PART 2: THE LITTLE BABIES1,000,000 ~ 10^10^1,000,000Here will be numbers that are bigger than a million but less than a millionduplex, including the googol and googolplex themselves. Be prepared for some numbers that aren't so wimpy, but still smallish in the world of googology.The Third-Block Range1,000,000 ~ 999,999,999Entries: 21
One million1,000,000 One million is a classic benchmark for large numbers, and probably associated with the idea of "really big number" more than any number. It's the first member of the -illion series, and the most widely used by far, with over a billion results on Google. The name is well-known and appears in all kinds of contexts, and it's also often used as hyperbole for a very large number
The word "million" came from an Italian word meaning "great thousand" - therefore it holds a unique place among the origin of number names in English. Here are some examples of how much a million is:
A million cookies would fill up a room or two, and a million seconds is 11.5 days. The Bible (which consists of 66 books) has about a million words. A million people is quite a populous city, although it's considered not a very populous. country or even US state. It would realistically take a year to count to a million if you only use free time for counting, so definitely people have done that. million has been described by Sbiis Saibian as being on the tenuous line between attainable and utterly out of reach. Overall, a million can be described as "truly large" - for a more detailed coverage of how much a million is look here.A million is the boundary between class 1 and class 2 numbers in Robert Munafo's idea of classes, because it's about the limit of physically perceivable - you can barely put a million dots on a paper and see each individual dot at once. While class 1 numbers are numbers between 6 and a million, class 2 numbers are from a million to 10^1,000,000 (a millionplex). In general, class 0 numbers can be recognized with our number sense alone, class 1 numbers can't be recognized but can by physically perceived, and class 2 numbers cannot be physically perceived, but can be exactly represented in decimal. With this idea, a billion, a quintillion, Avogadro's number, the number of molecules that make up Earth, a googol, a mililllion, and a googolgong are class 2 numbers. The SI prefixes for a million are mega- (large) and micro- (small).
Guppybit 1,048,576 This number is 2^20. It's the value of the mebi- binary prefix (see 1024), and therefore under some definitions it's the number of bytes in a megabyte. Sbiis Saibian gives this number the name guppybit. He defines a guppy as 10^20 and then coins a -bit suffix - x-bit is equal to x in binary, so while a guppy is 1 followed by 20 zeros in decimal, a guppybit is 1 followed by 20 zeros in binary. A guppybit is in fact one of Saibian's smallest googolisms. The next smallest number of his is a pipsqueak, which is another name for ten million. See also 32,768.
Googovi 2,985,984 The 6th member of the googo- series - it's equal to 12^6. The previous is googov and the next is googovij.
10! 3,628,800 The tenth factorial number. Only the tenth factorial is already bigger than a million!
Number of arrangements of a 2x2x2 Rubik's Cube 3,674,160 This is the number of arranmgements of a 2x2x2 Rubik's Cube, often called a Pocket Cube. It's equal to 7! * 3^6, leading to a number equal to about 3.67 million. It's the only number of arrangements of any size of a Rubik's cube that isn't out of human reach. But it's still a big number - to get an idea of its size, it takes a faucet 3.5 years running at full force to waste this many gallons. Rubik's Cube numbers:
This number is the
original Smith number, a number whose sum of digits is the same of the sum of the digits in its prime factorization. It was the phone number of Albert Wilansky's brother-in-law Harold Smith, and Albert Wilansky noticed Smith's phone number having this unusual property: 4,937,775 = 3*5*5*65,837, and 4+9+3+7+7+7+5 = 3+5+5+6+5+8+3+7 = 42.
π^^e (by linear approximation) ~ 5,328,483.61 This number is calculated with the linear approximation of tetration, the same method used to calculate e^^e ~ 2380. It's equal to π^π^π^(e-2), which evaluates to about five million.
7,500,000 This number is equal to 7 1/2 million. In The Hitchhiker's Guide to the Galaxy, it's the number of years it took a supercomputer to calculate 42 as the answer to life, the universe, and everything.
Ten million / crore / pipsqueak10,000,000 This number is normally called ten million, but it's called crore in the Indian number system, and there it’s written as 1,00,00,000. See also 100,000, which is called lakh in India. Sbiis Saibian gives this number an alternate name, pipsqueak, continuing the idea of using very small things to describe numbers that are small by googologists' standards, but are pretty scary in real life. The number is 50 times the clover mite and about 10 times the guppybit. See also little squeaker. For examples of its size, the state of Ohio (which I live in) has over 10 million people. The earth is 12 million meters wide. This many seconds takes up around three months. It may or may not be possible that people have counted to ten million, but that would require a lot of dedication.
Fzeight 16,777,216 This is 8^8 or 8^^2, also equal to 2^24 (a power of two). With the fz- prefix it can be called fzeight. This number is notable in googology because it used to be the number of possible numbers in Sbiis Saibian's Hyper-E Notation (see part 3 of this list) before new numbers were created like a gargantuul and a throodekol.
28,731,569.5 In a Calvin and Hobbes comic strip (link) which is part of a story arc, Calvin shows Hobbes a secret code he just finished, where he assigns each letter to be a completely random number. This is what the letter B becomes in the code. See 3,004,572,688 for more.
Number of seconds in a 365-day year 31,536,000 Another time-related figure similar in spirit to 86,400, the number of seconds in a day. This is a figure that occurs just as naturally as 86,400; however, a year is somewhat harder for us to comprehend than a single day.
Fifth perfect number 33,550,336 This number is the fifth perfect number, notable for being a big jump from the previous perfect number. Why is that? Remember the formula ((2 ^{p}-1)*2^{p-1}) discussed in the entry for 496? Well, that formula fails for p = 11, just as the Mersenne prime formula 2^{p-1} fails to generate a prime for p = 11. The perfect number formula, just like the Mersenne prime formula, works once again for p = 13, creating this number. As Mersenne primes become super-rare after a certain point, so do the perfect numbers.
73,939,133 In base 10, this number is the largest prime such that you can remove a digit from the right and it's still prime, AND that you can repeat that process and get a prime each time. In other words: 73,939,133 is prime 7,393,913 is prime 739,391 is prime 73,939 is prime 7393 is prime 739 is prime 73 is prime 7 is prime There are exactly 83 numbers in base 10 with this property. For prime numbers with related properties see 31, 43, and 137.
Five hyperfactorial 86,400,000 This number is the hyperfactorial of five (5 ^{5}*4^{4}*3^{3}*2^{2}*1^{1}, see 108), and strangely for a number this big has another unrelated property: it's the number of milliseconds in a day. This serves as an example of one of the nice coincidences time units can create.
Third taxicab number87,539,319 The third taxicab number (see 1729 for the backstory) is representible as a sum of two cubes in three different ways: 87,539,319 = 167^3+436^3 = 228^3+423^3 = 255^3+414^3. See also 24,153,319,581,254,312,065,344.
One hundred million / octad / myllion 100,000,000 This is 100 million, a number notable for several reasons. It was used by the Ancient Greeks as a myriad squared, and it's the largest number in the Bible, where the number of angels was said to be “ten thousand times ten thousand”. If not real this was probably meant to impress. In Chinese this is known as "yi" - the name is distinguished from the name "yi" for 1 only by the concept of words being differed by tones, a concept that is quite strange to an English speaker. Tones in Chinese are also the only way the words for "four" and "death" are distinguished, leading to 4's connotation of bad luck in Chinese culture. Counting to 100 million would take 9 years non-stop, but that’s unrealistic. It’s hard to say whether people have reached 100 million by cointing or not. Light moves at 300 million meters per second. America’s population is around 300 million. The moon is 400 million meters from the Earth. In the -yllion naming system (read this article for a detailed coverage of the system) proposed by Donald Knuth, this number is known as a myllion. With -yllions, zeros are grouped in fours instead of threes - 1 to 999 are named the same as in normal English, but after 1000 it gets different. 1000 is called ten hundred, 1100 is called eleven hundred, 1200 is called twelve hundred, 2000 is called twenty hundred, 3000 is called thirty hundred, etc. 1,0000 in the -yllion ststem is called a myriad. 1234,5678, as an example, is called twelve hundred thirty four myriad fifty six hundred seventy eight. A myriad myriad is called a myllion, and a myllion myllion is called a byllion (has its own entry) - continue with tryllion, quadryllion, etc. The “y” in the names is to be pronounced like “i” as in mile, so that myllion is pronounced mile-yun in contrast to million pronounced mill-yun. See also decyllion and vigintyllion.
Googovij 105,413,504 The 7th member of the googo- series, equal to 14^7.
299,792,458 This is a large number that shows up in the exact SI definition of a meter. A meter is defined as "the length of the path traveled by light in vacuum during a time interval of 1/299,792,458 of a second". The second is defined in purely scientific terms, not based on any units (or astronomical figures such as Earth's day for that matter), and its definition is discussed here.
Population of the USA (2014 estimate) 317,000,000 This is a commonly given large number figure, the population of the United States as of 2014. It's equal to roughly 300 million. See also the world population.
Fznine 387,420,489 9^9 or 9^^2, equal to about 387 million. Power towers of nines seem to show up sometimes in large number discussions (see 9^9^9). Personal: On Wikipedia, at one time (when I was 11 years old) my username was Cegalegolog99 (now it's Cookiefonster), and my signature there looked like this: Cega logLEGO9^{9}^{!} Cega was green becaues green's my favorite color, LEGO imitated the Lego logo, log was brown because logs are brown, and 9 ^{9}! represented my interest in mathematics. You could say that 9^9 appeared indirectly in my signature, but if the ! is interpreted as a factorial, we'd get the very large number that is nine to the power of the factorial of nine.
999,999,999This number is an example of a number someone would come up with in an attempt to create a large number - they hope to make it as big as possible by filling each place value with a nine, the largest number a place value has. Such a naive method won't work very well - you'll need to go a whole new way, by using exponentiation and then iterating that and stuff. Besides, since 1’s are faster to write and take less space than 9’s, you’d be better off writing a wall of ones than nines! However, the repeating-9’s method is still useful in things like Bignum Bakeoff where the goal was to make as a program that makes as large of a number as you can with 512 characters or less - I think all 20 entries in Bignum Bakeoff use strings of nines at one point or another. For more on Bignum Bakeoff see 14, the nubmer of entries submitted that produced a large number where you can jump to any of the entries for numbers submitted.
The Fourth-Block Range1,000,000,000 ~ 999,999,999,999Entries: 26
One billion 1,000,000,000 A very large number equal to 10^9, or a thousand millions. In the short scale this is known as a billion, while in the long scale (commonly used in countries like France and Germany) it is called a milliard. Like a million, it is a number that commonly crops up in day-to-day life, with billions of dollars and billions of computer operatons per second. While a million is impressively large, a billion is incredible and hard to wrap your mind around. You may be thinking that a billion is too small to be considered "very large". That’s because almost everyone underestimates the size of the number - with its prevalence in the human world, a billion just doesn't feel as big as it used to. So here are a few examples to make a billion feel as big as it really is: Saturn is just under a billion miles from the sun. If the oldest person to have ever lived spent all her life except for sleep counting, she would barely make it to a billion. Not like that’s evern possible, considering that your voice may wear out and you can’t count straight from birth, and that if you spent your whole life counting you probably wouldn’t even make it to the age of the oldest person ever. Therefore counting to a billion is realistically impossible. A billion cookies would cover six or seven houses in a cube. It would take almost a thousand years for a faucet to waste a billion gallons of water. As of September 2015 only eight videos on YouTube have a billion views or more, all of which are music videos (the highest by far is Gangnam Style, the only one with 2 billion or more). The number of webpages in the whole huge seemingly endless Internet is in the billions. India has a population of around a billion. So a billion is definitely a pretty big number alright, isn’t it? It can definitely be described as “very large” by ordinary standards. Really, a billion is large enough to have a definite connotation of "really big number" while still having a familiar name - therefore, it's very well suited as a hyperbole number, especially when "million" isn't quite big enough. See 3,000,000,000 for an example of billions' usage as hyperbole. The SI prefixes for a billion are giga- (gigantic) and nano- (tiny).
1,163,962,800 The entry for 7560 discusses that it is the first number that is highly composite but not superabundant. However, are there any superabundant numbers that are not highly composite? There are indeed such numbers - however you won't see them until you reach numbers in the billions order of magnitude! This makes for a naturally occurring very large number that makes 945's occurrence look trivial.
M_{31} / Eighth Mersenne prime2,147,483,647 This number is the eighth Mersenne prime, and for a time it was the largest known prime number (the current record is here). It was proven to be prime in 1772 by Euler. The exponent itself in this number is a Mersenne prime, making it a double Mersenne number. This number is additionally the maximum value of a 32-bit signed integer in computing. 32-bit integers are the most common type of number when you need to use an exact value. Therefore this number is important in computing, and the cause of such problems as the year 2038 problem (discussed at 2038). When you need exact numbers larger than 2.147 billion, that's when 64-bit integers come into play (see 127, 32,767, and 9,223,372,046,854,755,097). Gangnam Style view count (as of May 2015) 2,417,000,000 Gangnam Style, the most viewed video of YouTube, has an amazing 2.33 billion views and counting. It was uploaded July 15, 2012. It hit 1 billion views on December 21, 2012 (first to reach a billion), and it hit 2 billion on June 1, 2014. A bit later it passed 2,147,483,647 (the largest 32-bit integer) views, which is why some time before this point YouTube updated view counts from 32-bit to 64-bit integers.
Three billion 3,000,000,000 On one advertisement for the Rubik's Cube, it was said to have "over 3 billion combinations but only one solution" - though technically not wrong, it's quite the understatement, since the number is actually over 43 quintillion (that's over a billion TIMES bigger). However, the phrase "over three billion combinations" is enough to make the point about the challenge of solving a Rubik's Cube, and really this is a classic example of human innumeracy.
