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## Encoded

Num to English Decimal notation Numeral words Fraction System CW4EN Conway-Wechsler

English numerals represent numbers in English.

For example, the result of converting 123456789 to an English numeral is as follows.

`One Hundred Twenty-Three Million Four Hundred Fifty-Six Thousand Seven Hundred Eighty-Nine`

Values ​​after the decimal point can be expressed in word or fractional digits. For example, 0.99 is represented as "Zero point Nine Nine" in word and "Zero and 99/100" in fractions.

#### Large Numbers

For larger numbers, the digits are short scale or long scale and are expressed as follows: The name of the digit changes by 3 digits on the short scale and by 6 digits on the long scale. The long scale also has the Chuquet system, which represents digits of 106N+3 as "Thousand -illion", and the Peletier system, which represents "-illiard".

Short scale is mainly used in English-speaking countries such as the United States, Canada, and the United Kingdom (since 1974). In addition, the long scale Chuquet system was used in the United Kingdom before 1973, and the Peletier system is used in each language-specific notation in Europe, which is mainly non-English speaking countries such as France, Germany, and Italy.

DenCode uses the short scale that is common in modern English-speaking countries.

Short ScaleLong Scale (Chuquet)Long Scale (Peletier)
DigitN (103N+3)NameN (106N)NameN (106N)Name
1030Thousand0.5Thousand0.5Thousand
1061Million1Million1Million
1092Billion1.5Thousand Million1.5Milliard
10123Trillion2Billion2Billion
10185Quintillion3Trillion3Trillion
10216Sextillion3.5Thousand Trillion3.5Trilliard
10309Nonillion5Quintillion5Quintillion
103310Decillion5.5Thousand Quintillion5.5Quintilliard
103611Undecillion6Sextillion6Sextillion
103912Duodecillion6.5Thousand Sextillion6.5Sextilliard
104213Tredecillion7Septillion7Septillion
104514Quattuordecillion7.5Thousand Septillion7.5Septilliard
104815Quindecillion8Octillion8Octillion
105116Sexdecillion8.5Thousand Octillion8.5Octilliard
105417Septendecillion9Nonillion9Nonillion
105718Octodecillion9.5Thousand Nonillion9.5Nonilliard
106019Novemdecillion10Decillion10Decillion
106320Vigintillion10.5Thousand Decillion10.5Decilliard

The above digit names are common in current English dictionaries.

The name of the long scale Chuquet and Pelletier system was derived from French mathematician Nicolas Chuquet in 1484 defined until the "Nonillion" of N=9 (Byllion, Tryllion, Quadrillion, Quyllion, Sixlion, Septyllion, Ottyllion, and Nonyllion, in French), and Jacques Peletier du Mans spread "Milliard" (Milliart) in 1549 ("Milliard" was popularized in the sense of 1012 and Reduced to 109 later in the late 17th century).

#### Conway-Wechsler system

The Conway-Wechsler system defined by John Horton Conway and Allan Wechsler is a typical naming method for large digits with N=10 or more (1033 or more). The Conway-Wechsler system names digits according to the following rules:

Conway-Wechsler system
UnitsTensHundreds
1un(n) deci(nx) centi
2duo(ms) viginti(n) ducenti
3tre (s(x))(ns) triginta(ns) trecenti
5quin(qua)(ns) quinquaginta(ns) quingenti
6se (sx)(n) sexaginta(n) sescenti
7septe (mn)(n) septuaginta(n) septingenti
8octo(mx) octoginta(mx) octingenti
9nove (mn)nonagintanongenti

The Conway-Wechsler system was defined for short scales, but it can also be used for long scales. To get the digit name with this system, find the N of 103N+3 on the short scale and 106N on the long scale, and then use that N value to find the name in above table.

For example, 1096 is N=31 because it is 103*31+3 on the short scale, and it is combined from the lowest digit to the highest digit of N like "duo"(1) + "triginta"(30) + " illion" = "Duotrigintillion". If there is a vowel "aeiou" immediately before "illion", remove a vowel and combine them.

Also, when combining Units with Tens or Hundreds, the characters (mnsx) in parentheses in the above table are combined, including the characters if they match. This is called an assimilation rule. For example, for N=26, "se　(sx)"(6) + "(ns) triginta"(20) + "illion" = "Sestrigintillion".

