1.3 - The -illions

Go back to home page, or chapter 1.

Introduction

In this section, we will cover some of the -illions that we encountered in real life, plus some other generalizations of it, to extend the -illions to the limits.

To begin the section, we need to understand what is the meaning of the -illion.

"-illion" is the suffix which represents the powers of 1000 (which itself, the powers of 10), that is popular in most of the countries around the world. It is divided into two parts, short scale and long scale.

Short scale is widely used in English-speaking countries, while long scale is used in other languages.

"n-illion" is defined as 10^(3n+3) in short scale, and 10^6n in long scale, which means, short scale is also equal to 1000^(n+1), while long scale is equal to 1000000^n.

"n-illiard" is equal to 1000000^(n+0.5) = 1000^(2n+1) or 2n-illion in short scale.

The Basic -illions

The first -illions

Pre number 1. Thousand (1,000)

Even though we don't start with an -illion, but we just added thousand for comparison purposes on how big the -illions are in upcoming entries. SI prefix kilo- denotes a thousand times.



This number is definitely considered a milestone if anyone reaches this number by counting or viewing it. It serves as a boundary between class 1 and class 2. (for more info: check out Robert Munafo's page about details on classes)

There are multiple uses of "million" in real life:


The billion is the next milestone after million. It is also equal to a milliard in long scale. It is the 2nd -illion or 3rd power of 1,000. There are multiple examples of this number:


It's probably the largest -illion everyone knows, since most of them don't know what comes after trillion, some of them attempted to nest it by saying "thousand trillion", "million trillion", "trillion trillion", "trillion trillion trillion", etc., but then it will become more cumbersome to say. So, thankfully, there is a solution: quadrillion, (explanation in the next entry)

The SI prefix for trillion is tera- and the symbol is T.

The examples of trillion:


Beyond the first -illions

Here is the first -illion that is started to be less popular, but it is still known. In fact, the official English dictionary is listed until vigintillion (entry coming soon), which means there are 20 -illions (without taking into account of -illiard which is only used in long scale, but when taking that into account, it will have 40 -illions and -illiards combined).

The SI prefix is peta-, (symbol is P, not p, which is for pico-, which is for 10^-12)

The examples of quadrillion:


This -illion starts to become a point where the normal users couldn't figure how massive it is. (note: this -illion is the largest -illion that I knew until 2011, when I started to learn more about large numbers in that year, discovered larger -illions including centillion.) The reason it is rarely used: the numbers are started to get even BIGGER than before.

Moving on, the SI prefix of this number is exa- or E.

The uses of quintillion is still there, but it's getting less notable than before.


This number is started to be a hard level to imagine, as this number is ginormous, but it's actually 1,000 times a quintillion. During my childhood days, I thought that the number comes after quintillion is "a thousand quintillion", until I discovered this -illion along with some other large -illions until millinillion (millillion) (entry in the next tier).

The SI prefix of this number is zetta- or the symbol is Z.


It was the largest -illion which has an official SI prefix (until November 2022), which is yotta-, (symbol: Y), but some of the persons managed to extend the prefixes until 10^300, or even 10^3000. Yotta- was defined back in 1991.

The examples are far too uncommon, but one of them is that the mass of the Earth is 5.98 septillion kilograms, the diameter of the observable universe is approximately 900 septillion meters.


(until November 2022) It was the first power of 10, or more specifically, power of 1000, to not have any official SI prefix, but most of the people extend it by having the prefixes "xenna-", "bronto-", etc. The xenna- part is due to the fact that the previous prefixes have their letters decremented, so, Y, X, W,... up until A, and then wraps around with Z, combined with enna- which means 9 in Greek.

Some of the examples within this magnitude of -illion: Mass of Jupiter (in kg), mass of small stars (excluding white dwarf, or neutron star, which has higher mass than our Sun), number of atoms in our human body.

Update 2023-02-21: the SI prefix for 10^27 has been officially named "ronna-" in a CGPM resolution. The prefix breaks the trend of having the letters decremented. The symbol is R.


It is the 10th power of 1000, or 5th power of a million. The -illions are started to get repetitive, but worry not, things are started to get better. The solar mass in kg is within this magnitude of -illion (1.989 nonillion kg). The hottest temperature recorded is approximately 141 nonillion Kelvin (aka Planck temperature).