3,004,572,688In a Calvin and Hobbes comic strip (link) which is part of a story arc, Calvin shows Hobbes a secret code he just finished, where he assigns each letter to be a completely random number. This is what the letter A becomes in the code. Even though Hobbes says that's a "good code alright" to Calvin, it's worth nothing that the code, really, isn't any stronger than substituting each letter with, say, a random picture. However, this does go to show that people really do take large numbers to be difficult to work with to such an extent. For a bigger number to appear in Calvin and Hobbes, see 3*10^85, which is probably the largest number to appear in the entire comic.
4,294,967,295 If there are no Fermat primes larger than 65,537, then this is the highest odd number of sides of a regular polygon constructible with compass and straightedge (see 257 for details). Its prime factorization is all known Fermat primes, 3*5*17*257*65,537. It's also of note that this number is a product of numbers of the form 2^2^n+1, while the number itself is of the form 2^2^n-1.
Googoviji 4,294,967,296 The 8th member of the googo- series, equal to 16^8 = (2^4)^8 = 2^32 = 2^2^5 - two to the power of a power of 2. Because of that last property, this number sometimes shows up in computing (see 2,147,483,647) - it's the amount of possible 32-bit integers. Another computing-related property of this number is that it is the number of possible 32-bit IP addresses of the form x.x.x.x, since x is any number from 0 to 255 (256 possible x's), and there are four x's, so the number of possible IP addreses is 256^4 = 4,294,967,296. The number of IPs that can actually be used is actually a little lower since some are reserved for special purposes.
4,294,967,297 This number is the first composite Fermat number, expressible as 2^2^5+1. Its prime factorization is 641*6,700,417, making it a semiprime instead of prime.
Smallest odd-abundant number not divisible by 3 5,391,411,025 This is the smallest odd-abundant number that is not divisible by 3. It is an example of a fairly large number that occurs from a simple numerical property, much like 945.
World population as of Oct. 2014 7,268,000,000 This is roughly the current human population of the world, equal to about 7.268 billion. It is probably the single best-known example of a large number in a real world. Other beings have a larger population, such as fish in the trillions and ants in the quadrillions.
9,192,631,770 This is a number used to give the SI definition of the exact value of a second - it's defined as "the duration of 9,192,631,770 periods corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom". I don't care that I don't understand this physics mumbo jumbo, but I do find it cool just how it's exactly defined. A second is needed to precisely define a meter, whose definition also uses a largish number. A second was formerly defined as 1/86,400 of a day - however the SI later decided that it was better to define the second based on something more precisely determinable.
9,814,072,356 This is a large number that has several interesting properties - it's an example of a pandigital number, a number that contains each digit at least once. It's notable for being the largest pandigital square number without redundant digits, and the largest perfect power without a digit occurring more than once. This number is also the square of 99,066, which happens to be a stroborgammatic number (a number that reads the same upside down, see 69), making for a nice coincidence.
This number has achieved some fame through getting
its own Wikipedia article, and in fact it's one of very few numbers this big that gets one. The previous ones to get their own article are 100,000,000, 1,000,000,000, and 2,147,483,647, and the next ones are 9,223,372,046,854,755,097 and googol. However, there was some debate whether the number did indeed deserve its own article, even though now it is considered notable enough.
Ten billion / dialogue / fzten 10,000,000,000 or 10 ^{10 }This is ten billion or 10^10 = 10^^2, another small tetrational number. It can be called fzten with the fz- prefix. Sbiis Saibian calls it "dialogue", from "dia" meaning 2 plus the -logue suffix which names powers of ten. Ten to the power of this number is called a trialogue, a number that appears in a School House Rock song.
Age of the universe in years13,700,000,000 This well-known figure is our best guess at the amount of time since the Big Bang which started the universe. It is perhaps of note that this number is about 100 million times the physics cult number, 137.
e^^π (by linear approximation) ~ 19,337,456,547 This is e tetrated to pi in the linear approximation for tetration - it's equal to e^e^e^e^(π-3), creating a number equal to about nineteen billion. This number is only somewhat smaller than the famous 3^^3, but vanishingly smaller than π^^π by linear approximation.
Fifty billion / Little squeaker50,000,000,000 This number is called little squeaker by Sbiis Saibian. It's five times a dialogue and five thousand times a pipsqueak. A sphere of this many gallons would be 712.3 meters wide, or roughly half a mile. That means a "little squeaker sphere" with this many gallons of water would look pretty intimidating, as a casual walk around that sphere would take almost half an hour, about 27 minutes. Also, a little squeaker seconds (aka 50 billion seconds) is about 1584 years, meaning a little squeaker seconds ago was around the falling days of the Roman Empire. A little squeaker is a number that is starting to strike some fear into us, but it pales in comparison to a small fry, an alternate name for a quadrillion.
76,923,076,923 The fourth smallest number of the form (10^(p-1)-1)/p where p isn't 2 or 5 (this time it's 13) - this one also fails to be cyclic (see 142,857). However, the next such number (where p is 17) is cyclic.
80,313,433,200 This number is the "weak factorial" of 29. As discussed in the entry for 420, the smallest weak factorial that isn't highly composite, the next highly composite weak factorials larger than 420 are 840, 2520, 27,720, and 720,720, but the next one after those four is not until 80.3 billion. It's yet another example of sequences suddenly jumping to large values.
One hundred billion100,000,000,000 A common estimate of how many people have ever lived on Earth. It's an interesting figure, albeit one quite hard to estimate, for obvious reasons.
Googoix 189,359,290,368 The 9th member of the googo- series, equal to 18^9. Googo-x quickly outgrows 10^x, as you can see. If the Unix time system uses 64-bit integers (maxed out at 9.223 quintillion) instead of 32-bit integers (maxed out at 2,147,483,647), then it would roll over on the year 292,277,026,596 instead of on the year 2038 (the famous year 2038 problem). Since that year is well beyond the believed lifespan of Earth and the Sun, that "year 292,277,026,956 problem" is commonly used as a time formatting "problem" that isn't really a problem at all. Problems like this are often used as matters of theoretical interest.
Three hundred billion 300,000,000,000 A rough approximation for the number of stars in our galaxy. As you can see numbers like billions and trillions and even quadrillions/quintillions cover a broad range of scenarios. The Stellar Range10^{12} ~ 9.99*10^{29}Entries: 42
Trillion 1,000,000,000,000 or 10^12 A trillion is one followed by twelve zeros, or 10^12. In the long scale it's known as a billion. It's notable because for many people it's the largest "familiar" illion, the largest they usually hear of in life (usually with trillions of dollars, in national debts or the like). Quadrillion and quintillion are still used sometimes, but significantly less than trillion. A trillion is so big that it’s hard to come up with examples straight from the top of your head, but here are some: A trillion seconds is 32,000 years, so a trillion seconds ago humans started developing. Counting to a trillion nonstop would take 200,000 years - it’s pretty creepy to think about how a trillion could be reached by counting. Our body houses about a trillion bacteria. There are supposedly a few trillion fish in the world, and the human body contains 20-50 trillion cells. Here are some more examples: A tower of a trillion people would reach all the way to Saturn. A faucet would take nearly a
million years to waste a trillion gallons, and those trillion gallons in a sphere would be 1.2 miles wide. For more examples look at this article.The prefixes for a trillion are tera- (Greek, monster, also looks like a compaction of tetra-) and pico- (Spanish, bit). Prefixes beyond these aren’t regularly used, but still official and show up now and then.
Pi-illion 10^(3*π+3) ~ 2,959,365,073,955.536 By the sequence million, billion, trillion, quadrillion, etc, x-illion can be considered equal to 10^(3x+3). With that system, unusual -illions can be coined like this one, equal to about 2.96 trillion. See also piplex in part 1.
Megafuga-three 7,625,597,484,987 This is a number notable in googology because it shows up a lot when working with powers of three. It can be represented in up-arrow notation as 3^3^3, 3^^3, and 3^^^2. It's a very very small pentational number and a small tetrational number. An example of an occurrence of megafuga-three is that the size of the power tower of 3s representing the unfathomable number called tritri (equal to 3^^^3, seen in part 3) is 3^^3. The name comes from the prefix megafuga-, which means x^^x. This number can alternatively be represented as 3^27, 27^9, and 19,683^3 - it can be expressed in many ways.
Googox 10,240,000,000,000 The tenth member of the googo- series - this is notable for being a lot rounder than the other googo- numbers.
Twenty trillion 20,000,000,000,000 This number is an estimate on the number of red blood cells in the human body. Trillions are still small enough to occur everywhere in the world as you can see.
SpongeBob’s Number 29,998,559,671,349 SpongeBob SquarePants was my favorite TV show as a kid. In the episode “Have You Seen This Snail?”, SpongeBob is given a challenge of hitting a paddle ball this many times in a row, which causes him to neglect his pet snail, Gary. This makes Gary abandon SpongeBob. It’s interesting just how much large numbers show up even in pop culture. I knew of this number since I was a kid, but only recently gave it a name. Realistically, even with a fast four hits per second, this challenge would take 237,666 years. Probably by coincidence, SpongeBob's Number is a prime number. For another number to appear in SpongeBob see 46,853.
One hundred trillion100,000,000,000,000The current lower-bound to the solution of Skewes' problem - see Skewes' number (a famous old upper-bound) and 1.397*10^316 (the current upper-bound) for details.
Integral-dekapetaseptemdecile 588,235,294,117,647 This is the second smallest cyclic number, as its multiples are rearranged versions of its digits - the smallest is 142,857. It can be expressed compactly as (10^16-1)/17. The name "integral-dekapetaseptendecile" is from Andre Joyce.
Quadrillion / Small fry 1,000,000,000,000,000 or 10^15 A quadrillion is equal to 10 ^{15}, a million billions, or a thousand trillions. In the long scale it's known as a billiard. It's the smallest illion that mostly arises in science, and the name is a lot less known than trillion. However, in our modern world, number names like this are becoming more commonly used, as you could see if you were to Google search some of the names for -illions.A quadrillion protons in a chain would be a meter long. There are about 1-10 quadrillion ants on earth. It is said that the beach has about a quadrillion grains of sand, and Niagara Falls would take 210 years to use up a quadrillion gallons of water. All the world’s financial assets together are probably about a quadrillion dollars, and that many dollars in $100 bills packed tightly would look like the picture to the right, bigger than the Empire State Building. A quadrillion seconds is about 32 million years. A cube of a quadrillion cookies would cover aound 10 square miles. The Great Lakes have a volume of 6 quadrillion galllons. As you can see we're starting to blast into the astronomical-level numbers. Sbiis Saibian gives this number another name, "small fry" - a small fry is a name given to a newborn fish. It’s named “small fry” because it’s a small fry in the world of googology. Contrary to the name, by now the numbers are getting kind of dizzyingly huge, as said above. A sphere with this many gallons of water would be an ominous 12 miles or 19 kilometers. But that number is crushed by the guppy, equal to 100 quintillion! The prefixes for quadrillion are peta- (short for Greek penta-, five) and femto- (Danish femten, fifteen). For comparison a light-year is around 9 petameters.
Gogolbit 1,125,899,906,842,624 = 2^50 Sbiis Saibian defined a gogol to be equal to 10^50, and using the -bit suffix that turns 10^n into 2^n, he defines the gogolbit to be 2^50. This number is just over one quadrillion. To get a sense of its size, to waste this many gallons of water you'll need to run a faucet for a billion years! This is also the number of bytes in a pebibyte using the binary SI prefixes (see 1024).
10,000,000,000,000,000This is 10
^{16}, or ten quadrillion. It is known as a byllion in terms of Knuth's -yllions (see 100,000,000), and in Chinese it is known as jing.This is also cited by Andrei Linde and Vilaty Vanchurin as an estimate of how many bits of information a human being can absorb in their lifetime, and they use it in their estimate of the number of distinguishable parallel universes (10^10^16). Twelve quadrillion12,000,000,000,000,000The number of calculations Inspector Gadget can perform per second, thanks to a robotic implant in his brain. He could rewrite all of Minecraft's code while helping Penny with her homework and cleaning up Brain's doodie, all at the same time.
Number of words humanity speaks in a year 35,813,070,000,000,000 This number is equal to about 33.215 quadrillion, and it’s my estimate of how many words humanity speaks in a year. Here’s how I got that value: Studies have shown that men speak around 7,000 words per day, while women speak 20,000. This averages to 13,500 words per person per day. Now multiply that by 7.268 billion (around the population of the world as of November 2014), and we get 98.118 trillion words per day spoken by humanity. Already quite big. Multiply this by 365, and you get 35.81307 quadrillion words per year to come out of the mouths of the inhabitants of Planet Earth! So despite its staggering size, we can say that a quadrillion is still a number with real examples we can understand.
One hundred quadrillion 100,000,000,000,000,000 or 10^17 One hundred quadrillion is an estimate of the number of words printed on paper in the first 500 years of the printing press (1456 to 1956). This interesting figure reaches well into the astronomical range, and it doesn't even count the words printed from 1956 to today - the total number of words including today may be over a quintillion!
Age of universe in seconds 473,000,000,000,000,000 This is the age of the universe in seconds, equal to about 473 quadrillion, or half a quintillion.
Quintillion 1,000,000,000,000,000,000 or 10^18 A quintillion is 10 ^{18}, a million million million, or a billion billions. It's known as a trillion in the long scale. It is a sort of cut-off point for large numbers, as numbers beyond this point just don't get used much at all, and you rarely see those outside of lists. A quintillion was the largest -illion I knew of (other than vigintillion, trigintillion, and perhaps decillion) before discovering the name sextillion through the game Cookie Clicker, causing me to want to know the rest of the -illions. I knew of a quintillion (and quadrillion) since, at the latest, sixth grade, probably earlier. Below are some figures to get an idea of the size of a quintillion:A grain of salt is made of a quintillion atoms. A quintillion seconds is 32 billion years, around twice the age of the universe. Niagara Falls would take 210,000 years to use a quintillion gallons of water. A tower of a quintillion people would reach 180 million light years high, and it is 11 quintillion miles from Earth to the Andromeda Galaxy. With these impressive examples, a quintillion is an even better astronomical number. A well-known example of a figure in the quintillions is that there are 43 quintillion ways to arrange a Rubik's cube - a mind-boggling figure, especially since a quintillion is hard to wrap your mind around. That value has its own entry on this list (you can jump to it here). Also, based on the calculations a few entries ago, it would take about thirty years for us to utter a quintillion words! That’s ... a pretty long time. The prefixes for quintillion are exa- (Greek hexa-) and atto- (Danish atten, eighteen).
