by David C. Wise
When I worked at Hughes Aircraft Company, Ground Systems Group (GSG), I was given an internal reference publication, An Editor's Guide to Metric Units and Acronyms & Other Letter Symbols at GSG (a few of which are for sale on Amazon at the time of this writing). Remember that this was a decade before public access to the Internet and a decade-and-a-half before Wikipedia, so reference books were very valuable and useful items that you would collect at every opportunity (I have a couple books of tables which still come in handy).
On page 2-12 of my copy (dated 1982, so it might be on a different page in other editions) there's a list of large number names with their values and metric prefixes. Of course you couldn't find names like jillions, zillions, gadzillions, or bajillions, because those simply don't exist; people just make them up because they don't know the large numbers' actual names that we've had since the 15th Century. Instead, the names are derived from the Latin names for numbers with a "-illion" suffix. Very systematic and logical (and described below).
OK, that list does include two made-up numbers, but only because Carl Sagan had just popularized them in the original Cosmos (1980, around the time that the Editor's Guide had been written by engineering types who tend to be professional nerds): the googol (10100) and the googolplex (10googol, also expressed as 1010100).
But then the page's footnote for the "Multiplication Factor" column taught me something I had never known before:The factors given here are for the U.S. system of naming magnitudes, which progresses upward in sets of 3 zeroes from one thousand to vigintillion. In Europe a totally different system is employed in which 109 is one milliard, 1012 is one billion, and the system continues with the same names as the U.S. system but progressing upward in sets of 6 zeroes, so that vigintillion is 10120.
That footnote planted in my mind the seed of curiosity, but I couldn't really act upon it at the time since I was working for a living, raising a family, and the Internet and Wikipedia had not arrived yet. It turns out that there have been over 20 numbering systems. The two systems used by most of the world are the subject of this page. They have a shared European history (primarily from France) which is complicated by some countries having switched from one to the other and even back to the first one again.
My page examines these events and supporting issues concluding with a fairly comprehensive table of large number names.
I would like to recommend this British YouTube video, though I cover a lot of its content below: How big is a billion? - Numberphile with Dr. James Grime, mathematician.
This is where I learned that the UK had switched numbering systems in 1974 over to the American system, something which British curmudgeons still complain about bitterly. And how the number names make enormous mathematical sense in the original system, but no longer in the American system (which we Americans had originally gotten from the French who then switched back later, but that's too much foreshadowing at this point).
Part of the fallout from that 1974 switch in the UK created a minor crisis in terminology. Up to that point, we in the USA would contrast the "American System" with the "British System", but then when the British System was suddenly the same as the American System, we had to come up with different names, such as "European System". I have no way to confirm that this was the cause of the next development, but in the following year, 1975, a French mathematician, Geneviève Guitel, presented the geographically neutral terminology of long and short scales.
Another resource is the Wikipedia article, Names of large numbers.
The basic system is fairly simple: construct a number name by prepending "-illion" with a Latin number name, adjusting spelling by reducing "ii" to "i", etc. The Latin number names are based on:
Units Tens Hundreds 1 Un Deci Centi 2 Duo Viginti Ducenti 3 Tre Triginta Trecenti 4 Quattuor Quadraginta Quadringenti 5 Quinqua Quinquaginta Quingenti 6 Se Sexaginta Sescenti 7 Septe Septuaginta Septingenti 8 Octo Octoginta Octingenti 9 Nove Nonaginta Nongenti
For an idea of the possible combinations, here is a list of number name examples with their Arabic numeral equivalents:
Assigning values to these names is fairly simple: to get the next higher number name, multiply the current number name's value by a fixed factor.
That is where we get to a minor complication and source of confusion which is the second theme of this page: the differences between the short scale (AKA "American") and the long scale (AKA "British" (though no longer), "European", "Continental"). The short scale uses a factor of one thousand (1,000) while the long scale uses a factor of one million (1,000,000). If you are working in powers of ten (or just adding zeros, with is equivalent), the multipying by 1,000 is adding three (3) to the exponent (ie, adding three zeros) and the multiplying of 1,000,000 is adding six (6) to the exponent (ie, adding six zeros).
