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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
10³	0	Thousand	0.5	Thousand	0.5	Thousand
10⁶	1	Million	1	Million	1	Million
10⁹	2	Billion	1.5	Thousand Million	1.5	Milliard
10¹²	3	Trillion	2	Billion	2	Billion
10¹⁵	4	Quadrillion	2.5	Thousand Billion	2.5	Billiard
10¹⁸	5	Quintillion	3	Trillion	3	Trillion
10²¹	6	Sextillion	3.5	Thousand Trillion	3.5	Trilliard
10²⁴	7	Septillion	4	Quadrillion	4	Quadrillion
10²⁷	8	Octillion	4.5	Thousand Quadrillion	4.5	Quadrilliard
10³⁰	9	Nonillion	5	Quintillion	5	Quintillion
10³³	10	Decillion	5.5	Thousand Quintillion	5.5	Quintilliard
10³⁶	11	Undecillion	6	Sextillion	6	Sextillion
10³⁹	12	Duodecillion	6.5	Thousand Sextillion	6.5	Sextilliard
10⁴²	13	Tredecillion	7	Septillion	7	Septillion
10⁴⁵	14	Quattuordecillion	7.5	Thousand Septillion	7.5	Septilliard
10⁴⁸	15	Quindecillion	8	Octillion	8	Octillion
10⁵¹	16	Sexdecillion	8.5	Thousand Octillion	8.5	Octilliard
10⁵⁴	17	Septendecillion	9	Nonillion	9	Nonillion
10⁵⁷	18	Octodecillion	9.5	Thousand Nonillion	9.5	Nonilliard
10⁶⁰	19	Novemdecillion	10	Decillion	10	Decillion
10⁶³	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 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
	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, 10⁹⁶ 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" = "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.

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).

CW4EN 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", "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.

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