1,111,111,111,111,111,111 This number, equal to a string of 19 1's or 1.111 quintillion, is the second smallest repunit prime - a repunit prime is a prime which has no digits other than 1. It can be written 1[19] in SpongeTechX's copy notation (see 22). Repunit primes are also trivial cases of permutable primes. Behavior of repunit primes varies a lot between bases. For example, in binary, repunit primes are the famous Mersenne primes, and there is never more than one repunit prime when the base is a perferct power (e.g. base 4, 9, or 27). Also, some bases (such as base 18) seem to have larger first repunit primes with over 2 digits than others, though I can't make out any pattern in that sort of thing. In base 18, the smallest repunit prime with over 2 digits is a string of 25,667 ones!
Guppybyte 1,152,951,504,606,846,976 = 8^20 = 2^60 Remember guppybit, equal to 2^20? Well, another small Saibianism, the guppybyte, is equal to 1 followed by 20 zeros in base 8 (octal). Just as the -bit suffix turns a number into a power of 2, the -byte suffix turns it into a power of 8. 1,638,365,164,028,614,560According to Sbiis Saibian (source) this is the number of years it would take all planets to align within 4 degrees of precision. He brings it up in his article on large numbers in probability, and says that although this may seem more like a duration (a quantity) than a probability, it reflects the vast amount of states the solar system can take - because of this you'd need to waite about 1.6
quintillion years for all the planets to align within 4 degrees of precision. Compare this number (1,638,365,164,028,614,560) to the age of the universe in years (13,700,000,000).
Eighth perfect number 2,305,843,008,139,952,128 On the entry for 496, I discussed how a number x is a perfect number if and only if it is of the form 2 ^{p-1}*(2^{p}-1), where p is a prime number and 2^{p-1} is prime. When 2,147,483,647 (2^{31}-1) was proven to be prime by Euler, it quickly followed that this number, equal to about 2.305 quintillion, was a perfect number. When this number was shown to be perfect from Euler's proof that 2 ^{31}-1 was prime, English mathematician Peter Barlow wrote in a math journal that this number will probably be the largest perfect number anyone would discover. And how wrong he was! (see 1.447*10^{77}) Also, this number falls just under Pervushin's Number.
Pervushin's Number 2,305,843,009,213,963,951 This number is the ninth Mersenne prime, expressible as 2^61-1, or M61. It's equal to about 2.3 quintillion, and was proven to be prime by Russian mathematician Ivan Pervushin in 1883, hence the name "Pervushin's number". Pervushin's number is a turning point in Mersenne primes since trial division is no longer feasibly doable, as you'd need to divide this number by over 75 million different primes (for trial division, you need to see if number X is divisible by any primes less than the square root of X)! Therefore more clever techniques need to be used to see if a number is prime.
Pervushin's number never was the largest known prime - that record had been already broken by M127 in 1876. However, Pervushin's number was the second largest known prime from its discovery up until it was dethroned by M89 in 1911.
9,223,372,046,854,755,097This is a large number equal to 2 ^{63}-1, or about 9.223 quintillion. It is important in computing since it's the maximum value of a 64-bit signed integer. 64-bit integers are used if you need to use an exact value and 32-bit integers aren't wide-ranged enough. However, if you don't need exact values it's much more common to use the floating-point numbers, which overflow at 1.7976*10^{308}. See also 127, 32,767, and 2,147,483,647.This is also the only number between 9,814,072,356 and a googol to get its own Wikipedia article.
12,157,692,622,039,623,539This number, equal to about 12.157 quintillion, is equal to 1
^{1}+2^{2}+1^{3}+5^{4}+6^{6}+9^{7} ... +3^{19}+9^{20}, sum of consecutive powers of its digits, and the largest number to have this property. See also 135, another number with this property.
2^64 / Fzsixteen18,446,744,073,709,551,616 This is the number of different values that can be stored as a 64-bit integer (see 9,223,372,046,854,755,097). It's equal to 2^64 (a power of two) or about 18.446 quintillion, also expressible as 4,294,967,296^2, 65,536^4, 256^8, 16^16 4^32, and 2^64. This was also the largest number listed in
an early version of Robert Munafo's number list - the largest is now Steinhaus's mega. Robert Munafo points out that this happens to be the largest integer that he has memorized (not counting trivial things like vigintillion).
Number of positions of a Rubik’s cube 43,252,003,274,489,856,000 This is a well-known large number that appears in combinatorics - it's the number of ways you can legally arrange a normal (3x3x3) Rubik's cube. Almost everyone knows what a Rubik's cube is - it's a puzzle made of 26 little cubes connected with a center piece, where the goal is to configure it such that each face has only one color. The puzzle looks deceptively simple, but it's actually quite a difficult puzzle if you don't know the trick to solve it. That little cube turns out to have 43 quintillion possible configurations. If you're allowed to disassemble and reassemble it you get a number 12 times larger.To get an idea of this number's size, this many gallons of water would cover up the state of Ohio 3 miles deep. It would take about 90.8
million years for all those gallons of water to go down Niagara Falls. Think about that for a second - this means that if water started running down the Niagara Falls when the dinosaurs went extinct (the Niagara Falls didn't actually exist back then!) and continued running up to the present day, we still would have to wait 25 million more years till a Rubik's Cube number of gallons would go down the falls.See also 3,000,000,000 and the other Rubik's Cube numbers:
Guppy / one hundred quintillion100,000,000,000,000,000,000 = 10^20 A guppy is defined by Sbiis Saibian as equal to ten to the twentieth power, or alternatively a hundred quintillion. A guppy is a type of fish known for its small size, and true, by googological standards it's pretty small. Sbiis Saibian says about the guppy that it can be thought of as a miniature version of a googol. However, in the physical world the guppy is still a very large number - for example it's about 1/3 of the number of gallons of water on Earth, and that many pennies could cover the state of Maryland one mile deep! Additionally a sphere of a guppy gallons of water would be about 897 km wide, which is 1/14 of the diameter of Earth! It hasn't even been a guppy seconds since the Big Bang - a guppy seconds is the unfathomably long 3.16 trillion years!! Sbiis Saibian says that the guppy is only a guppy in comparison to the numbers we'll see later (see minnow)! Number of gallons of water on Earth 326,000,000,000,000,000,000 This is an estimate of the number of gallons of water on all of Earth's oceans. It's equal to about 326 quintillion and it serves as a benchmark for the size of other numbers around this range.519,024,039,293,878,272,000This is the number of ways to arrange a Rubik's cube if you're allowed to disassemble and reassemble it. It's equal to about 519
quintillion and it's exactly 12 times bigger than the number of arrangements if you aren't allowed to take it apart (see this entry).
Sextillion 1,000,000,000,000,000,000,000 or 10^21 A sextillion is 10 ^{21}, or a billion trillions. In the long scale it's known as a trilliard. It is a number that is rarely heard of at all, even though it's still the legitimate and real continuation of after the "cut-off -illion" quintillion. However a sextillion, like basically all larger -illions, is quite freely used by some people when talking about large numbers.Here are some examples of numbers in the sextillions: Earth is about a sextillion cubic meters in volume. It would take 30,000 years for us to speak a sextillion words, which is longer than all of recorded history! Even with proto-humans, in the past 30,000 years we haven’t spoken a sextillion words, considering how much smaller the population once was. Also, the earth weighs about 6 sextillion tons. There are 6 sextillion cups of water in all the oceans in the world. The distance between one end of the universe and the other may be 88 sextillion miles. The prefixes for sextillion are zetta- (based on Greek hepta-, seven) and zepto- (based on Latin septem).
Weight of Earth in elephants 1,194,600,000,000,000,000,000 Or 1.1946*10 ^{21} in scientific notation. This value was given by Lawrence Hollom on the beginning of his old site, when he discusses large numbers in the real world. In one part, he brings up the weight of Earth in kilograms, and then in terms of elephants - he sarcastically says that putting Earth's weight in terms of elephants makes it much easier to picture.
Dutrimevalka 2,222,222,222,222,222,222,222 This is another copy notation googolism by SpongeTechX. It is noted 2[2,3], which evaluates to 2[[3]] = 2[2[2]] = 2[22] = 222...222 with 22 2's. 6,670,903,752,021,072,936,960This is the number of possible solved grids with each number filled in the well-known puzzle Sudoku, where you start off with a 9x9 grid with some of the numbers filled and the goal is to fill the grid such that each number 1 through 9 occurs once in each row, column, and 3x3 grid. This number is equal to about 6.67 sextillion, but if you lift the restriction that you need to have a 1 through 9 once in each 3x3 grid you get a number about a million times larger.
Largest known / sixth taxicab number 24,153,319,581,254,312,065,344 The sixth taxicab number is the largest known one - it's the first number expressible as a sum of two cubes in six different ways. It's equal to about 24.153 sextillion. For other taxicab numbers (and more about them), see 1729 and 87,539,319.
Three hundred sextillion 300,000,000,000,000,000,000,000 = 3*10^23 This number is the current estimate on the number of stars in the observable universe. This value is difficult to estimate, since many stars cannot be observed at all from Earth due to technical restrictions, so much guess-work needs to be done to get such a value. Previous estimates for this figure have been smaller.
Avogadro’s number602,214,076,000,000,000,000,000 = 6.02214076*10^23 Avogadro's number is defined as the number of carbon-12 atoms that would weigh exactly 12 grams, or at least it was defined as that until 2019, when its value was declared to be exactly the number shown above. In science, it's convenient to define constants in terms as numerical and unambiguous as possible, which is why the definition of Avogadro's number was changed to be a precise, specific number. In chemistry, Avogadro's number of atoms or molecules is known as a
mole. A mole of any substance weighs the same number of grams as the molecule or atom's atomic mass, and a mole of a substance will always have precisely Avogadro's number of molecules or atoms of that substance. With Avogadro's number we can easily determine the number of molecules in n grams of a substance. For example, since carbon's atomic mass is 12.011, three grams of carbon would be made of (Avogadro's number)*3/12.011 atoms, or about 1.50*10^{21} (150 sextillion atoms).Avogadro's number is notable because it is a large number that is used in standard science education, and because it's much larger than our everyday millions, billions, and trillions. However, there are numbers in science and in mathematics that make Avogadro's number look quite humble, for example the number of atoms or Planck volumes in the observable universe, or the largest known prime number.
Septillion1,000,000,000,000,000,000,000,000 or 10^24 A septillion is 10^24 or a trillion trillions, and in the long scale it's known as a quadrillion. This is a pretty insane number alright, and by now it's getting hard to keep track of all these zeros. Here are some examples of a septillion: A septillion cookies would be about the size of the planet Mercury. Earth weighs five septillion tons. A liter of water has about 33 septillion molecules. The prefixes for septillion are yotta- and yocto-, both based on Greek/Latin “octo”, meaning eight. A septillion is notable because it's the largest -illion to get an SI prefix - yotta- and yocto- are the largest and smallest official SI prefixes, respectively. Beyond this point there are no officially recognized SI prefixes, but there are many extensions to them people have made (see octillion).
Minnow / Ten septillion10,000,000,000,000,000,000,000,000 = 10^25 This number is defined by Sbiis Saibian as 1 followed by 25 zeros, and it still continues the idea of naming numbers after small fish. However it's a dizzyingly huge number - for example, the observable universe only has about 3% of a minnow stars, and a minnow gallons of water would be about the size of Neptune! Pretty scary right? Well, a goby, 1 followed by 35 zeros, will make THAT look like a minnow.
25! 15,511,210,043,330,985,984,000,000 This is 25 factorial, or about 15.511 septillion. It's notable as the first number of the form x! which is more than 10^x; therefore, if x ≥ 25, 10^x serves as a lower bound for x!. It’s also quite easy to see that x^x is always an upper bound for x!, making the factorial boundable by two simple functions. This makes it a bit easier to bound numbers like the googolbang.
M_{89} / Tenth Mersenne prime 618,970,019,642,290,147,449,562,111 This number is the tenth Mersenne prime. It's equal to about 619 septilion and can be expressed as 2^89-1. It was proven by Ralph Ernest Powers to be prime in 1911, and held the record for second largest known prime for three years until Powers discovered another prime, 2^107-1.
Octillion1,000,000,000,000,000,000,000,000,000 or 10^27 An octillion is 10^27, known as a quadrilliard in a long scale. It is a billion billion billions, which is almost impossible to even comprehend! At this point people would almost certainly prefer scientific notation instead of even saying the name of this number, so in real life situations that will involve saying/writing “octillion” are very rare. Most people, as a matter of fact, haven’t even heard of an octillion! Some examples of octillions: The sun has a volume of 1.4 octillion cubic meters and weighs 2 octillion tons. Earth weighs 6 octillion grams. The human body has about 7 octillion atoms. An octillion is the first -illion that does not get an official SI prefix. It has many different unofficial SI prefixes, such as novetta- (Robert Munafo), xenna- (Sbiis Saibian), xova- (Andre Joyce), xona- (Jim Blowers), harpi- (Morgan Burke), hella- (jocular but recognized in Google's unit conversion, like meter to hellameter), xenta- (hoax by Jeff Aaronson), bronto- (origin unknown but widespread on the Internet), xotta- (Paul Schuch), xera- (another hoax), and probably even more. Many seem to be based on x and words for nine, extrapolating from zetta- and yotta-. In my own system, the prefix for one octillion is bronta-, based on the popular hoax prefix bronto-. For more about my own system of extended SI prefixes, see
here.5,524,751,496,156,892,842,531,225,600This is the number of possible Sudoku grids with the restriction that each 3x3 grid must have each number 1 through 9 once lifted. This is therefore the number of 9x9 Latin squares, a number equal to about 5.524 octillion. This is roughly a million times more than the number of possible Sudoku grids. For more on Latin squares see 56.