To try to have that make more sense, here are some sequential number names showing their values in terms of powers of ten in both scales:
1 Million 106 106 2 Billion 109 1012 3 Trillion 1012 1018 4 Quadrillion 1015 1024 5 Quintillion 1018 1030 6 Sextillion 1021 1036 7 Septillion 1024 1042 8 Octillion 1027 1048 9 Nonillion 1030 1054 10 Decillion 1033 1060
If you want to find the value of a given number name, you do not need to construct the entire list leading up to that name. Instead, you could apply a simple formula given the prefix value of n for that name:You can verify these methods against the sequential list of names above.
- Long Scale
- Raise one million (106) to the power of n. That is the same as taking the power of ten to n multiplied by six (6).
For example:Nonillion, n = 9, 109×6 = 1054
Duotrigintillion, n = 32, 1032×6 = 10192
Sexoctingentillion, n = 806, 10806×6 = 104836
- Short Scale
- This is a bit less straight forward. The power of ten, x, is calculated with the formula: x = 6 + 3(n - 1).
For example:Nonillion, n = 9, x = 6 + 3(9-1) = 6 + 3(8) = 6 + 24 = 30, 10x=1030
Duotrigintillion, n = 32, x = 6 + 3(32-1) = 6 + 3(31) = 6 + 93 = 99, 10x=1099
Sexoctingentillion, n = 806, x = 6 + 3(806-1) = 6 + 3(805) = 6 + 2415 = 2421, 10x=102421
In that list you should have noticed that both scales use the exact same names just with different values. You should have also noticed that while all the values in the long scale are represented in the short scale (or rather you would see that if I had extended the list far enough), about half the values in the short scale appear to have no names in the long scale.
Actually, the long scale does have names for those values; they are just not common in English -- I first heard of "milliard" for the short-scale billion when learning German. While non-English languages lean towards the "-illiard" formulation, when the British were still using the long scale (ie, pre-1974). they would use the form, "thousand <previous number name>".
Value Short Scale Name Long Scale Name Old British Name 109 Billion Milliard Thousand million 1015 Quadrillion Billiard Thousand billion 1021 Sextillion Trilliard Thousand trillion 1027 Octillion Quadrilliard Thousand quadrillion 1033 Decillion Quintilliard Thousand quintillion
I've complied a more complete table at the bottom of this page.
Fortunately, the International Standard (SI) values of metric prefixes are not affected by which scale is being used; eg:
Value Short Scale Name Long Scale Name SI Symbol SI Prefix 106 Million Million M Mega- 109 Billion Milliard G Giga- 1012 Trillion Billion T Tera- 1015 Quadrillion Billiard P Peta- 1018 Quintillion Trillion E Exa- 1021 Sextillion Trilliard Z Zetta- 1024 Septillion Quadrillion Y Yotta-
There are about 20 native numbering systems in use in the world, but most of the world uses one or both variants of the European system, which are the subject of this page.
The first attempt at devising large number names appears to have been by 15th century French mathematician Jehan Adam who was secretary to Nicholle Tilhart, notary, secretary and auditor of accounts to King Louis XI of France. Jehan Adam published a manuscript in 1475 containing the first use of the terms "bymillion" and "trimillion", which gave rise to the modern terms billion and trillion. Their values were 1012 and 1018, respectively, which correspond to the long scale values for billion and trillion.
Jehan Adam's work appears to have been extended by French mathematician Nicolas Chuquet in his 1484 article, Triparty en la science des nombres, for which he is credited with creating the basis of our current system. However, that article was not published until it was rediscovered in 1880, so its contents were only known about through Estienne de La Roche's 1520 textbook, l'Arismetique, in which he copied Chuquet's system without attribution.
