Number Names

by David C. Wise

If you are impatient and just want to look at the big number names, then you could jump directly to Number Names at the bottom of this page.

But if you were to do that, then you would miss the chance to learn a few things, like what the long and short scales are and why they matter -- at the very least, you would be able to understand the columns of that table of number names at the bottom of this page.

Introduction

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 15^th 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 (10^googol, 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.

Naming Theory

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:

2. Billion 3. Trillion 4. Quadrillion 5. Quintillion 6. Sextillion 7. Septillion 8. Octillion 9. Nonillion 10. Decillion

11. Undecillion 12. Duodecillion 20. Vigintillion 32. Duotrigintillion 33. Trestrigintillion 40. Quadragintillion 50. Quinquagintillion 60. Sexagintillion 70. Septuagintillion

80. Octogintillion 86. Sexoctogintillion 90. Nonagintillion 100. Centillion 102. Duocentillion 121. Primo-vigesimo-centillion 133. Trestrigintacentillion 200. Ducentillion 207. Septenducentillion

300. Trecentillion 400. Quadringentillion 500. Quingentillion 600. Sescentillion 700. Septingentillion 800. Octingentillion 806. Sexoctingentillion 900. Nongentillion 1000. Millillion

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:

Prefix Value	Name	Short Scale	Long Scale
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:

• 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, 10^9×6 = 10⁵⁴
  Duotrigintillion, n = 32, 10^32×6 = 10¹⁹²
  Sexoctingentillion, n = 806, 10^806×6 = 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, 10^x=10³⁰
  Duotrigintillion, n = 32, x = 6 + 3(32-1) = 6 + 3(31) = 6 + 93 = 99, 10^x=10⁹⁹
  Sexoctingentillion, n = 806, x = 6 + 3(806-1) = 6 + 3(805) = 6 + 2415 = 2421, 10^x=10²⁴²¹

You can verify these methods against the sequential list of names above.

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>".

For example:

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-

 

History

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:

• Milliard = one thousand millions
• Billiard = one thousand billions
• Trilliard = one thousand trillions
• Quadrilliard = one thousand quadrillions
• Quintilliard = one thousand quintillions

As a result, the long scale is also known as the Chuquet-Peletier system.

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.

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First uploaded on 2019 July 08.
Updated on 2020 January 02.