Meroogol
(8/45) ^{2}*(152,587,890,625,000/7,623)^{3}*128/5445
~5.85*10^27 This is an unusual number I coined (name comes from merology + googol). It's the result of plugging in "googol" to Joyce's first, more complicated merology system, which gives instructions for turning a word into a number by turning each letter or group of letters into a number, and then adding and/or multiplying all the numbers in the end, depending on the letters used in the word. His system is intended to turn "zero" into 0, "one" into 1, "two" into 2, etc, but it doesn't even remotely work. When plugging in "googol" to the system, it comes out as ... 5.85 octillion. That's much less than the googol and barely in the same class of numbers. See also -9. The Galactic Range10^30 ~ 9.999*10^99Entries: 57
Nonillion 10^30 A nonillion is 10^30, known as a quintillion in the long scale. A nonillion is a quadrillion quadrillions, or a million trillion trillions. It's also the last -illion number recognized in Microsoft Office 2010's spell check, but Office 2013 added a decillion to the list of -illions it recognizes. Examples of nonillions: The sun weighs 2 nonillion kilograms, and there are about 5 nonillion bateria on Earth. Most unofficial SI prefixes for a nonillion are either based on w- and/or a root for 10 (like weka- or decetta-), or completely arbitrary in origin. In my own system, the prefix for a nonillion is geopa- (pronounced /joh-pa/), based on the hoax SI prefix for a nonillion, geop-. See here for more on my system of extended SI prefixes.
Belphegor's prime 1,000,000,000,000,066,600,000,000,000,001 This is a prime number notable for its decimal expansion - it's one, thirteen zeros, 666, another thirteen zeros, and one. Since it contains occurrences of thirteen (bad luck) and 666 (the beast number), it can be said to be an evil-themed number that also happens to be prime. It was named by Clifford Pickover after a demon in Christianity and Judaism.
Googolbit / Little googol / Bingol 2^100 ~ 1.2676*10^30 This number is written as one followed by 100 zeroes in binary - it looks exactly like a googol in binary, but it's actually much smaller than a googol. The number has several different names. Traditionally it's been called little googol or also bingol (binary+googol), but Sbiis Saibian gave this number the name googolbit using his -bit suffix (see 32,768).
Googoxex 1.100*10^32 This is the twentieth member of the googo- series. It's equal to aboet 110 nonillion.
f_{2}(100) ~ 1.26*10^32 This is a number used as an example of the fast-growing hierarchy in my list. The fast-growing hierarchy is a googological notation very commonly used for approximating large numbers. Here are some examples of how it works: f _{0}(x) = x+1 f _{1}(x) = f_{0}(f_{0}(f_{0}(......f_{0}(x)).....))) nested x times = 2x f _{2}(x) = f_{1}(f_{1}(f_{1}(......f_{1}(x)).....))) nested x times = x*2^x etc. f _{w}(x) = f_{x}(x) f _{w+1}(x) = f_{w}(f_{w}(f_{w}(......f_{w}(x)).....))) nested x times etc. f _{w2}(x) = f_{w+x}(x) f _{w3}(x) = f_{w2+x}(x) etc. f _{w^2}(x) = f_{wx}(x) f _{w^2*2}(x) = f_{w^2+w^2}(x) f _{w^3}(x) = f_{w^2*x}(x) etc. Those are just a few examples that only scratch the surface of what the fast-growing hierarchy has to offer - Googology Wiki's article is helpful for understanding how the fast-growing hierarchy works. and I plan to have several articles covering the notation starting in section 3.This number is a staggering 33-digit number equal to about 1.276 nonillion, showing how powerful even the beginning of the fast-growing hierarchy is - f
_{3}(x) can produce truly gigantic numbers with input values as small as 4! f_{3}(3) and f_{3}(4) can be found later on this list, and larger values of the fast-growing hierarchy can be found on the list as well. ~ 1.4168*10^32This is the value of the Planck temperature in degrees Kelvin (degrees Celsius above absolute zero). Unlike the Planck length and Planck time, the Planck temperature is the
highest possible measurable temperature. It's equal to about 141.68 nonillion degrees Kelvin, and about the same amount of degrees Celsius since the Celsius scale is just offset 273 units from the Kelvin scale.The Planck temperature was the temperature of the universe a Planck time after the Big Bang, the same time the forces started to become distinct.
M_{107} / Eleventh Mersenne prime 2^107-1 ~ 1.623*10^32 This number is the eleventh Mersenne prime, equal to about 1.623 nonillion. It was discovered in 1914 by Ralph Ernest Powers. It was the second largest known prime for forty years, until it was knocked down to third in 1951. Planck temperature in degrees Fahrenheit~ 2.5502*10^32This is the Planck temperature in degrees Fahrenheit (see this entry for more).
Decillion10^33 A decillion is 10 ^{33}, known as a quintilliard in the long scale. It was the smallest -illion that was extrapolated from Chuquet's names (read this page for details). It's one of my personal favorite illions, along with the vigintillion, and it was the largest my mom taught me as a kid.Walt Whitman has used numbers up to a decillion in his poetry, showing that numbers this big, even as names, have appeared in literature. A decillion is also the last -illion recognized in many spell checks, showing that much like a quintillion it serves as a sort of cut-off point for -illions.
Examples of decillions: The sun weighs 2 decillion grams. The area of the Milky Way Galaxy is 203 decillion cubic meters. Also, a decillion gallons of water in a sphere would be 19 million km wide, 15 times larger in diameter than the sun itself! In my extended SI prefix system, the prefix for a decillion is geopakila-, combining geopa- (a nonillion) + kila- (modifying kilo- for 1000). A decillion is also (according to Clifford Pickover) the largest power of 10 that can be expressed as a product of two numbers with no zero digits: 2
^{33}*5^{33} = 8,589,934,592*116,415,321,826,934,814,453,125 = 10^{33}.31,691,265,005,705,730,374,175,801,344,000,001This number is the smallest known factor of a googolplex plus one (see that entry for more). It's equal to about 31.7 decillion.
Goby / one hundred decillion10^35 A goby is defined by Sbiis Saibian as 1 followed by 35 zeros, alternately named 100 decillion. It's ten billion times larger than a minnow. To get an idea of how ridiculously big it is, a sphere with a goby gallons of water would be 89 million kilometers (O_o) wide, and that sphere, if it were centered at the sun, would fit just within the orbit of Mercury. That goby-sphere would be just a little smaller than the giant star Rigel, which is 80 times bigger in diameter than the sun. A minnow seems tiny now, doesn't it? And you ain't seen nothing yet! (see gogol)
Undecillion10^36 An undecillion (sextillion in the long scale) is 10^36, or a trillion trillion trillions. It's the first non-”primitive” -illion, constructed out of a the root un- for one plus decillion. The volume of the supergiant star Betelgeuse is 1.2 undecillion cubic meters. The electromagnetic force is an undecillion times more powerful than the gravitational force between two protons. Two undecillion2*10^36This number is notable as part of a lawsuit, where someone sued the company Au Bon Pain for this many dollars. It's an INSANE amount of money - even the total economic production of the human race so far is not even a
quintillionth as big as this number. Even if you had an Earth or even a sun made of gold that would cost much less than this many dollars. Randall Munroe wrote a blog post on his "What If?" blog that goes into detail on how much money those 2 undecillion dollars here.
Twelfth Mersenne prime / M_{127}2^127-1 ~ 1.7014*10^38 This number is the twelfth Mersenne prime, equal to about 340 undecillion. It was proven to be prime in 1876 by Edouard Lucas, before the previous three Mersenne primes were discovered, so it was the largest known prime for 75 years until it was broken in 1951 by a non-Mersenne prime. This number is a member of Catalan's sequence (see 127), and is a double Mersenne number (a number of the form 2 ^{M}-1 where M is a Mersenne prime) - that's why it was proven to be prime so much sooner than the previous Mersenne primes. It's the largest known prime double Mersenne number.
Fugafour((4^4)^4)^4 = 4^64 ~ 3.4028*10^38 This number is formed by applying the fuga- prefix to 4. It evaluates to 4^4^3 = 4^64 ~ 3.4028*10^38 - it's amazing that by plugging 4 into this function we already end up with this huge number! This number can also be expressed as 2^128 or as 2^2^7, and is equal to about 340 undecillion. It's almost exactly twice as large as the previous number.
Duodecillion10^39 A duodecillion (sextilliard in the long scale) is equal to 10^39. Jonathan Bowers strangely calls this number "doedecillion". This is also the first -illion listed on Wiktionary as rare - it's truly rare to actually use these number names. The Great Lakes have 53 duodecillion water molecules.
π^^π (by linear approximation) ~ 2.7777166 * 10^40 Equal to π^π^π^π^(π-3). After the progression e^^e ~ 2380, π^^e ~ 5,000,000, e^^π ~ 19,000,000,000, it may come as surprising that π^^π is not equal to something like sixty trillion as one might expect, but a 41-digit number, about 27 duodecillion! It's even crazier because even though pi is a little more than three, pi^^pi is bigger than the cube of 3^^3. This number is a great example of the counter-intuitivity of tetration (I myself expected pi^^pi to maybe have about 20 digits at most), but hold on now ... we are just getting started with really big numbers!
Tredecillion10^42 A tredecillion (septillion in the long scale) is equal to 10^42. Earth’s atmosphere contains 200 tredecillion molecules. All the stars in the Milky Way weigh 100 tredecillion grams combined.
(2^148+1)/17~ 2.098*10^43 This number was proven to be prime in 1951, breaking the previous record of M127 after a long 75 years. It's notable for not being a Mersenne prime, and the last record-breaking prime number to be discovered without a computer. It was proven to be prime with a variant of Fermat's Little Theorem (see 142,857).
One hundred tredecillion / Zai10^44 This number is called "zai" in the modern Chinese numeral system. It's the largest officially recognized Chinese number name (for details on Chinese numerals see 10,000) See also vigintillion, centillion, and 10 ^{4096}.Quattuordecillion10^45 A quattuordecillion (septilliard in the long scale) is equal to 10^45, and yes this is the official name for this number - quattuor means four and deci means ten. With the unwieldy names here, you can see why scientists prefer scientific notation for numbers this big. There are about 42 quattuordecillion water molecules in all of Earth's oceans. A quattuordecillion is the largest number of dimensions namable in Jonathan Bowers' n-dimensional figure naming system which continues polygon for 2 dimensions, polyhedron for 3 dimensions, polychoron for 4 dimensions. A 10^45 dimensional figure has the name in Bowers' system: polynonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxenovnonecxenirm nonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonecxenon
Gogolbyte 8^50 = 2^150 ~ 1.427*10^45 This number is the -byte version of a gogol, equal to 1.427 quattuordecillion, 8 to the 50th power, or 2 to the 150th power. It's a ridiculously big number but still small by googology standards.
Number of arrangements of a 4x4x4 Rubik's cube~ 7.401*10^45 A 4x4x4 Rubik's cube can be arranged in far more ways than a normal Rubik's cube - the value is on the scale of very scarcely used -illion names, and on the high astronomical level. To get an idea of this number's size, a sphere of this many gallons of water centered at the sun would exceed the orbit of Pluto at its furthest point from the sun 41.4 times.
Quindecillion10^48 A quindecillion (octillion in the long scale) is equal to 10^48. It is equal to a trillion trillion trillion trillions! It's sometimes referred to as a quinquadecillion, as this is its name using Conway and Guy's illions. It's a really awesome number in my opinion, largely because it's the logarithmic halfway point between two of my favorite -illions, the decillion and the vigintillion.It's been estimated that the Earth is made of about 89 quindecillion molecules (see 2 entries later). 5.2320*10^49This is Robert Munafo's upper bound on the number of possible chess positions. He says on his number list that it allows 2 to 32 pieces on the board, including 1 king of each color, and up to 8 pawns (unlike with other estimates, Munafo allows pawn promotion). This is equal to about 52 quindecillion and it's comparable to a
gogol.The number of possible chess games is much larger - you can look at my upper-bound here.
8.9*10^49This number is 89 quindecillion, and it's an estimate of how many molecules make up the Earth itself. It's a good benchmark for large numbers when comparing them against the physical world - this number, on the scale of atoms, can be seen as the boundary between "dwarfed by Earth" and "dwarfing Earth".
Gogol / Lcillion10^50 This number is equal to 10 to the 50th power, or the square root of a googol. It's the logarithmic halfway point to between one and a googol and it can be named 100 quindecillion with the short scale of illions. Sbiis Saibian gives this number a name, "gogol".To get a sense of how big the gogol is, imagine a gogol gallons of water in a sphere. If that sphere is centered around the sun, the sphere of a gogol gallons of water would exceed Pluto's orbit, 613 times!! That is just RIDICULOUSLY bigger than the measly goby, and the minnow and guppy now look super-tiny. The next googolism of Sbiis Saibian's is a jumbo shrimp. This number has also been named lcillion (pronounced /el-sil-yun/), a name based on the Roman numeral "L" for 50.
Here's another interesting thing about 10
^{50}: For a long time it was unknown if you could cut a circle into a finite number of pieces and assemble them to form a square of the same size. The problem was first solved with a decomposition into about 10^{50} pieces, making heavy use of the axiom of choice and the pieces are not pieces the way you can cut them with scissors. Nonetheless, what's interesting is that this gives an example of a problem whose premise a child could understand gives an example of a fairly large number.
Sexdecillion10^51 A sexdecillion (octilliard in the long scale) is equal to 10^51. In Conway and Guy's illions it's known as a sedecillion. The Milky Way has a volume of a sexdecillion cubic kilometers.
Septendecillion10^54 A septendecillion (nonillion in the long scale) is equal to 10^54, or a quintillion quintillion quintillions. The Pleiades star cluster has a volume of about 250 septendecillion cubic centimeters.
