Names for large numbers

Extending the million, billion, trillion system to much, much bigger numbers National Debt Clock

What’s bigger than a trillion? Bigger than a quadrillion? Can we give names for large numbers which are snappier than $2^{74,207,281}−1$? As we have found the need to use large numbers in our lives, various interesting systems have been proposed. Impress your friends with some of these!

Extending million, billion, trillion…

An extension to ‘million, billion, trillion’ was proposed in John Conway and Richard Guy’s book, The Book of Numbers (not to be confused with the slightly older book sharing its name).

With the English language finally having settled on the traditionally American designation that 1 billion = 1 thousand million (thanks finance!), Conway & Guy extended the system that already exists up to $10^{30}$. They take the number $n$ occurring in $10^{3n+3}$:

 $10^6$ million $n=1$ $10^{21}$ sextillion $n=6$ $10^9$ billion $n=2$ $10^{24}$ septillion $n=7$ $10^{12}$ trillion $n=3$ $10^{27}$ octillion $n=8$ $10^{15}$ quadrillion $n=4$ $10^{30}$ nonillion $n=9$ $10^{18}$ quintillion $n=5$ $10^{33}$ decillion $n=10$

and use its Latin translation as a prefix for $n \geq 10$. So we get

• $n = 11$: undecillion
• $n = 18$: octodecillion
• $n = 25$: quinvigintillion

In case your Latin needs a refresher, here’s the table to create the prefix, as refined by Olivier Miakinen. It works left-to-right, so for your $n$, you do the units first, then the tens, then the hundreds.

Units Tens Hundreds
0
1 un (n) deci (nx) centi
2 duo (ms) viginti (n) ducenti
3 tre (s) (ns) triginta (ns) trecenti
5 quin (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
When placed before a component marked (s) or (x), “tre” becomes “tres” and “se” becomes “ses” or “sex”. Similarly, placed before a component marked (m) or (n), “septe” and “nove” become “septem” and “novem” or “septen” and “noven”.

Once you’ve combined these parts together in units–tens–hundreds order, remove the final vowel, and add -illion. Easy! Let’s go and name some real-life big numbers.

Names for real-life big numbers

• Old Zimbabwean dollars in new Zimbabwean dollars

$2 \times 10^{35} = 200 \times 10^{3\times\mathbf{10} + 3} =$ 200 decillion

Large numbers are often seen in terms of money. But how large? Estimates suggest the total amount of money in the world is around \$1 quadrillion: this is only$\$10^{15}$.

However, we can find some larger numbers due to hyperinflation! The Zimbabwean dollar started life as the Rhodesian dollar in 1970, at the rate of 10 shillings (half a pound) = \$1, in the same style as other Commonwealth nations converting from pounds, shillings and pence. Accelerated erosion of the currency in the 2000s lead to three revaluations of the currency (knocking noughts off), with the final dollar being worth$2 \times 10^{35}$of the old dollar. That’s a pretty big number, but of course, the process of knocking noughts off meant that largest number ever seen on a banknote was \$100 trillion. These days, Zimbabweans use a mixture of other currencies.

• Number of atoms in the universe

$10^{80} = 100 \times 10^{3\times\mathbf{25} + 3} =$ 100 quinvigintillion

Surely that must be the physical limit of large numbers?

• A googol

$10^{100} = 10\times10^{3\times\mathbf{32} + 3} =$ 10 duotrigintillion

• One hundred factorial

$100! \approx 9.3 \times 10^{157} = 93 \times 10^{3\times\mathbf{51} + 3} =$ 93 unquinquagintillion

The factorial function grows incredibly quickly. Classic maths challenge question: how many zeros are on the end of this number?

• The largest known prime

$2^{74,207,281}−1 \approx 3.004… × 10^{22,338,617} \approx 300 \times 10^{3\times\mathbf{7,446,204} + 3} =$ ???

Fret not! The system expands further. Run the naming system for each group of thousands and put an illi between them. So,

$2^{74,207,281}−1 \approx 3.004… × 10^{22,338,617} \approx 300 \times 10^{3\times\mathbf{7,446,204} + 3} =$ 300 septillisesquadragintaquadringentilliquattuorducentillion!

• The number of years you’d have to wait for the universe to regenerate itself in a similar state to now, if you let the universe repeat its history arbitrarily many times: $10^{10^{10^{10^{10^{1.1}}}}}$

OK, what? The big bang theory gives us an age for the universe of 13.8 billion years: that’s $4.355 \times 10^{17}$ seconds. Still not bigger than the number of atoms in the universe, though. But in order to get that incredibly large number—in fact, probably the largest number you’ve ever seen—we need to get a bit physical. Read about it here. This number even dwarves a googolplex, the largest number to have a name.

The naming of this number is left as an exercise to the reader.

Where do we get the names million, billion, trillion from?

The Latin mille for 1000—from which we get millennium (1000 years) and mile (1000 paces)—is thought to have come from pre-13th century Italian. As the Romans had no names for numbers larger than 100,000, the Italians added the ending -one to mille to make it larger: milione.

The words bymillion and trimillion were first recorded in 1475. Subsequently, the French mathematician and chief notation-creator Nicolas Chuquet wrote in his book, Triparty en la science des nombres,

The first mark can signify million, the second mark byllion, the third mark tryllion, the fourth quadrillion, the fifth quyillion, the sixth sixlion, the seventh septyllion, the eighth ottyllion, the ninth nonyllion and so on with others as far as you wish to go.

Chuquet records the usage here as using the ‘long scale‘: where a billion equals a million million, instead of a thousand million. This usage was common in British English up until the mid-1970s, when pressure from American English (‘a billion dollars’) tipped the balance. Most European languages still use the long scale: ironic, given that the short scale was adapted in the US from a 17th century French convention.

Competing systems for large number names

The Conway system for naming large numbers expands the current system in a logical way, but suffers from the possibility of long scale/short scale confusion, as well as inconsistency between $n<10$ and $n\geq10$. Russ Rowlett’s 2001 suggestion is to use Greek prefixes to create new, unambiguous numbers:

 $10^3$ thousand $10^6$ million $10^9$ gillion $10^{12}$ tetrillion

The Conway system also suffers from a bad starting point, which makes it difficult to work out from the naming system what, say, 1 million × 1 billion is. (It’s a quadrillion). Donald Knuth (of TeX fame) invented an exponential system of ‘-yllions’ where a new name is only introduced at $10^2, 10^4, 10^8, 10^{16}$ and so on.

 $10^1$ ten $10^2$ hundred $10^4$ myriad $10^8$ myllion $10^{16}$ byllion

So $10^3 =$ ten hundred and $10^6 =$ a hundred myriad

Names for large numbers only catch on when they appear in our daily lives. In measurement they’re avoided by the use of SI prefixes. Barring any hyperinflation, the highest we’d expect to see for a while is the world wealth of \$1 quadrillion. For everything else, we can be pretty happy with standard form. For more large number fun:

• Robert Munafo has a terrific (but long) read on names for large numbers on his website,
• and see how far Louis Epstein has extended the long-scale system on his website.

Top picture credit: Benoît Prieur (Agamitsudo) – CC-BY-SA. Images of the banknote, Mersenne and Chuquet’s millions are public domain. Adam is an assistant professor at Durham University, where he investigates weird, non-Newtonian fluids. If he’s not talking about the maths of chocolate fountains he is probably thinking about fonts, helping Professor Dirichlet answer your personal problems, and/or listening to BBC Radio 2.
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