"tre (s(x))"(3) is special, and "s" is added regardless of which character follows (sx). For example, for N=83, It will be "tre (s(x))"(3) + "(mx) octoginta"(80) + "illion" = "Tresoctogintillion".

For more larger numbers with N=1,000 and above, combine N every 3 digits with the above procedure to derive the name and finally join. If N=1,000,000X + 1,000Y + Z and the name of each digit is "Xillion", "Yillion", "Zillion", they are combined like "Xilliyillizillion" and omit "on" of "-illion" in the middle. For example, if N=1,003, then "Million"(1) + "Trillion"(3) = "Millitrillion". Also, when N=12,210, "Duodecillion"(12) + "Deciducentillion"(210) = "Duodecillideciducentillion".

Also, if the 3-digit value is 0, it will be "Nillion". For example, if N=1,000,003, then "Million"(1) + "Nillion"(0) + "Trillion"(3) = "Millinillitrillion".

The Conway-Wechsler system is basically Latin-based, so for example the following names may differ from the names defined in the English dictionary.

NConway-Wechsler systemEnglish dictionaryLatin word
16SedecillionSexdecillion"Sedecim" is more common than "sexdecim".
19NovendecillionNovemdecillionNormally it is "undeviginti", but it may be written as "novendecim" or "novemdecim". A similar assimilation rule, N=17, is more commonly "septendecim" than "septemdecim".

The "quinqua" for 5 is "quinque" in Latin, but 15 is "quindecim" in Latin and "quindecillion" in English. Therefore, only "quinqua" in the Conway-Wechsler system may be replaced with "quin". This replacement was presented by Olivier Miakinen (See：Olivier Miakinen. Les zillions selon Conway, Wechsler... et Miakinen, 2003 (French web page)). DenCode also uses "quin", which is closer to the name of the English dictionary.

#### CW4EN system

DenCode supports the above Conway-Wechsler system, but it defines its own system that is more in line with the English dictionary and uses it as the default conversion system. For convenience, we will refer to it as "CW4EN system" (Conway-Wechsler for English system).

CW4EN system
UnitsTensHundreds
1undeci(s) centi
2duovigintiducenti
3tre (s)trigintatrecenti
5quinquinquagintaquingenti
6sexsexagintasescenti
7septenseptuagintaseptingenti
8octooctogintaoctingenti
9novemnonagintanongenti

"Tre (s(x))", "se (sx)", "septe (mn)", and "nove (mn)" of the Conway-Wechsler system are fixed at "tre", "sex", "septen", and "novem" on the CW4EN system. Only when N=103 is "Trescentillion" instead of "Trecentillion". This is to avoid duplication with "Trecentillion" with N=300.

There are examples of adopting systems similar to the CW4EN system, but they do not take into account or mention the difference between "Trescentillion" / "Trecentillion". (e.g. Glossary of Stock Market Terms & Definitions | Nasdaq)

The following is a list of typical names that differ between the Conway-Wechsler system and the CW4EN system.

NConway-Wechsler systemCW4EN system
16SedecillionSexdecillion
19NovendecillionNovemdecillion
23TresvigintillionTrevigintillion
26SesvigintillionSexvigintillion
27SeptemvigintillionSeptenvigintillion
33TrestrigintillionTretrigintillion
36SestrigintillionSextrigintillion
39NoventrigintillionNovemtrigintillion
53TresquinquagintillionTrequinquagintillion
56SesquinquagintillionSexquinquagintillion
59NovenquinquagintillionNovemquinquagintillion
66SesexagintillionSexsexagintillion
69NovensexagintillionNovemsexagintillion
76SeseptuagintillionSexseptuagintillion
79NovenseptuagintillionNovemseptuagintillion
83TresoctogintillionTreoctogintillion
87SeptemoctogintillionSeptenoctogintillion
96SenonagintillionSexnonagintillion
97SeptenonagintillionSeptennonagintillion
99NovenonagintillionNovemnonagintillion
109NovencentillionNovemcentillion
206SeducentillionSexducentillion
209NovenducentillionNovemducentillion
303TrestrecentillionTretrecentillion
306SestrecentillionSextrecentillion
309NoventrecentillionNovemtrecentillion