Octodecilion10^57 An octodecillion (nonilliard in the long scale) is equal to 10^57. The Pleiades star cluster may have about 800 octodecillion hydrogen atoms. The sun is made of about an octodecillion atoms.
Novemdecillion10^60 A novemdecillion (decillion in the long scale) is equal to 10^60. In Conway and Guy's illion system it is known as a novendecillion. The Milky Way is about 150 novemdecillion cubic feet, and the universe is 8.03 novemdecillion Planck times old.
In my extended SI prefix system, a novemdecillion, being the 20th power of 1000, has the next prefix after geopa- to get a unique name - the SI prefix in my system for a novemdecillion is amosa-.
Vigintillion10^63 A vigintillion (decilliard in the long scale) is 10^63. It is a personal favorite of mine, along with the decillion. A vigintillion is notable because it is the second largest officially recognized -illion - the largest is a centillion, and there are no canonical -illions between a vigintillion and a centillion. Numbers beyond this have unofficial but often recognized names, that are not officially part of the English language (though they might as well be). The closest we can get to naming them using only what is recognized as part of English language is (for example) calling 10^66 a thousand vigintillion, and combining names from there. That kind of thing is pretty common in English - less familiar -illions can be expressed in terms of more familiar -illions, e.g. million trillion trillion in place of nonillion. A vigintillion is also used in the calculation of the ever-growing number called the lynz. The Virgo supercluster (where we live) has a volume of 3.5 vigintillion cubic kilometers. Personal: I read about this number in a math book for kids in second grade at school - one of the things, together with a description of the googol and googolplex, was as a table of the -illions. The googol and googolplex stayed in my head due to their fascinating size, but only a few -illion numbers did. Two -illions that did remain in my head were vigintillion and trigintillion. I actually thought they were vingtillion and tringtillion until 2013. A vigintillion is still one of my favorite illion numbers.
Jumbo shrimp / one hundred vigintillion10^65 A jumbo shrimp is another jump from the gogol, defined by Sbiis Saibian as equal to 10 to the 65th power. A jumbo shrimp gallons of water would be about as big as the Milky Way! It's very hard now to even appreciate these numbers, and STILL haven't reached a googol!
10^66-1 This is the largest number that can be counted to in English if combining names of powers of 1000 (like a thousand vigintillion for 10^66) is not allowed. It representes the problem discussed in the entries for 54 and a vigintillion. It's named 999 vigintillion 999 novemdecillion 999 octodecillion ............ 999 million 999 thousand 999.
Unvigintillion 10^66 This is the smallest power of 1000 that doesn't have an official name in English. Using only canonical -illions we would have to call it a "thousand vigintillion", though at this point most people are fine with just calling it "ten to the power of sixty-six". However, people interested in extended -illions refer to this number as an unvigintillion, an obvious continuation from "vigintillion". In the long scale this is known as an undecillion. Our galaxy has an unvigintillion atoms.
2*10^66+5 This number is the number of possible integers namable in English if combining powers of 1000 (like thousand vigintillion for 10^66) is not allowed. If you sort all of those numbers in numerican order, it would look like this: -googolplex, -centillion, -googol, -10^66+1 ......... -4, -3, -2, -1, 0, 1, 2, 3, 4 ......... 10^66-1, googol, centillion, googolplex
Duovigintillion 10^69 10^69, known as a million vigintillion using only canonical illions, is often known by number enthusiasts as a duovigintillion. In the long scale it's known as an undecilliard. The volume of the observable universe is roughly a duovigintillion cubic miles. The Virgo Supercluster may have 200 duovigintillion hydrogen atoms in all of its stars.
Trevigintillion 10^72 10^72 is known as a duodecillion in the long scale, or a trevigintillion using the extended short scale. The Sloan Great Wall superstructure in the universe hasa volume of about 2 trevigintillion cubic meters. It's one of the largest structures in the observable universe, and we are not in it.
Calvin and Hobbes Gazillion10^74 A "gazillion" is a slang term that refers to any indefinitely large number. However, the famous comic strip Calvin and Hobbes hints at a definition for a gazillion: Calvin: Psst! Susie, what's 7+6? Susie: Three hundred billion gazillion. Calvin: Oh, thanks for the big help! Susie: That's a three, followed by 85 zeroes. Calvin: Ah! I knew that. In this case, gazillion would be equal to 10^74, which unlike the regular -illions doesn't have a number of zeroes divisible by three; clearly, Bill Watterson isn't much of a googologist. A gazillion, being an indefinite -illion used figuratively to represent a large amount, would probably be much larger than this if it were to be a real number. Really, I’d say even a zillion is bigger than this. Also, if 10^74 is a gazillion, then we can define “gaz” as 23 ⅔, since a gazillion can be thought of as the "gaz"th -illion. For another proposed definition of a gazillion see 10^86,340.
Number of arrangements of a 5x5x5 Rubik's Cube ~ 2.827*10^74 This is the number of arrangememnts of a 5x5x5 Rubik's Cube. It's equal to about 282 trevigintillion, and is a staggeringly large number - it's far far more than the number of atoms in Earth, or even in the entire solar system - hell, it's more than the number of atoms in our entire galaxy! To get an idea of this number's size, a sphere of this many gallons would have a diameter of 108 million light years, which is a thousand times larger than the diameter of our galaxy, and somewhat larger than the distance from us to the Eridanus galaxy cluster. Rubik's Cube numbers:
Lightweight / Quattuorvigintillion10^75 10^75 is known as a duodecilliard in the long scale and a quattuorvigintillion in the extended short scale. Sbiis Saibian gives this number the alternate name "lightweight". The number is so big that there aren't a lot of examples of this insane number's size. A sphere of this many gallons of water would have a diameter of 204 million light years, dwarfing even the Virgo Supercluster of galaxies! The value seems god-like now, but hold on.... we still have 25 orders of magnitude just to reach a googol!
Twelfth perfect number ~ 1.44701*10 ^{77 }After 2 ^{127}-1 was proven to be prime in 1876, it quickly followed that this number, which is equal to 2^{126}*(2^{127}-1), was a perfect number. This easily disproved Barlow's assertion that 2,305,843,008,139,952,128 would be the largest perfect number anyone would know.
Quinvigintillion 10^78 10^78 is known as a tredecillion in the long scale and a quinvigintillion in the extended short scale. This is fairly close to the number of atoms in the observable universe.
180*(2^{127}-1)^{2}-1 ~5.21*10^78 This number was proven to be prime in 1951 shortly after the previous record-breaker, and once again it's not a Mersenne prime. It was the first prime number discovered with an electronic computer, using EDSAC, an early computer in Cambridge University.
Ogol10^80 An ogol is defined by Sbiis Saibian as equal to one followed by 80 zeros. As with the guppy and gogol, he says this number can be seen as a contraction of the googol. This many gallons of water is a sizable fraction of the observable universe, taking up about 1/1000 of it. This means that we'll only need a thousand ogol gallons of water to fill the observable universe!
10^80 is also the standard estimate the number of atoms in the observable universe - despite its unimaginable size, it's still well under a googol (100 quintillion times less). Large numbers are often compared to this figure, usually to show how easy it is to dwarf insane figures like this just with some simple large numbers, such as 9 ^{9^9}.Sexvigintillion 10^81 10^81 is known as a tredecilliard in the long scale and a sexvigintillion in the extended short scale. The observable universe has a volume of a sexvigintillion cubic feet. It’s more than the number of atoms in the observable universe because most of it isn’t packed with atoms.
Septenvigintillion 10^84 10^84 is known as a quattuordecillion in the long scale and a septenvigintillion in the extended short scale. The observable universe has a volume of a septenvigintillion cubic inches.
Tiny twerpuloid10^85 Sbiis Saibian defines a tiny twerpuloid to be 10 to the 85th power, or a quadrillionth of a googol. This many gallons of water would finally dwarf the observable universe, being since a sphere of that many gallons would be about 5 times as large in diameter! So NOW we're getting somewhere with large numbers. However we still are 15 orders of magnitude away from a googol!
Calvin and Hobbes Number
3*10^85 In a Calvin and Hobbes comic, Susie told Calvin during a math test that 300 billion gazillion is equal to 3 followed by 85 zeroes; I call this number the "Calvin and Hobbes number". See 10^74, the implied value of a gazillion in that Calvin and Hobbes comic, for more. Though I'm pretty confident it's the largest number in Calvin and Hobbes, this number isn't the largest number to appear in comics. It wasn't even the largest during the strip's publication - the record had already been broken twenty-something years ago in a Peanuts comic strip with the googol.
Octovigintillion 10^87 10^87 is known as a quattuordecilliard in the long scale and an octovigintillion in the extended short scale. The observable universe has a volume of an octovigintillion cubic millimeters.
Googolspeck / Novemvigintillion10^90 10^90 is known as a quindecillion in the long scale and a novemvigintillion in the extended short scale. Sbiis Saibian gave 10 to the 90th power the name "googolspeck". You will find out why it's called a googolspeck in a little bit. For now the only thing there is to say about a googolspeck is that it's ten billion times less than a googol!
Googolbyte 8^100 = 2^300 ~ 2.037*10^90 This number, coined by Sbiis Saibian, is eight to the 100th power. It's about twice as large as a googolspeck. It's just under a googol in exponential terms, and written out it's almost as long, but it's still 4.9 billion times less than a googol!
Trigintillion10^93 10^90 is known as a quindecilliard in the long scale and a trigintillion in the extended short scale. The name comes from Latin triginta meaning 30, since it's the the 30th -illion. It is logarithmically close to a googol (though it's actually ten million times smaller). It is continued with untrigintillion, duotrigintillion, etc. To get an idea of how big a trigintillion is, consider this: if you divided the observable universe into a trigintillion equally sized pieces, each would be half as small as the width of a human hair! Personal: A trigintillion was probably the largest -illion number (along with the vigintillion) I knew before 2013; I used to think it was spelled "tringtillion" (see vigintillion for how I knew of the vigintillion and trigintillion). A trigintillion is the largest number that I consider to have any sort of childhood significance.
Googolcrumb10^95 A googolcrumb, which can alternately be called one hundred trigintillion, is defined by Sbiis Saibian as equal to 10 to the 95th power. It's 100,000 times smaller than a googol.
Untrigintillion 10^96 10^96 is known as a sexdecillion in the long scale and an untrigintillion in the extended short scale. This number is 10,000 times smaller than a googol.
Googolchunk / Duotrigintillion10^99 10^99 is known as a sexdecilliard in the long scale and a duotrigintillion in the extended short scale. It's is the largest -illion under a googol, and Sbiis Saibian named it a googolchunk. This number is exactly 10% of a googol, and a sphere of this many gallons would be about 46% as large in diamater as a googol sphere. To get an idea of what that means, look at this picture (credit goes to Sbiis Saibian for image and description): This image shows us just why Sbiis Saibian chose the names: he has said that while a googolchunk is a sizable chunk of a googol, a googolcrumb is just a little crumb of a googol, a googolspeck is just a speck next to a googol, and a tiny twerpuloid looks adorably small, as does the observable universe.
9.999...*10^99 This number, just under a googol, is the overflow value of most scientific calculators. Some more expensive calculators overflow at 9.999...*10^999, just below a googolchime. The Googol Range10^100 ~ 10^999Entries: 45
Googol10^100 Main article: Googol and GoogolplexA googol is a well-known large number equal to 1 followed by 100 zeros. Here it is written out in full: 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
It was coined in 1928 by 9-year-old Milton Sirotta as a name for 1 followed by 100 zeros, and it was shared to the world in 1936 by his uncle Edward Kasner along with its infamous big brother, the googolplex. Today it is the single best known googolism and the quintessential large number. It can also be given a more techinical name, ten duotrigintillion. The googol become very significant in our day, and a lot of googology originated from it, including the name "googology" AND the whole idea of naming large numbers. In some sense this
is the smallest googological number.To get an idea of the size of a googol, imagine dividing the observable universe, with a volume estimated to be 3*10
^{80} cubic meters into a googol equal-sized portions. Then each portion would be 310 nanometers wide, which is very small, comparable to the size of some of the smallest known bacteria. Thus we can say that about a googol tiny bacteria would fill up the observable universe. For a more detailed review of how much a googol is, read my article on the googol and googolplex.See also googolplex.
Googolteen 10^100 + 10 Fourteen is 10+4, and fifteen is 10+5. You can extrapolate this to get a suffix -teen which you can apply to any number n, defined as 10+n. Andre Joyce applied this -teen suffix to the googol, creating a humorous number called googolteen. Here is googolteen written out in full: 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,010
Googol plus thirty-three10^100 + 33The smallest semiprime (product of two prime numbers) larger than a googol.
Gooprol 10^100 + 267 This is the smallest prime number larger than a googol. Its name was coined in the inactive blog "A googol is a tiny dot" - see also the next few entries and gooprolplex. Here is gooprol written out in full:
10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,267
Booprol 10^100 + 949 The second smallest prime number larger than a googol.
Trooprol 10^100 + 1243 The third smallest prime number larger than a googol.
Quadrooprol 10^100 + 1293 The fourth smallest prime number larger than a googol. For its magnitude it's unusually close to the previous prime.
70! ~ 1.198*10^100 This number is 70 factorial. It's notable for being the smallest larger than a googol, and it's quite close to the googol in size.
57^57 ~ 1.216*10^100 The smallest number equal to a whole number raised to its own power which is larger than a googol.
8*10^100+1 This number is equal to 8 times a googol plus 1. It is the number of letters (without the space) in the name for a googolduplex in Conway and Guy's -illion system - for more on that system see here.
Googolbunch / Googolty 10^101 = 10^100 * 10 Forty is 10*4, and fifty is 10*5. We can extrapolate this to get a suffix -ty, which you can apply to any number n to multiply it by ten. Andre Joyce applied this prefix to the googol forming this humorous number along with googolteen. Sbiis Saibian gives the name "googolbunch" for this number, in analogy to the -speck, -crumb, and -chunk suffixes but with naming multiples of a number instead of fractions of the number. It's equal to exactly ten times a googol.
