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.

1. Million (1,000,000)

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:

• A million seconds is approximately equal to 11.5 days, or 1.7 weeks, while a million minutes is approximately 694.44 days, slghtly more than a year in Mars, or nearly 2 Earth years, and then a million hours is approximately 114 years, an average age of the oldest persons ever lived.

• A million $1 bills can cover a big stadium.

• The largest country in the world, Russia, covers over 17,000,000 (17 million) square kilometers / kilometres.

• The diameter of the Earth is 12.742 million meters / metres (12,742 kilometers / kilometres or 7,919.2 miles)

• A prime with at least a million digits is known as a megaprime.

• SI prefix of a million: mega- (with the symbol M, capital letter to distinguish it with m, which used for milli-, prefix for 10^-3)

• A million is used as a basis for the Aarex's Forcal series.

2. Billion (10^9) (or milliard in long scale) (1,000,000,000)

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:

• A billion seconds is approximately equal to 31.7 years, a billion minutes is equal to 1,901.3 years (Roman Empire ruled during that number of minutes ago), a billion hours is approximately 114,000 years. A billion days is equal to 2.736 million years (quarternary era)

• The largest signed 32-bit integer is 2.147483647 billion (an exact value, not an approximation).

• The world population is currently 7.876 billion (as of 2021-06-30).

• The distance between the Earth and the Sun is 149.7 billion metres / meters (149.7 million kilometers / kilometres or 1 astronomical unit).

• SI prefix of a billion: giga- (symbol: G)

• Counting from 1 to a billion takes a long lasting 31.7 years (assume that the number is counted 1 per second)

• A billion $1 bills (=$1 billion) can fit a supertall skyscraper (skyscraper that is over 300 m / 984 ft tall). Coincidentally, some of the supertall buildings cost that amount of money to build it.

• Our galaxy, Milky Way, has approximately 500 billion stars.

3. Trillion (10^12) (or billion in long scale) (1,000,000,000,000)

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:

• A trillion $1 bills actually could fit any states between 100 km^2 and 1000 km^2

• A human cell contains about 50 trillion cells.

• RTX 3090 has a theoretical performance of 35.7 TFLOPS of single-precision floating point format.

• The number of known digits of pi (π) is 50 trillion, as of 2021.

• In the human brain, it has approximately 100 trillion synapses.

Beyond the first -illions

1. Quadrillion (10^15) (billiard in long scale)

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:

• A light year is equal to 9.46073 quadrillion meters / metres (9.46073 trillion kilometers/kilometres or 5.87989 trillion miles).

• A parsec is 3.26157 light years, which is also equal to 30.86 quadrillion meters.

• Sbiis Saibian coined "small fry" and "guppycrumb" for this number. Note that guppy is 10^20, and the suffix -crumb divides the number by 100,000.

• In computing, when represented in IEEE double precision floating-point format, all integer values can be expressed until 2^53 (9.0072 quadrillion).

2. Quintillion (10^18) (trillion in long scale)

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.

• A large asteroid or a small moon (diameter of 50 km to 200 km) has a mass of a quintillion kilograms.

• The largest signed 64-bit integer is 9.223 quintillion (exact value: 2^63-1).

• The wheat and chessboard problem can be expressed as 2^64-1 (or 18.446 quintillion)

• The possible combinations of a 3x3x3 Rubik's Cube is 43.3 quintillion, without cheating. (note: cheating is not allowed when solving it, unless if you have a hard time to solve it)

• 100 quintillion is known as "guppy", and is equal to E20 in Hyper-E notation, E means 10^.

3. Sextillion or a thousand quintillion (10^21) (trilliard in long scale)

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.

• The volume of the Earth is 1.085 sextillion cubic meters (since the Earth is a sphere, we can calculate it as follows: 4/3 * (pi) * (6.371*10^6)^3, note: the radius is in meters).

• We can fit a whopping 6 sextillion cups of water to fill the oceans of the Earth.

• Sudoku has a possible 6.6709 sextillion grids (9x9 grid).

• The Avogadro's number (defined by the number of particles in a mole of substance), NA (A is supposed to be subscripted, but I can't see the option to do that), is approximately 602.214076 sextillion. 

• There is a power of 2 which is comparable to the number above; 2^79, which is approximately 604 sextillion, the ratio between this and the number above it is 1.0037.

4. Septillion (10^24) (quadrillion in long scale)

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.

5. Octillion (10^27) (quadrilliard in long scale)

(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.

6. Nonillion (10^30) (quintillion in long scale)

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).

7. Decillion (10^33) (quintilliard in long scale)

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.

8. Undecillion (10^36) (sextillion in long scale)

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.