Update 2023-02-21: 10^30 is the largest -illion which has an official SI prefix thanks to a CGPM resolution. It is named "quetta-" (symbol: Q).


The -illions are the most well known large ones. It is the 10th -illion, or the 11th power of 1000.

Goby is within this magnitude of -illion.

Since November 2022, it is the smallest power of 1000 that does not receive any official SI prefix.

Note: It looks like: you think that it's just a small step above decillion, but it is actually 50-50. It is still 1,000 times larger than a decillion.

The largest known Double Mersenne prime (2^(2^7-1)-1) is within this magnitude of -illion, and also 2^128 lands here.

Jonathan Bowers coined "doedecillion" for this number.

As some of the -illions are started to get less and less popular, I might as well skip the description of each -illions after duodecillion:


Finally, we reached the second largest official -illion recognized by a dictionary (the largest is centillion), but there are extensions to that (none of these are in the dictionary)

It is the largest known -illion recognized in the official dictionary, in fact, it is equal to 1000 times the cube of a googol.

Extending the -illions

Conway-Wechsler System

This system is actually based on Chuquet's system. Note that this system is extensively used and researched, and some of the known mathematicians proposed to standardize the usage of -illion.

Unfortunately, there are some gaps between vigintillion and centillion, and beyond.

So, it was developed by John Horton Conway and Allen Wechsler, as the name above suggests. The system uses short scale by default, (so billion is 1,000,000,000, instead of 10^12), but it can also be used in long scale as well.

(Unfortunately, since the new sites doesn't give an option to create a table, so I needed to find a way of the table (from https://sites.google.com/site/largenumbers/home/2-4/2-4-6-conway-guys-latin-based-illions))


Note: the "table" below has been modified to be compatible with Bowers' illions as well.


Units                                     Tens                                       Hundreds

1                    un                                                 (n)deci                                      (nx)centi

2                   duo                                          (msx)viginti                               (n)ducenti

3               tre(*)                                           (mnsx)trigint(ai)                             (ns)trecenti

4              quattuor                                    (mnsx)quadragint(ai)                    (ns)quadringenti

5              quinqua (**)                             (mnsx)quinquagint(ai)                  (ns)quingenti

6              se(sx)                                        (mnx)sexagint(ai)                               (n)sescenti

7              septe(mn)                                 (mnx)septuagint(ai)                       (n)septingenti

8             octo                                         (mx)octogint(ai)                         (mx)octingenti

9                nove(mn)                                   nonagint(ai)                                (m)nongenti

Here are some rules for the -illions:

For example, for the 869th -illion, we can combine nove(mn) + (mnx)sexaginta / sexaginti + octingentillion, it will become novensexagintioctingentillion.

So, the limit of the normal -illion is novenonagintanongentillion, which is the 999th -illion (10^3000 in short scale, 10^5994 in long scale).

This system can be extended, however. After the 999th -illion, Conway and Guy suggested "millinillion" for the 1000th -illion.

So, the next -illion after millinillion is millimillion. After that, it is millibillion, millitrillion, and so on. Practically, x-illi-y-illion is the (1000x+y)th illion.

I will list the -illions below:

10^66 = unvigintillion

10^69 = duovigintillion

10^72 = tresvigintillion

10^75 = quattuorvigintillion

10^78 = quinvigintillion

10^81 = sesvigintillion / sexvigintillion

10^84 = septenvigintillion

10^87 = octovigintillion

10^90 = novemvigintillion

10^93 = trigintillion

10^100 = googol (10 duotrigintillion)

10^123 = quadragintillion

10^153 = quinquagintillion

10^183 = sexagintillion

10^213 = septuagintillion

10^243 = octogintillion

10^273 = nonagintillion

10^303 = centillion

10^309 = duocentillion

10^333 = centidecillion

10^500 = googolding (100 quinsexagintacentillion)

10^603 = ducentillion

10^903 = trecentillion

10^1203 = quadringentillion

10^1503 = quingentillion

10^1803 = sescentillion

10^2103 = septingentillion

10^2403 = octingentillion

10^2703 = nongentillion

10^3003 = millinillion

10^6003 = billinillion

10^9003 = trillinillion

10^12,003 = quadrillinillion

10^15,003 = quintillinillion

10^18,003 = sextillinillion

10^21,003 = septillinillion

10^24,003 = octillinillion

10^27,003 = nonillinillion

10^30,003 = decillinillion

10^60,003 = vigintillinillion

10^90,003 = trigintillinillion

10^300,003 = centillinillion

10^3,000,003 = millinillinillion

and so on...