Googolcrowd 10^105 Once again, this number by Sbiis Saibian is an analogy to the -speck, -crumb, and -chunk suffixes. It's equal to exactly a hundred thousand times a googol, and therefore it can be thought of as a "crowd of googols".
Googolswarm / Eleventyplex 10^110 This number has been named "eleventyplex" by Andre Joyce as an example of one of the many numbers that can be named with the -plex suffix. Sbiis Saibian gives it another name, "googolswarm" - here, "swarm" means multiplying by 10 billion, in contrast with -speck which divides a number by ten billion. This number is a swarm of googols in comparison to the googol, and makes a googolspeck look unimaginably small. This difference is only twenty orders of magnitude, and pretty soon we'll be blasting into the stars where the differences between two numbers on this list have no analogy we can use at all! As rule number 1 of googology says, it only gets worse!!! This number is also an estimate of how many atoms would be needed to fill up the observable universe - therefore after this point we can no longer use a "sphere of this many atoms" analogy to capture the size of these numbers.
Number of ways a 6x6x6 Rubik’s cube can be arranged1.57*10^116 This is the number of ways a 6x6x6 Rubik's cube can be arranged, and it's notable for being larger than a googol. We can say that if you ever get your hands on a 6x6x6 Rubik’s cube, you have a real exmple of a number over a googol, right at your fingertips! To get an idea of how big this value is, gallons of water will no longer help, since this many gallons of water easily dwarfs the observable universe. So we'll need to describe its size some other way, by dividing the universe into this many portions. But even that is a very difficult analogy. Helium atoms are the smallest out of all atoms, and if this many helium atoms filled up the observable universe, we STILL wouldn't reach this gigantic number. So we need to put this number in terms of something smaller like, say, protons. This many protons in a sphere would be about 115,000 light years, bigger than the Milky Way galaxy. Rubik's Cube numbers:
Common estimate of all possible chess games 1.47*10^118 This number, equal to 1.47 octotrigintillion or about 1.5 quintillion times a googol, is a common estimate of al possible chess games. It was estimated with very simple methods and guesses, assuming that each player gets 40 turns and that on average there are 30 moves you can make, giving you 30^80 ~ 1.47*10^118. My upper-bound on the number of possible chess games is much larger, and likely a more reasonable esitmate considering that a game can easily last much longer than 80 turns.
Quadragintillion 10^123 The unofficial 40th -illion, from Latin "quadraginta" meaning 40. The observable universe has a volume of 47 quadragintillion cubic femtometers. Protons and neutrons are about a femtometer wide. This is also known as a "vigintilliard" in the long scale, the largest power of 1000 to get its own name in the long scale other than the much larger long-scale centillion and centilliard.
Great googol (alternate) 10^150 An alternate definition of Joyce's great googol (see 10 ^{1000}). Logically, this should be the great googol, since a great gross is a gross raised to the 1.5th power, and a googol raised to the 1.5th power is 10^{150}. However, defining it as 10^{1000} is more aesthetically appealing and still makes sense in a way, if you consider how the exponents compare. This is equal to a novemquadragintillion in the -illion system.
Quinquagintillion 10^153 The unofficial 40th -illion, from Latin "quinquaginta" meaning 50. This is approximately the volume of the observable universe in cubic yoctometers (the smallest metric unit of volume).
4^4^4 This is 4^^3, a smallish tetrational number notable for being larger than a googol. This is a better tetrational number but still not quite great, and logarithmically not too far from a googol. It's approximately 13 quinquagintillion and it has exactly 154 digits. Written out in full it's equal to:
13,407,807,929,942,597,099,574,024,998,205,846,127,479,365,820,592,393,377,723,561,443,721,764,030,073,546,976,
801,874,298,166,903,427,690,031,858,186,486,050,853,753,882,811,946,569,946,433,649,006,084,096
M_{521} / Thirteenth Mersenne prime ~ 6.865*10^156 This number is the thirteenth Mersenne prime, discovered on January 30, 1952 by Raphael M. Robinson. It is 157 digits long, and it's the smallest one with more than 100 digits. It held the record for largest known prime for a few hours (?) until Robinson discovered another prime number that same day. Five Mersenne primes were discovered in 1952, all by Robinson.
Number of arrangememts of a 7x7x7 Rubik's Cube~ 1.95*10^160 This is the number of arrangements of a 7x7x7 Rubik's Cube. The number has exactly 161 digits and it's equal to about 19.5 duoquinquagintillion. It's really too big for us to actually comprehend. If you do want an idea of its size, this number is so big that this many cubic yoctometers would fill up ten million observable universes. However, it’s still less than the volume of the observable universe in Planck volumes. Rubik's Cube numbers:
Sexagintillion 10^183 The unofficial 60th -illion (from Latin "sexaginta" meaning 60). 300 sexagintillion is approximately the number of Planck volumes in the observable universe - see next entry.
Observable universe in Planck volumes 3*10^185 A Planck length is about 1.6*10 ^{-35} meters, or 16 geptomicrometers with my extension of SI prefixes, and it's considered the smallest physically meaningful length; a Planck volume is a cubic Planck length. By measuring the volume of the whole observable universe in Planck volumes, we get what is arguably the largest number that actually exists. What's interesting about this figure is that it easily shows us that as big as a googol is, it still has tangible meaning.In a sense this is the largest number that actually exists in the physical world. Beyond this point it's debatable whether any number actually "exists", though the physical world allows us to generate numbers far larger than this as we will see. See also the quartic hypervolume of the universe in Planck units.
Gargoogol / Fzhundred 10^200 A gargoogol is the square of a googol, formed by applying the gar- prefix (squares the number it's applied to) to a googol. For more on the gar- prefix see gargoogolplex. A gargoogol is also equal to a fzhundred, using the fz- prefix which takes a number to its own power. It's interesting to note that the fz- prefix achieves the same value as the gar- prefix does for a much larger number.
Septuagintillion 10^213 The unofficial 70th -illion (from Latin "septuaginta" meaning 70). 47 of this would be the volume of the universe in Planck volumes 150 quintillion years after the Big Bang.
Googoc 200^100 ~ 1.2676*10^230 This is the best-known example of a number named with the googo- prefix, a prefix that Andre Joyce made from a curious observation about the googol. He noticed that googol can be expressed as 100^50, and googol ends with a letter L, which is 50 in Roman numerals. Therefore, he made a prefix googo-, wheere googo-x is (2x)^x and x is in roman numerals. For example, googoc = (2*100)^100 = 200^100 ~ 1.27*10^230, since C is 100 in Roman numerals. The number has exactly 231 digits and falls between a septuagintillion and an octogintillion.If I were to only mention one of the googo- numbers, this would be it. Interestingly, it's equal to the product of two other googol-related numbers (googolbit and gargoogol).
Octogintillion 10^243 The unofficial 80th -illion (from Latin "octoginta" meaning 80). 47 of this is the volume of the universe in Planck volumes 1.5 nonillion years after the Big Bang. Some time after that point, the universe may start decomposing, if the theory about proton decay is true.
Quartic hypervolume of the observable universe in Planck hypervolumes (birth-present) 8.01*10^245 Quartic hypervolume is like volume, but it adds time as a 4th dimension. We can get some really big numbers using quartic hypervolume in Planck lengths*Planck lengths*Planck lengths*Planck times, for example of the universe from birth to present. Here’s how I got to this tremendous number: 1.855*10 ^{43} Planck times/second * 3.1536*10^{7} seconds/year * 1.37*10^{10} years of the universe * 3*10^{185} Planck volumes/universe = 2.40*10^{246} quartic Planck unitsBut the universe has expanded since it started, akin to a 4-dimensional hypercone. The volume of a 4-dimensional hypercone is ⅓π*r^4*h, so we need to divide all that by 3 to get this still enormous number.
Nonagintillion 10^273 The unofficial 90th -illion (from Latin "nonaginta" meaning 90). By the time the universe is this many Planck volumes it may have already decayed if the theory of proton decay is correct. But we could still get real-world numbers this big using quatrtic hypervolume of the universe, not to mention the numbers we could get from probability.
2^1000 ~ 1.0715*10^301 2^1000 is used in the definition of a transcendental integer, an exotic range of uncomputable huge numbers devised by Harvey Friedman. Currently, no integers have been proven to be transcendental. I honestly have no clue why 2^1000 was chosen - may have to do with binary numerals (see the page for 2), but don't ask me why it was chosen.
Centillion10^303 A centillion is the largest official -illion, exactly equal to 1 followed by 303 zeros. For a long time it was the last number to get an official name in the English language, before the googol and googolplex came along. It's still the largest -illion that gets an official name. It's unusual that a centillion gets an official name but no -illions between a vigintillion and it does. Nonetheless a centillion is a pretty awesome number. A centillion is close to too large to represent anything in the universe and much larger than ANYTHING you’ll see in science, but it’s still somewhat in the practical number range. This number is a bit more logarithmically than the number of ways 1000 coins can land (2^1000), and it’s smaller than the probability of certain unspeakably rare occasions as we will see.
Sbiis Saibian has strong personal connections to a centillion - while reading the dictionary, he was fascinated by the names of large numbers, especially the centillion. Therefore a few of his googologisms, like ecetonplex, are based on a centillion. A way to get the idea of the size of a centillion is to consider quartic hypervolume, which is like volume but adding time as a 4th dimension. In that case, from the start of the universe to 3.778 septillion years in the future, the quartic hypervolume of the universe is around a centillion quartic Planck units.
Largest number in double floating point~ 1.7976931*10^308 This is a number with special significance in computing: it's the largest integer that can be stored in the commonly used IEEE754 double-floating-point system, and therefore the largest number storable in double floating point in Java and several other programming lanauages. It's the largest number that can be directly stored in most programming languages at all. This gives it quite some significance in many Javascript applications, such as Google's built in calculator, and indirectly in Robert Munafo's Hypercalc. The value is defined like this because 53 bits are used to represent the part before the exponent (e.g. 3.5 in 3.5*10^40), 10 to represent the exponent, and one used to say if it's positive or negative, adding up to 64 bits but far more wide ranged than just plus-minus nine quintillion. This number, therefore, is 1.11...(53 1's)...111*10^1111111111 (10 1's's in the exponent) in binary, or just under 2 ^{1024}. It can therefore be expressed in base 10 as ((2^{53}-1)/2^{52})*2^{1023}.Current upper-bound to Skewes' problem1.397*10^316This is the current upper-bound to the solution of the problem of what the first number where π(x) > li(x), the problem which gave birth to the very large outdated upper-bound known as Skewes' number (see that entry for details). The current lower-bound to the solution is 10
^{14} or 100 trillion.Quartic hypervolume of the observable universe (birth to proton-decay death) 5.07*10^341 From birth to proton-decay death of the universe (assuming the theory of proton decay), the quartic hypervolume of the universe would be 507 duodecicentillion quartic Planck units. Sbiis Saibian has said that this number is important because it's probably the limit of what we can actually use to measure the universe. Beyond this, numbers can only be physically interpreted in combinatorics and listing possibilities, as well as hypothetical sizes of the whole universe.
Faxul 200! ~ 7.886*10^374 788657867364790503552363213932185062295135977687173263294742533244359449963403342920042840119846 239041772121389196388302576427902426371050619266249528299311134628572707633172373969889439224456
214516642402540332918641312274282948532775242424075739032403212574055795686602260319041703240623
5170085879617892222789623703897374720000000000000000000000000000000000000000000000000 in full.
This number was coined by Lawrence Hollom, creator of hyperfactorial array notation, a notation which extends the factorial into the realm of googology, taking unusually powerful approaches. It's the smallest of Hollom's googolisms, and it's the only one that can be written out in decimal, as shown above. This number, being the factorial of 200, represents the number of ways 200 books/apples/whatever can be ordered. Its name was chosen arbitrarily, akin to the googol. Fugafive5^5^4~ 7.1821*10^436This is the fifth member of the fuga- series, a function which achieves staggering hyper-exponential growth. It has exactly 437 digits, and as such it falls between the faxul and the output of pete-3.c.
Output of pete-3.c9*2^1566 ~ 2.3292*10^472This is the output of the smallest entry in Bignum Bakeoff (the 2001 competition of who could write the 512-character C program with the largest output) that produces a number other than 1. It is a simple program submitted by a guy named Pete that "takes 9 and shifts it left 163 times by 9 and once by 99" (according to David Moews, the man who held Bignum Bakeoff), producing an exponential sized number. Pete submitted nine different entries (including this one, pete-3) to Bignum Bakeoff, the most of anyone who participated. pete and pete-2 didn't terminate, but pete-4, pete-5, pete-6, and pete-7 produced some very large numbers ranging from w2 to w^w in the fast-growing hierarchy, or from 4-entry arrays to high linear arrays. pete-8 and pete-9 were meant to be refinements, but a bug caused them to only be in the high tetrational range.
Bignum Bakeoff entries:
< (n/a) |
pete-3 | pete-9 >
Googolding 10^500 A googolding is a number coined by Sbiis Saibian in analogy to the googolgong (10^100,000), equal to 1 followed by 500 zeros. It's a ridiculously huge number far bigger even the quartic hypervolume of the entire run of the universe, and needs to be compared to entirely different things! If you want an example of its size, the probability that 196 random people in a room all share the same birthday is about 1 in a googolding.
Googocci 402^201 ~ 2.814*10^523 The 201st member of the googo- series - this one is most notable for having an Italian-like name, as stated by Joyce himself.
