## Decoded

Num from English |
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## Encoded

Num to English |
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### About English numerals

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 10^{6N+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 Scale | Long Scale (Chuquet) | Long Scale (Peletier) | ||||
---|---|---|---|---|---|---|

Digit | N (10^{3N+3}) | Name | N (10^{6N}) | Name | N (10^{6N}) | Name |

10^{3} | 0 | Thousand | 0.5 | Thousand | 0.5 | Thousand |

10^{6} | 1 | Million | 1 | Million | 1 | Million |

10^{9} | 2 | Billion | 1.5 | Thousand Million | 1.5 | Milliard |

10^{12} | 3 | Trillion | 2 | Billion | 2 | Billion |

10^{15} | 4 | Quadrillion | 2.5 | Thousand Billion | 2.5 | Billiard |

10^{18} | 5 | Quintillion | 3 | Trillion | 3 | Trillion |

10^{21} | 6 | Sextillion | 3.5 | Thousand Trillion | 3.5 | Trilliard |

10^{24} | 7 | Septillion | 4 | Quadrillion | 4 | Quadrillion |

10^{27} | 8 | Octillion | 4.5 | Thousand Quadrillion | 4.5 | Quadrilliard |

10^{30} | 9 | Nonillion | 5 | Quintillion | 5 | Quintillion |

10^{33} | 10 | Decillion | 5.5 | Thousand Quintillion | 5.5 | Quintilliard |

10^{36} | 11 | Undecillion | 6 | Sextillion | 6 | Sextillion |

10^{39} | 12 | Duodecillion | 6.5 | Thousand Sextillion | 6.5 | Sextilliard |

10^{42} | 13 | Tredecillion | 7 | Septillion | 7 | Septillion |

10^{45} | 14 | Quattuordecillion | 7.5 | Thousand Septillion | 7.5 | Septilliard |

10^{48} | 15 | Quindecillion | 8 | Octillion | 8 | Octillion |

10^{51} | 16 | Sexdecillion | 8.5 | Thousand Octillion | 8.5 | Octilliard |

10^{54} | 17 | Septendecillion | 9 | Nonillion | 9 | Nonillion |

10^{57} | 18 | Octodecillion | 9.5 | Thousand Nonillion | 9.5 | Nonilliard |

10^{60} | 19 | Novemdecillion | 10 | Decillion | 10 | Decillion |

10^{63} | 20 | Vigintillion | 10.5 | Thousand Decillion | 10.5 | Decilliard |

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 10^{12} and Reduced to 10^{9} 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 (10^{33} or more). The Conway-Wechsler system names digits according to the following rules:

Units | Tens | Hundreds | |
---|---|---|---|

1 | un | ^{(n)} deci | ^{(nx)} centi |

2 | duo | ^{(ms)} viginti | ^{(n)} ducenti |

3 | tre ^{(s(x))} | ^{(ns)} triginta | ^{(ns)} trecenti |

4 | quattuor | ^{(ns)} quadraginta | ^{(ns)} quadringenti |

5 | quin(qua) | ^{(ns)} quinquaginta | ^{(ns)} quingenti |

6 | se ^{(sx)} | ^{(n)} sexaginta | ^{(n)} sescenti |

7 | septe ^{(mn)} | ^{(n)} septuaginta | ^{(n)} septingenti |

8 | octo | ^{(mx)} octoginta | ^{(mx)} octingenti |

9 | nove ^{(mn)} | nonaginta | nongenti |

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 10^{3N+3} on the short scale and 10^{6N} on the long scale, and then use that N value to find the name in above table.

For example, 10^{96} is N=31 because it is 10^{3*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" = "Se**s**trigintillion".

"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" = "Tre**s**octogintillion".

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.

N | Conway-Wechsler system | English dictionary | Latin word |
---|---|---|---|

15 | Quinquadecillion | Quindecillion | 5 is "quinque", but 15 is more commonly "quindecim" than "quinquadecim". |

16 | Sedecillion | Sexdecillion | "Sedecim" is more common than "sexdecim". |

19 | Novendecillion | Novemdecillion | Normally 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).

Units | Tens | Hundreds | |
---|---|---|---|

1 | un | deci | ^{(s)} centi |

2 | duo | viginti | ducenti |

3 | tre ^{(s)} | triginta | trecenti |

4 | quattuor | quadraginta | quadringenti |

5 | quin | quinquaginta | quingenti |

6 | sex | sexaginta | sescenti |

7 | septen | septuaginta | septingenti |

8 | octo | octoginta | octingenti |

9 | novem | nonaginta | nongenti |

"Tre ^{(s(x))}", "se ^{(sx)}", "septe ^{(mn)}", and "nove ^{(mn)}" of the Conway-Wechsler system are fixed at "tre", "se**x**", "septe**n**", and "nove**m**" on the CW4EN system. Only when N=103 is "Tre**s**centillion" 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 "Tre**s**centillion" / "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.

N | Conway-Wechsler system | CW4EN system |
---|---|---|

16 | Sedecillion | Sexdecillion |

19 | Novendecillion | Novemdecillion |

23 | Tresvigintillion | Trevigintillion |

26 | Sesvigintillion | Sexvigintillion |

27 | Septemvigintillion | Septenvigintillion |

33 | Trestrigintillion | Tretrigintillion |

36 | Sestrigintillion | Sextrigintillion |

39 | Noventrigintillion | Novemtrigintillion |

43 | Tresquadragintillion | Trequadragintillion |

46 | Sesquadragintillion | Sexquadragintillion |

49 | Novenquadragintillion | Novemquadragintillion |

53 | Tresquinquagintillion | Trequinquagintillion |

56 | Sesquinquagintillion | Sexquinquagintillion |

59 | Novenquinquagintillion | Novemquinquagintillion |

66 | Sesexagintillion | Sexsexagintillion |

69 | Novensexagintillion | Novemsexagintillion |

76 | Seseptuagintillion | Sexseptuagintillion |

79 | Novenseptuagintillion | Novemseptuagintillion |

83 | Tresoctogintillion | Treoctogintillion |

87 | Septemoctogintillion | Septenoctogintillion |

96 | Senonagintillion | Sexnonagintillion |

97 | Septenonagintillion | Septennonagintillion |

99 | Novenonagintillion | Novemnonagintillion |

109 | Novencentillion | Novemcentillion |

206 | Seducentillion | Sexducentillion |

209 | Novenducentillion | Novemducentillion |

303 | Trestrecentillion | Tretrecentillion |

306 | Sestrecentillion | Sextrecentillion |

309 | Noventrecentillion | Novemtrecentillion |

403 | Tresquadringentillion | Trequadringentillion |

406 | Sesquadringentillion | Sexquadringentillion |

409 | Novenquadringentillion | Novemquadringentillion |

503 | Tresquingentillion | Trequingentillion |

506 | Sesquingentillion | Sexquingentillion |

509 | Novenquingentillion | Novemquingentillion |

606 | Sesescentillion | Sexsescentillion |

609 | Novensescentillion | Novemsescentillion |

706 | Seseptingentillion | Sexseptingentillion |

709 | Novenseptingentillion | Novemseptingentillion |

803 | Tresoctingentillion | Treoctingentillion |

807 | Septemoctingentillion | Septenoctingentillion |

906 | Senongentillion | Sexnongentillion |

907 | Septenongentillion | Septennongentillion |

909 | Novenongentillion | Novemnongentillion |