' centi and 87= septe' me octoginta .
In the same way, nove becomes novem in front of the words preceded by a m , and noven in front of the words preceded by a N : thus, 109= nove' centi and 89= nove' me octoginta .

The figures are stated in the order unit, ten, hundred; and when the figure is one zero, the corresponding term is simply omitted.

With this construction, a 421-lion is called a a-vinginti-quadringenti-lion .

Extension propoposée by Conway

In the same publication, Conway proposes to build the Latin radicals for the numbers higher than thousand in the following way:

Is NR the required Latin prefix to write a NR - illion.
To gather the figures of NR per blocks of three digits.
Utiliser preceding coding for each block of three digits, or nor - lli if the three figures are null.
To intercalate lli between each block thus obtained.

Thus, with this method, a 3_000_102-llion is called a tri - lli - nor - lli - duet-centi - lli - one .

Other systems of great numbers

Gillion system

Proposed by Russ Rowlett, based on the Greek numerical Prefixes, and the powers of thousand:

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System Myriad

Proposed by Donald E. Knuth, this system is another manner of generalizing the Greek myriads: with the place that each " order of grandeur" corresponds to a regrouping of four digits, for Archimedes, Knuth considers that each order of magnitude can have twice more figures than the precedent.

Beyond the names where the presence of the " is recognized; y" characteristic, it uses separators different for groups of 4,8,16,32 or 64 digits (respectively the comma, the semicolon, and the two points, space and the apostrophe ; the decimal separator remains the point in this notation). They are formed on powers of two successive powers of ten thousand (myriad). This system makes it possible to write and name enormous numbers (the first great number which cannot be expressed with the traditional denominations is the oktyllion, the thousand-twenty-fourth power of the myriad). However, the name " myriade" remain most known because it corresponds to a historical denomination.

However names are seldom used because they are often homonymous and homophons of other numbers (including in English where these names were defined), and create new ambiguities with the short and long scales.

The Googol system

The terms googol and googolplex were invented by Milton Sirotta, nephew of the mathematician Edward Kasner, who introduced them into a publication of 1940, Mathematics and the Imagination, where it describes this invention:

The " term; googol" was invented by a child, the nephew of Dr. Kasner, then eight years old. One had asked him to imagine a name for a very large number, for example one 1 follow-up of a hundred of zeros. It was sure that this number was not infinite, and quite as certain that it did not have a proper name. It suggested the " term; googol" , and in the tread another for a number even larger proposed some: the " googolplex". A googolplex is much larger than a googol, but remains finished, which the inventor of the term quickly pointed out. At the beginning, the definition suggested one 1, was followed of as much of zero which one could write without falling from tiredness. It is certainly what would be likely to arrive if somebody tries to write a googolplex, but two different people would be tired at the end of a different time, and that would not have a direction that Carnera is a better mathematician than Einstein simply because it has a better endurance. For this reason, the googleplex is a specific number, but with so much of zeros behind its " un" that the number of zeros is itself of a googol.

Thereafter, Conway and Guy suggested as extension that a N-plex corresponds by convention to 10^N. With this system, a googol-plex is worth 10^googol well, and a googolplexplex is worth 10^googolplex.

Other authors proposed the forms googolduplex , googoltriplex , etc, to indicate 10^googolplex, 10^googolduplex respectively, and so on.

See too

Order of scientific magnitude
Notation
Prefix of the international system
Scales long and short

Numbers in French
List of the numbers
List of the great numbers

External bonds

zillions Them according to Conway, Wechsler… and Miakinen (Miakinen).
Broad Numbers article by Robert Munafo
'' How high edge you count? '' by Landon Curt Noll .
Full list off broad number names list sorted by 10ⁿ and by Word length

Refer

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