Tupper's number ~ 4.8585*10^543This number is the 544-digit number that makes Tupper's self-referential formula work - I call it "Tupper's number". Tupper's self-referential formula is a formula that, when graphed, can visually reproduce the formula itself - however upon further examination it's really more of a magic trick. Here's how it works: Start with the formula:
1/2 < [mod([y/17]2
Let N = the 544-digit number:
48584506361897134235820959624942020445814005879832445494830930850619347047088099284506447698655243648499972470249151191104116057 39177407856919754326571855442057210445735883681829823754139634338225199452191651284348332905131193199953502413758765239264874613 39490687013056229581321948111368533953556529085002387509285689269455597428154638651073004910672305893358605254409666435126534936 36439571255656959368151843348576052669401612512669514215505395545191537854575257565907405401579290017659679654800644278291314885 48259914721248506352686630476300
When graphing the set of points (x,y) satisfying the inequality, in the area of the graph where 0<x<106 and N<y<N+17, the graph, when turned upside down, produces the formula:
This may seem amazing, but actually this formula is a lot more like the infinite monkey theorem than a self-referential formula. The formula, as it turns out, can graph anything depending on what value of N you pick!
And also, think back to the instructions for the formula. In particular, take note that it requires making use of a specific 544-digit number, not just the formula itself! Because of this, the graph shown above of the formula is not so much in the formula as it is in the definition of the 544-digit number. This means that the formula isn't so much of a self-referential formula as it is a "magic trick" so to speak.
If you want a formula that actually is self-referential in this sense, check
this out.
Centillion (long scale) 10^600 This is what a centillion is equal to using the long scale instead of the short scale. In the long scale, which is used in Europe but obsolete in English-speaking countries, x-illion = 10^6x instead of 10^3x+3. x-illiard = 10^6x+3. Therefore in the long scale a centillion would be this value instead of 10^303, with about twice as many digits.
Ducentillion (short scale) / Centilliard (long scale) 10^603 This is the largest official long scale -illion/-illiard, and also the unofficial 200th short scale -illion.
256^256 ~ 3.2317*10^616 This number is an example of a very large solution to the four fours puzzle, expressible as (4^4)^(4^4). It isn't quite as big as the gigantic 4^^4, the only solution to transcend a googolplex! 256^256 also shows up in the computation of Steinhaus's mega.
Trecentillion 10^903 The 300th -illion. This number falls just under a googolchime.9.999...*10^999Some calculators max out not at 9.999...*10^99, but at 9.999...*10^999, which is just under a
googolchime. That is quite a wide range, considering that it's more than can be directly stored in most programming languages (see 1.7976*10^{308}). The Chime-Gong Range10^1000 ~ 10^999,999Entries: 41
Googolchime / Joyce's great googol 10^1000 This number is called googolchime by Sbiis Saibian, with the chime suffix similar in spirit to the -gong suffix. It's equal to one followed by a thousand zeros, similar to how a googol is one followed by 100 zeros. It's the tenth power of a googol and the squared of a googolding, and can be thought of as a "bigger version" of the googol just as the guppy, gogol, and ogol can be thought of as "smaller versions". Andre Joyce has decided that since a gross is 12^2 and a great gross is 12^3 and a googol is 10^10^2, a great googol should be 10^10^3. Actually, this is wrong: a great googol, with this logic, should be only 10^150! But since 10^1000 can be considered a valid interpretation, we can let it slide. In fact, Joyce has also defined it not only as 10^150 and 10^1000, but also two more broken variants (10^101 and 10^300). Where do numbers this big crop up in the real world? One place where numbers this big has shown up is in studying the distant future. For example, it's been estimated that somewhere around 10^1500 years in the future may be an age where all matter has turned into iron-56 in the form of iron stars - in that time the stars would be unimaginably far apart from each other and the universe would be a very lonely place.
Quadringentillion 10^1203 The 400th -illion.
M_{4253} / 19th Mersenne prime ~ 1.908*10^1280 This is the nineteenth Mersenne prime, discovered by 1961 by Alexander Hurwitz with an IBM 7090 computer. It's the first Mersenne prime with over a thousand digits.
Googod1000^500 = 10^1500 On the entry for 496, I discuss how even perfect numbers are all known to have a specific pattern, but how there are not known to be any odd perfect numbers. If there are any such numbers, the smallest one is at least this big, i.e. it has more than 1500 digits. What is known about odd perfect numbers (if there are any), e.g. at least 9 distinct prime factors and a largest prime factor greater than 100,000,000, is quite strange results, leading many people to believe that there are no odd perfect numbers. Also, this number can be named googod, the 500th member of the googo- series, with Joyce's googo- naming system.
Quingentillion 10^1503 The 500th -illion.
Sescentillion 10^1803 The 600th -illion.
Septingentillion 10^2103 The 700th -illion.
5^5^5 19110125......08203125 with 2185 digits This is another rather small tetrational number, expressible as 5^^3 using tetration. Here it is written out in full: 19110125979454775203564045597039645991980810489900943371395127892465205302426158030120593865197398502655864401557944622353592127
88673806972288410146915986602087961896757195701839281660338047611225975533626101001482651123413147768252411493094447176965282756
28519673751439535754247909321920664188301178716912255242107005070906467438287085144995025658619446154318351137984913369177992812
74338404315492368555267835963741021053315460313537253257486369091597786903282664591829838152302869365728736914226481312917437621
36325730321645282979486862576245362218017673224940567642819360078720713837072355305446356153946401185348493792719514594505508232
74922160584891291094518995994868619954314766693801303717616359259447974616422005088507946980448713320513316073913423054019887257
00383298012460501970134673971759090273894939238173157869968458997947810680428224360937839463352654228157043028324423855150823164
90967285712171708123232790481817268327510112746782317410985888683708522000711733492253913322300756147180429007527677793352306200
61828601245525424306100689480544658470482065098266431936096038873625851074707434063628697657670269925864995355797631817390255089
13312232947439303439561613283340728316634982581452268620043077990846881038041873683248009038735962129196336025831207816736737425
33322879296907205490595621406888825991244581842379597863476484315673760923625090371511798941424262270220066286486867868710182980
87280256069310194928083082504419842479679205890881711232719230145558291674679519743054802640464685400273399386079859446596150175
25869658114475685100415686877309037124825353438392853975987494584970500382250124892840018265900562512861876299380444073401423470
62055785305325034918189589707199305662188512963187501743535960282201038211616048545121039313312256332260766436236688296850208839
49614283048473911399166962264994856368523471287329479668088450940589395110465094413790950227654565313301867063352132302846051943
43813998105614006525953007317907727110657834941746426847209561346473277485842382748996687550525043942182321913572230540667153733
74248543645663782045701654593218154053548393614250664498585403307466468541890148134347714650315037954175778622811776585876941680
908203125
We're getting into the somewhat better tetrational numbers, although this number can still be realistically computed, as it still has only about two thousand digits. That means that it can be realistically written by hand; when you write 2 digits per second it would take about 18 minutes. Nonetheless it's still a very big number, far beyond what we can appreciate and somewhere between a septingentillion and an octingentillion in terms of the Latin-based -illions.
777^777~ 7.19667*10^2245The 800th -illion.
Nongentillion 10^2703 The 900th -illion.
Fzthousand / novemnonagintinongentillion 1000^1000 = 10^3000 An example of a number nameable with the fz- prefix. It's also the -illion right beofre a millillion.
Millillion10^3003 A millillion is the thousandth -illion, and it's one of the few -illions after the centillion that has a generally agreed-upon name - that's because millillion is the obvious Latin-name continuation from "vigintillion" and "centillion". In fact "millillion" is probably the largest -illion that has an agreed-upon name among number enthusiasts. Even though in Conway and Guy's system it has the slightly different name "millinillion", most other -illion systems either instead opt for "millillion" or use an entirely diferent name.
Milli-untllion 10^3006 This is the 1001st -illion with -illions extended past a millillion - reversing the order of the terms needs to be done for the most logical continuation.
Googom 2000^1000 10^4096This number is known as a
decyllion in terms of Donald Knuth's -yllions (see 100,000,000). In one ancient version of the Chinese numeral system this was known as zai (in modern Chinese zai is 10^44), and zai is the largest officially recognized number name in Chinese. For more on Chinese numerals see 10,000.
Googolbell 10^5000 A googolbell is defined by Sbiis Saibian as 1 followed by 5000 zeros. It's a super-huge number that falls somewhere above a millillion, and it's the fiftieth power of a googol and the fifth power of a googolchime. However it's still less than some very huge combinatorical numbers like the famous Hamlet monkey number! Platillion10^6000A
platillion is a peculiar illion name found on Jonathan Bowers' -illions page. Bowers says that this name was suggested to him but he has no idea where it came from.I don't consider "platillion" to be part of Bowers' illion system, since it's the 1999th illion in the short scale and Bowers' system obviously extends upon the short scale, and the 1999th name shouldn't be a milestone name.
10^6000 would be the value of "millillion" (the 1000th illion) using the long scale of -illions instead of the short scale, so perhaps the person who suggested "platillion" to Bowers had the long scale in mind with those names. In any case, this seems like an arbitrary name that is little more than a peculiarity.
M_{19,937} / 24th Mersenne Prime~ 4.315*10^6001This is the 24th Mersenne prime, a number with exactly 6002 digits. It was discovered in 1971 by Briant Tuckerman on an IBM 360 computer.
This number is notable as the period of the
Mersenne twister (Wikipedia page), which is by far the most widely used pseudorandom number generator. This means that numbers the Mersenne twister sequence generates repeat with a period of this number. This period is an absurdly huge number (even this many Planck times is about 10^5940 times longer than the age of the universe), and it passes many statistical randomness tests, meaning that the numbers it generates are random for all practical purposes.
Dumillillion 10^6003 The 2000th -illion. Fugasix6^6^5~ 8.0191*10^6050This is the sixth member of the fuga- series. It has exactly 6051 digits, and it falls somewhere between a
googolbell and a googoltoll in terms of Sbiis Saibian's googolisms.
10^8196 This number is equal to one myriad undecyllion in terms of Donald Knuth's -yllion system (see this page). One myriad undecyllion was erroneously given by Andre Joyce (most likely actually Michael Halm) as what a googol is named in terms of Knuth's -yllions. However, this value is much larger than a googol - it's more than raising a googol to the 81st power! In Knuth's -yllions, a googol would actually be called one myriad tryllion quadryllion.
Tremillillion 10^9003 The 3000th -illion. We can continue with "quattuormillillion", "quinmillillion", and so on.
Googolpleiji 3^3^9 ~ 1.505*10^9391 This is a number nameable with the googolple- prefix, a prefix that is similar to the googo- prefix, which was invented by Andre Joyce, but has a far more convoluted history. There are three ways to interpret it: 1. The original verbatim way, where googolple-x = x^(2x)^x where x is written in Roman numerals - that way does not work right.
2. The corrected way, where googolple-x = x^x^x^2 where x is written in Roman numerals - this is the way I normally use googolple-.
3. The -ple- idea, where x-ple-y = y^x where y is written in Roman numerals. This is probably the most logical and the one Joyce used in the revised page, but not as interesting as the second way.
With the second way, which I'm using, we can name this number "googolpleiji" (pronounced /goo-gol-play-jee/), which is just a pronunciation respelling of the unwieldy "googolpleiii".
This is the third member of the googolple- series, and an interesting one in my opinion. It begins 15054164... and ends ...17859227. It can be expressed as 3 to the 19,683rd power, and is larger than googomemem (4000^2000) but less than googomememem (6000^4000), and it's 9392 digits long.
Googoltoll / Great great googol 10^10,000 This number is one followed by ten thousand zeros, a number Sbiis Saibian calls googoltoll, from "toll", the sound bells make at a funeral. It's the tenth power of a googolchime and the 100th power of a googol. It's far far larger than numbers we can actually comprehend, but we can try to imagine it as starting with a sphere of a googol objects, then drawfing the sphere by a factor of itself 99 times. That's really impossible to actually understand, since even a googol objects is well beyond what you can really grasp. Therefore a googoltoll can be seen as a number that is, well, pretty fucking huge ... but tiny in comparison to some FAR MORE MIND-BLOWING numbers!!!Andre Joyce continues his great googol with the great great googol, great great great googol, etc, allowing us to call this number great great googol. A googoltoll is also notable for being about as far as we can currently go with testing any single number to be prime - beyond that, special tests need to be used used, which is part of why Mersenne primes are special - they're a lot easier to test for primality than most primes.
M_{44,947} / 27th Mersenne Prime ~ 8.545*10^13,394 This Mersenne prime was discovered on April 8, 1979 by Harry Lewis Nelson and David Slowinski using a Cray-1 supercomputer. It's the first Mersenne prime to have over 10,000 digits, and therefore the smallest Mersenne prime to be a gigantic prime.
3.5*10^18,726 A lower bound for BB(6) using the busy beaver function (see 4098). This value falls between the 27th Mersenne prime and 2^^5.
2^^5 2003529930................5719156736 with 19,729 digits This is 2 tetrated to the fifth or 2^2^2^2^2 = 2^65,536. It's a gigantic number with exactly 19,729 digits, and the only non-trivial integer of the form n^^5 that can be written out in full. Full decimal expansion:
http://sites.google.com/site/largenumbers/home/a-1/LNL/265536
Decimillillion / Myrillion 10^30,003 The 10,000th -illion. It's sometimes called myrillion because of the name "myriad" for 10,000, but that name is nonstandard.
Smallest more-than-two-digit repunit prime in base 18 ~ 7.06219*10^32,217 This prime number is a string of 25,667 ones in base 18 (equal to 18^25,666 + 18^25,665 + 18^25,664 ... ... + 18^3 + 18^2 + 18 + 1), and it's the smallest string of more than two ones to be prime in base 18 - see the entry for 25,667, the number of digits in this prime in base 18, for why this is special. 2.5427*10^33,201This is my upper-bound on the total number of possible chess games, using the fifty-move rule. I calculated this by first using a maximum of 218 possible moves every turn (218 is the current record most possible moves in any single known chess possition). Then, based on the fifty-move rule and total maximum number of possible captures or pawn moves per player (71), I got 14,296 as an upper-bound of the maximum number of turns a chess game can last (see that entry for a more detailed explanation of how I got that number). Thus we calculate 218^14,296 ~ 10^33,201 as an upper-bound of all possible chess games.