Chuquet's system was basically as I have described in the previous section. The names are based on prefixes constructed from Latin number names attached to the suffix, "-illion" and their values are the result of raising a million to the power of the value of the prefix (again, as described in the section above). Another less mathematical way to describe the system is that each sequential number name's value is one million times greater than the previous name's value.
Chuquet's system has been expanded upon since then. Jacques Pelletier du Mans (1517 – 1582) added names for the intermediate values; eg:
As a result, the long scale is also known as the Chuquet-Peletier system.
- Milliard = one thousand millions
- Billiard = one thousand billions
- Trilliard = one thousand trillions
- Quadrilliard = one thousand quadrillions
- Quintilliard = one thousand quintillions
In the 17th century, usage of large numbers caused the assignment of values to vary, such that billion came to mean 109 instead of the original 1012. That usage was brought to the American colonies where it is still used today.
By early in the 18th century, France had completely switched over the the short scale, calling the Chuquet-Peletier system the "now obsolete system". But then in 1948, France switched back to the long scale, making that move official in 1961. Conversely, the UK, which had been using the long scale, switched over to the short scale in 1974. Most other English-speaking countries followed suit shortly thereafter, though the official status of the short scale is not always clear.
All these changes created a minor naming crisis for us (ie, the USA). We had fallen into the habit of calling our system the "American System" and the other the "British System," or alternatively the "European System." But now that the UK had switched over to the "American System", we could no longer refer to the "British System", now could we? It almost feels like it was in response to that problem that yet another French mathematician, Geneviève Guitel, came up with a new naming convention of the long and short scales, a nomenclature which I have been using on this page.
The Wikipedia page on the long and short scales lists which countries use which. For the most part, usage follows language as well as colonial pasts, such that most countries use the system of the mother country. Even when the mother country has switched systems (eg, the UK, France), countries that use the same language tend to switch as well, likely because of ongoing communication and commerce with each other. There are also seven (7) countries using both systems because they straddle or encompass two cultures (eg, Canada and Puerto Rico). And due to the US dollar's traditional dominance in the financial world, the short scale was adopted for official United Nations documents.
Now the Internet makes it more important that we be clear about which scale we are using. When we read or write "billion", we can no longer assume that everybody will know what we mean. For that reason, on my other pages which use that number I try to make it clear that I'm using the short scale by explicitly giving the value of billion as 109.
Fortunately, the values of metric prefixes are not affected by which scale is being used. So "giga-" will always be 109, "nano-" always 10-9, "kilo-" always 103, and "peta-" always 1015.
Here is a more complete table which I copied from somewhere of large number names and their values.
Value US, Canada and modern British
Traditional European (Peletier)