(UNDER CONSTRUCTION)


Rowlett's -illions

This -illion is based off of the Greek numerals. This name of -illions suggested in 2001 to replace the Latin-based -illions by Russ Rowlett, which sometimes can be ambiguous.

The system started with 1,000,000,000, and it's applicable for higher types of powers of 1000 (the index must be a non-negative integer). Note that all of these are in short scale.

So, the first -illion that used this system is named "gillion". The fact that it is derived from the SI prefix "giga" in order to prevent ambiguity because, logically speaking, triaillion, which sounds similar to trillion, when equal to 1,000,000,000, can make confusions even worse, as trillion is already used for 10^12 in short scale.

After "gillion", he proceeded to add a Greek prefix to the -illions (or powers of 1000). For example, tetrillion, pentillion, etc.

Below are the names used in Rowlett's illions (up to 10):

The numbers here are based on powers of 1000:

For 11 to 19, just append "hen", "duo", "tria", "tetra", "penta", "hexa", "hepta", "okto", "ennea" to dekillion.

Note that for larger -illions, most of the names are forked from Greek numerals, which Bowers also used for Tier 2 -illions in his system (which is much larger) (marked with *), we will see it later.

According to Saibian, whom wrote the article, provided intermediate -illions based on the table given in the page (sorry for being messy)

To generate the -illions between 1-999, here are tables for ones, tens and hundreds. (courtesy of Saibian, direct copy of the table does not work, so, a screenshot of the table is made)


Here are some examples:

Note that the ones section, the vowels at the end of the word will be removed, otherwise, it will sound more complicated, and the tens section, the "i" is replaced with "a", when linking with ones, except for icosi. As for hundreds section, the "o" in hecto is replaced with "a" when linking with other parts.

To generate these -illions, the tens section has been modified to match Rowlett's pattern. Normally, names for 30 to 90 (count of 10; 30,40,50...90) in Greek (transliterated) are:

trianta, saranta, peninta, exinta, evdominta, ogdonta, eneninta

Obviously, the names above are somewhat inconsistent. So, to resolve that, Saibian made a modification for the tens roots and the result looks like this:

triaconta, tetraconta, pentaconta, hexaconta, heptaconta, oktaconta, enneaconta (Cookiefonster uses ennaconta)

Here are more numbers:

So, the limit is enneahectaenneacontaennillion, so far. So, what is the 1000th -illion for this system? In Greek, a thousand is known as "chilias", but Saibian opted to use kilo- instead, which forked from SI prefix. The "ch" pronunciation can be confused, so, people think that "chilillion" comes from the Mexican cuisine, rather than the Greek "ch", which pronounced as k. Because of that, the 1000th -illion is known as "kilillion".

Without further ado, let's continue the journey.

We can continue by appending the existing prefixes, based off the 1-999 root table on top of kilo-, which results in duokilo-, triakilo-, until enneahectaenneacontaenneakilo-.

So, what's next after 1000^999,999? It's the millionth -illion. So, in this system, it is called ekatommyrio, which means 100 myriad (100*10,000).

Because of that, it is known as ekatommyrillion.

To append the first until 999,999th -illion to the million, it will become ekatommyria-.

We can continue with milestones like ekatommyrillion (10^3,000,000), duoekatommyrillion (10^6,000,000), triaekatommyrillion (10^9,000,000), dekaekatommyrillion (10^30,000,000), hectaekatommyrillion (10^300,000,000), or some of the complex -illions like:

1000^4,070,182 = 10^12,210,546 = tetraekatommyria-heptacontakilo-hectaoktacontadillion

Obviously, I will not list the -illions fully, but, I will have a separate page to list some of the -illions that are unique.

To push the limits, Saibian also extended the -illions by forking the numbers directly from Greek, like:

Normally, modern Greek numbers only established and recognized up to septillion, at least, not until further -illions (powers of 1000) are defined.

Bowers' -illions

This -illion has a dedicated page here. Among all of the -illion series, this series of -illions is the most popular, but also most complicated ones. The original system has 4 tiers that ended with multillion. There are several attempts to extend it, including myself.

Conclusion