9. Duodecillion (10^39) (sextilliard in long scale) 

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:

10. Tredecillion (10^42) (septillion in long scale)

11. Quattuordecillion (10^45) (septilliard in long scale)

12. Quindecillion (10^48) (octillion in long scale)

13. Sexdecillion (10^51) (octilliard in long scale)

14. Septendecillion (10^54) (nonillion in long scale)

15. Octodecillion (10^57) (nonilliard in long scale)

16. Novemdecillion (10^60) (decillion in long scale)

17. Vigintillion (10^63) (decilliard in long scale)

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)

18. Centillion (10^303)

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

Here are some examples:

1. For the 21st up to the 99th power of 1000, simply append (tens group) + (ones group) + "-illion" . 

•  For example, to obtain the 45th power of 1000, "tetraconti"  + "penta" + "-illion" - "a" = tetracontapentillion.

2. To generate the 100th to the 999th power of 1000, (hundreds group) is added to it.

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:

1. 1000^101 = 10^303 = hectahenillion

2. 1000^110 = 10^330 = hectadekillion

3. 1000^120 = 10^360 = hectaicosillion

4. 1000^130 = 10^390 = hectatriacontillion

5. 1000^140 = 10^420 = hectatetracontillion

6. 1000^150 = 10^450 = hectapentacontillion

7. 1000^160 = 10^480 = hectahexacontillion

8. 1000^170 = 10^510 = hectaheptacontillion

9. 1000^180 = 10^540 = hectaoktacontillion

10. 1000^190 = 10^570 = hectaenneacontillion

11. 1000^201 = 10^603 = duohectahenillion

12. 1000^250 = 10^750 = duohectapentacontillion

13. 1000^350 = 10^1050 = triahectapentacontillion

14. 1000^470 = 10^1410 = tetrahectaheptacontillion

15. 1000^999 = 10^2997 = enneahectaenneacontaennillion

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.

1. 1000^1000 = 10^3000 = kilillion (Bowers used killillion to continue, note the additional l that has been italicized)

2. 1000^1001 = 10^3003 = kilohenillion

3. 1000^1002 = 10^3006 = kilodillion (duo changed to di)

4. 1000^1010 = 10^3030 = kilodekillion

5. 1000^1020 = 10^3060 = kiloicosillion

6. 1000^1100 = 10^3300 = kilohectillion

7. 1000^1200 = 10^3600 = kiloduohectillion

8. 1000^1999 = 10^5997 = kiloenneahectaenneacontaennillion

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-.

1. 1000^2000 = 10^6000 = duokilillion

2. 1000^2500 = 10^7500 = duokilopentahectillion

3. 1000^3000 = 10^9000 = triakilillion

4. 1000^4000 = 10^12,000 = tetrakilillion

5. 1000^5000 = 10^15,000 = pentakilillion

6. 1000^6000 = 10^18,000 = hexakilillion

7. 1000^7000 = 10^21,000 = heptakilillion

8. 1000^8000 = 10^24,000 = oktakilillion

9. 1000^9000 = 10^27,000 = enneakilillion

10. 1000^10,000 = 10^30,000 = dekakilillion

11. 1000^20,000 = 10^60,000 = icosakilillion

12. 1000^50,000 = 10^150,000 = pentacontakilillion

13. 1000^100,000 = 10^300,000 = hectakilillion

14. 1000^200,000 = 10^600,000 = duohectakilillion

15. 1000^999,999 = 10^2,999,997 = enneahectaenneacontaenneakiloenneahectaenneacontaennillion

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:

• billion = disekatommyria, hence, 1 billionth -illion is disekatommyrillion (10^3,000,000,000)

• trillion = trisekatommyria, hence, 1 trillionth -illion is trisekatommyrillion (10^3*10^12)

• quadrillion = tetrakis ekatommyria, hence, 1 quadrillionth -illion is tetrakisekatommyrillion (10^3*10^15)

• quintillion = pentakis ekatommyria, hence, 1 quintillionth -illion is pentakisekatommyrillion (10^3*10^18)

• sextillion = exakis ekatommyria, hence, 1 sextillionth -illion is exakisekatommyrillion (10^3*10^21)

• septillion = eptakis ekatommyria, hence, 1 septillionth -illion is eptakisekatommyrillion (10^3*10^24)

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

References of this section

1. https://www.worldometers.info/world-population/ (Real-time world population clock)

2. https://sites.google.com/site/largenumbers/home/4-3/4-3-2-hyper-e-numbers (List of Hyper-E numbers by Sbiis Saibian)

3. https://sites.google.com/site/largenumbers/home/2-4/2-4-7-russ-rowletts-greek-based-illions

4. https://sites.google.com/site/pointlesslargenumberstuff/home/1/extendedillions1