1 1 106 Million Million Million 2 1 109 Billion Thousand million Milliard 3 2 1012 Trillion Billion Billion 4 2 1015 Quadrillion Thousand billion Billiard 5 3 1018 Quintillion Trillion Trillion 6 3 1021 Sextillion Thousand trillion Trilliard 7 4 1024 Septillion Quadrillion Quadrillion 8 4 1027 Octillion Thousand quadrillion Quadrilliard 9 5 1030 Nonillion Quintillion Quintillion 10 5 1033 Decillion Thousand quintillion Quintilliard 11 6 1036 Undecillion Sextillion Sextillion 12 6 1039 Duodecillion Thousand sextillion Sextilliard 13 7 1042 Tredecillion Septillion Septillion 14 7 1045 Quattuordecillion Thousand septillion Septilliard 15 8 1048 Quinquadecillion Octillion Octillion 16 8 1051 Sedecillion Thousand octillion Octilliard 17 9 1054 Septendecillion Nonillion Nonillion 18 9 1057 Octodecillion Thousand nonillion Nonilliard 19 10 1060 Novendecillion Decillion Decillion 20 10 1063 Vigintillion Thousand decillion Decilliard 21 11 1066 Unvigintillion Undecillion Undecillion 22 11 1069 Duovigintillion Thousand undecillion Undecilliard 23 12 1072 Tresvigintillion Duodecillion Duodecillion 24 12 1075 Quattuorvigintillion Thousand duodecillion Duodecilliard 25 13 1078 Quinquavigintillion Tredecillion Tredecillion 26 13 1081 Sesvigintillion Thousand tredecillion Tredecilliard 27 14 1084 Septemvigintillion Quattuordecillion Quattuordecillion 28 14 1087 Octovigintillion Thousand quattuordecillion Quattuordecilliard 29 15 1090 Novemvigintillion Quindecillion Quindecillion 30 15 1093 Trigintillion Thousand quindecillion Quindecilliard 31 16 1096 Untrigintillion Sedecillion Sedecillion 32 16 1099 Duotrigintillion Thousand sedecillion Sedecilliard 33 17 10102 Trestrigintillion Septendecillion Septendecillion 34 17 10105 Quattuortrigintillion Thousand septendecillion Septendecilliard 35 18 10108 Quinquatrigintillion Octodecillion Octodecillion 36 18 10111 Sestrigintillion Thousand octodecillion Octodecilliard 37 19 10114 Septentrigintillion Novendecillion Novendecillion 38 19 10117 Octotrigintillion Thousand novendecillion Novendecilliard 39 20 10120 Noventrigintillion Vigintillion Vigintillion 40 20 10123 Quadragintillion Thousand vigintillion Vigintilliard 50 25 10153 Quinquagintillion Thousand quinquavigintillion Quinquavigintilliard 60 30 10183 Sexagintillion Thousand trigintillion Trigintilliard 70 35 10213 Septuagintillion Thousand quinquatrigintillion Quinquatrigintilliard 80 40 10243 Octogintillion Thousand quadragintillion Quadragintilliard 90 45 10273 Nonagintillion Thousand quinquaquadragintillion Quinquaquadragintilliard 100 50 10303 Centillion Thousand quinquagintillion Quinquagintilliard 101 51 10306 Uncentillion Unquinquagintillion Unquinquagintillion 102 51 10309 Duocentillion Thousand unquinquagintillion Unquinquagintilliard 103 52 10312 Trescentillion Duoquinquagintillion Duoquinquagintillion 110 55 10333 Decicentillion Thousand quinquaquinquagintillion Quinquaquinquagintilliard 111 56 10336 Undecicentillion Sesquinquagintillion Sesquinquagintillion 120 60 10363 Viginticentillion Thousand sexagintillion Sexagintilliard 121 61 10366 Unviginticentillion Unsexagintillion Unsexagintillion 130 65 10393 Trigintacentillion Thousand quinquasexagintillion Quinquasexagintilliard 140 70 10423 Quadragintacentillion Thousand septuagintillion Septuagintilliard 150 75 10453 Quinquagintacentillion Thousand quinquaseptuagintillion Quinquaseptuagintilliard 160 80 10483 Sexagintacentillion Thousand octogintillion Octogintilliard 170 85 10513 Septuagintacentillion Thousand quinquaoctogintillion Quinquaoctogintilliard 180 90 10543 Octogintacentillion Thousand nonagintillion Nonagintilliard 190 95 10573 Nonagintacentillion Thousand quinquanonagintillion Quinquanonagintilliard 200 100 10603 Ducentillion Thousand centillion Centilliard 300 150 10903 Trecentillion Thousand quinquagintacentillion Quinquagintacentilliard 400 200 101203 Quadringentillion Thousand ducentillion Ducentilliard 500 250 101503 Quingentillion Thousand quinquagintaducentillion Quinquagintaducentilliard 600 300 101803 Sescentillion Thousand trecentillion Trecentilliard 700 350 102103 Septingentillion Thousand quinquagintatrecentillion Quinquagintatrecentilliard 800 400 102403 Octingentillion Thousand quadringentillion Quadringentilliard 900 450 102703 Nongentillion Thousand quinquagintaquadringentillion Quinquagintaquadringentilliard 1000 500 103003 Millinillion Thousand quingentillion Quingentilliard
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First uploaded on 2019 July 08.
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