This number is just crazy big, as it has exactly 33,202 digits. This means that it's roughly a millillion times the cube of a googoltoll. Actually it's about 10
^{198} times bigger than that! This is an utterly unbelievable huge number, but small in comparison to numbers we'll examine later.For more numbers like this see my article on
large numbers in probability.
6^6^6 = 6^^3 2659119772................7863878656 with 36,305 digits This is six tetrated to the third. This number has 36,305 digits, so at this point all the digits placed here would be a bit of a drag to scroll through. Nonetheless it's still small enough to be realistically computed, but don’t expect that to last for much longer. Full decimal expansion: http://sites.google.com/site/largenumbers/home/a-1/LNL/E6_46656
Hitchhiker's Number 2^276,209, ~ 5.11764*10^83,297 This is a number that appears in The Hitchhiker's Guide to the Galaxy, the same book that made 42 famous. In one chapter, it says that one can survive 30 seconds within the vacuum of space, and the probability of being picked up by a passing spaceship in those thirty seconds is "two to the power of two hundred seventy-six thousand two hundred nine against". The value is much larger than 6^6^6 but smaller than Greg's gazillion. Sbiis Saibian comments on this number, saying that the improbability seems to be higher than it "should" be - 10^80, the number of atoms in the observable universe, can be considered a very generous upper bound for the probability - even if you make the universe much much bigger, it must be a pretty lonely place for the probability to be this big. See also 9^9^9, an even larger number that appeared in a work of fiction.
Full decimal expansion: https://sites.google.com/site/largenumbers/home/appendix/a/numbers/E2_276709
Greg's Gazillion 10^86,340 A guy named Greg ( source) gave an interesting definition of a "gazillion". He claims that since "gaz" is Latin for "earthly edge", he assumed this to mean it's the Earth's circumference in Greek miles, which is 28,810. Therefore he says that a gazillion should be 1 followed by 28,810 groups of 3 zeros, aka 1 followed by 86,340 zeros. He even bothers to write it out! However, I feel like a "gazillion", if it were a number, should be too big to even write out and have at least a "zillion" zeros. Nonetheless it's an interesting interpretation.The value is slightly larger than the Hitchhiker's Number. Actually it's just over a millillion times bigger. However on this scale that's considered pretty close. For another definition of a gazillion see 10^74. Fugaseven7^7^6~ 8.6958*10^99,424This is the seventh member of the series of numbers formed with the fuga- prefix. It's notable for being quite close to the
googolgong and it has nearly 100,000 digits.
Googolgong 10^100,000 A googolgong is 1 followed by 100,000 zeros. It's the thousandth power of a googol, meaning that it's a googol googol googol ... ... ... googol where you say "googol" a thousand times, which is pretty insane. It's a googolism with a rather unsual story.It all happened when Sbiis Saibian was kid and had recently gotten into large numbers and always talked about them. One day, he mentioned the centillion to his best friend’s father (ironically, he didn’t find out about the googol and googolplex until much later). His best friend’s father then told him that scientists had come up with an enormous number called a “googolgong” which was something like 1 followed by 100,000 zeroes. Saibian was floored and in his mind popped a huge gong that, when struck, would ring for a googolgong years!
The funny thing is that there never even was such a thing as a googolgong; only a googol and a googolplex actually existed and the googolgong was just made up! He was actually incorrectly explaining the much much larger googolplex to Saibian and made that number up out of his mixed-up thoughts. Scientists didn’t even come up with a googolplex; a single mathematician did. I like to imagine that the man thought that scientists came up with a googolgong something like this:
Scientist 1: I finally figured it out! Scientist 2: Oh goody! What did you find? Scientist 1: I found out the number of ways this experiment could have gone! I’ve been trying to solve this equation for months! Scientist 2: Just how big is the number? Scientist 1: I calculated it as one, followed by one hundred THOUSAND zeroes! Scientist 2: Oh my goodness! That number is monstrous. We should make a name for this huge number! What are your ideas? Scientist 1: We should name it after something big! Like ... Google perhaps! Scientist 2: Wow, great idea! But we should change the spelling so they don’t sue us. Scientist 1: How about g-o-o-g-o-l? Scientist 2: That sounds great, but we should end it with something that emphasizes its hugeness! Scientist 1: I know! We should call it a googolgong! Scientist 2: This idea is pure gold! We need to publish our studies and allow the world to hear about this huge number we discovered!
Of course, that’s nowhere near the true story of a googolplex. Nonetheless, Sbiis Saibian has now appropriated a googolgong as his own googolism and uses it as the base of and inspiration for many of his bigger numbers.
Five expofactorial 5!1 = 5^4^3^2^1 = 5^262,144 ~ 6.20607*10^183,230 This number is the exponential factorial of 5, compactly expressible as 5 to the power of 262,144. The number has exactly 183,231 digits, so it falls between a googolgong and the 32nd Mersenne prime. This means that it can still be realistically written out, and in fact it's the largest expofactorial that can be written out. It starts 6206069878... and ends ...8212890625.
M756,839 / 32nd Mersenne Prime ~ 1.74*10^277,831 This is the 32nd Mersenne prime, discovered by David Slowinski and Paul Gage on February 17, 1992. The number is the first Mersenne prime with over 100,000 digits, and logarithmically it is fairly close to the famous Hamlet monkey number.
Centimillillion
10^300,003 The 100,000th -illion. Hamlet monkey number (my version)~ 1.0381*10^307,383 You've probably heard of the infinite monkey theorem before. It states that a monkey on a typewriter pressing random keys will, after a long bsymbolut finite time, type absolutely every possible string of text at one point or another.
Now imagine a monkey at a typewriter pressing random keys. It would usually type out gibberish but occasionally an intelligible word. This number is the odds against a monkey completely and perfectly typing out the Skhakespearean play Hamlet on its first try. In other words, if we put this many monkeys in a room (we’d need a pretty big room), probably one of them would type Hamlet out on the first try. Many, however, would still type it out, but with a few mistakes. This number is pretty easy to calculate: my version of this number uses a
47-key typewriter (alphabet, numbers, space, .,?!();:'"/-) and 182,831 characters (including spaces) to get the odds that the monkey would type all the characters right: 47^{182,831} ~ 10^{307,383} to one against. There are several variants of this number, some of which only use 35- or 27-key typewriters, and some that use much smaller (likely erroneous) numbers of characters in Hamlet.Now this number is absurdly huge! Compare it with something like a
centilion, itself an unfathomable number. This is about 307,080 orders of magnitude larger than a centillion - the Hamlet monkey number is itself 307,383 orders of magnitude away from one, which seems only a bit bigger than the ratio between it and a centillion! Even multiplying a centillion by itself a thousand times (imagine dwarfing a sphere of a centillion particles by a factor of itself 1000 times) wouldn't quite take you to this number! See my article on large numbers in probability for more on this number, and other numbers like this.
9^(9!) ~ 3.325*10^346,275 One of my old Wikipedia signatures (see 9^9) ended with 9 ^{9!}, which may be interpreted as this big number, though I don't recall ever thinking of it that way. I only interpreted it as the modest 9^9, equal to about 387 million. This number falls between the Hamlet monkey number and the 35th Mersenne prime.
M1,398,269 / 35th Mersenne Prime ~ 8.15*10^420,920 This Mersenne prime was discovered on November 13, 1996 by Joel Armengaud. It's notable because it's the first Mersenne prime to be discovered on GIMPS, the Great Internet Mersenne Prime Search. All Mersenne primes hereafter were discovered by participants in GIMPS.
7^7^7 3759823526................2870132343 with 695,975 digits 7^^3 or 7^7^7. This number has close to a million digits. All of its digits in a text file would take up over a megabyte of space, pretty damn big for a text file. But this number can still be humanly computed, keep that in mind. Googology Wiki has its full decimal expansion. The Megadigit Range10^1,000,000 ~ 10^10^99.999Entries: 61
Millionplex / Milliplexion / Maximusmillion / Googolplux (anonymous)10^1,000,000 1 followed by a million zeroes is a number that has several different names. I prefer the name "millionplex" which is of course from "million" plus the -plex suffix. Sbiis Saibian suggests an alternate name, "milliplexion", because he never thought millionplex sounded very good. In my opinion "milliplexion" is a cool-sounding name but just not as natural as "millionplex". Two alternate names for this number coined by an unknown author are "googolplux" and "maximusmillion"; the former is also the name of a much larger Joycian googolism. This number is a borderline case of a hyper-exponential number, a number on the order of a^b^c where c is a reasonably smallish number. The hyper-exponential range (10^1,000,000 - 10^10^1,000,000) corresponds to class 3 numbers, a range from a millionplex to a millionduplex, so this is the boundary between class 2 and class 3 numbers. Class 3 numbers are numbers that can't easily (or at all, depending on the size) be written out but can be approximated pretty precisely with the leading digits and number of digits - for example, 4^^4 can't be written in full but can have its leading digits and number of digits computed pretty easily. Examples of class 3 numbers are the largest known prime number, trialogue, googolplex, 4^^4, and promaxima. Fugaeight8^8^7~ 10^1,813,917This is the eighth member of the fuga- series, a number with about 1.8 million digits. It falls between a millionplex and Borges' number.
~ 1.6950*10^1,834,097This is the number of books in the
Library of Babel, sometimes known as Borges' number (named by Robert Munafo). It came from a story of the same name by Jorge Luis Borges, about a universe which is made entirely of a giant library, containing every possible book with 410 pages, 40 lines per page, and 80 characters per line, with 25 different characters (22 letters, space, period, comma), giving 25^{410*40*80} ~ 10^{1,834,097} books in the Library of Babel. The library at first seems glorious because among those books is information on literally everything - your biography, the biography of anyone you've met, news reports on anything that has happened, a chronology of Earth, your house, etc, written in any language. However, many more of those books would contain false information, ranging from books that botch up a fact or two to books whose information is complete bullshit that seems like someone pulled it out of their ass! Even worse, most of the books would in fact be completely meaningless nonsense to anyone, being a long string of completely random letters. Because of the peculiarities in the Library of Babel, the Library of Babel has been valued as a mathematical thought experiment.This number is just INSANELY HUGE. It has 1.8 million digits, meaning that you'd need something like a novel to store all its digits! It's a small class 3 number, and it's larger than the number of Planck volumes in the observable universe ...
. That means you'd need dwarf the observable universe by a factor of its volume in Planck volumes raised to the 995th powera thousand times to get a universe with Borges' number of Planck volumes! But, as I've said a bunch of times already, this is quite modest to numbers we'll explore soon, not even as big as a googolplex!See also the Hamlet monkey number and promaxima.
M6,972,563 / 38th Mersenne Prime ~ 4.37*10^2,098,960 This Mersenne prime was discovered on June 1, 1999 by Nayan Hajartwala on GIMPS, the Great Internet Mersenne Prime Search. It's the smallest Mersenne prime to be a megaprime, meaning it's the smallest with over a million digits.
Micrillion / milli-millillion / millinillinillion 10^3,000,003 This number is the millionth -illion. It's a number that has been given multiple names in extensions to the -illions. It's named "milli-millillion" in Professor Henkle's system for naming one million -illions, and it's the largest -illion in that system. It's a natural-sounding name because milli-milli literally means "thousand thousands", aka "million".
In Conway and Guy's system it's called millinillinillion, which is a name that is certainly less ad hoc than any of the names in Henkle's, but just not as natural of a name as milli-millillion.
In Bowers' system, this number is named micrillion. Why is that?
Notice these numbers and the SI prefixes hidden in them: decillion is the 10th -illion, deci- divides by 10centillion is the 100th -illion, centi- divides by 100millillion is the 1000th -illion, milli- divides by 1000Therefore it’s natural to continue the pattern call the millionth -illion a micrillion since the prefix micro- divides by a million. Although the name makes it sound like a small number, this system allows for a natural continuation of the -illion names as we will see with later numbers on this list. 10^4,194,304A vigintyllion is the largest -yllion Knuth mentioned in the article where he introduced his -yllions (see
this page for more about them). It's one followed by 4,194,304 (2^{22}) zeros. It's notable because it's only the 20th -yllion, and yet it's larger than numbers like Henkle's milli-millillion, the largest number named in his system! See my article on them for how Knuth's system is able to work like that.See also decyllion and centyllion.
Fzmillion 1,000,000^1,000,000 = 10^6,000,000 This number is a million raised to its own power. It was given by Cockburn himself as an example of a number nameable with the fz- prefix, in the article where he introduces the gar-, fz-, and fuga- prefixes.
8^8^8 6014520736..........5421126656 with 15,151,336 digits 8^^3 (eight tetrated to three) or 8^8^8, a number with fifteen million digits. Storing this number would take up about 100 MB of space, and the value is larger than a fzmillion but less than a googolbong. Click here for all of its digits.
Largest known prime number (as of 2013-15)2 ^{74,207,281}-1~ 5.82*10^17,425,170 This number ( click here for all the digits) is the 48th known Mersenne prime. For three years it was the record holder of largest known prime number. It was discovered January 25, 2013 by Curtis Cooper, a prolific contributor to GIMPS (the Great Internet Mersenne Prime Search). It broke the previous 12-and-a-bit-million digit record of 2008. The number is gigantic with over seventeen million digits, making it a small class 3 number.Largest known prime number (as of 2016)2 ^{74,207,281}-1~ 3.00*10^22,338,617This number ( click here for all the digits) is the 49th known Mersenne prime. It's the current record holder of largest known prime number, the largest number that we know for sure is prime. It was discovered September 17, 2015 on Curtis Cooper's computer software, though a bug prevented the email about the new largest known prime from being sent, so it was not discovered by human beings until the prime-finding computer was checked for maintenance on January 7, 2016. Although the prime was found by a computer in September 2015, the discovery date is marked as four months later because that date was the first time a human discovered the number as prime.This record-holding prime has five million more digits than the previous record holder, meaning that it's not just 5 million times larger than the previous number, but 10
^{5,000,000} times larger. It is known that there are infinitely many prime numbers, but testing a number to be prime becomes famously difficult as a prime gets very large - the largest known primes have been Mersenne primes because Mersenne primes are a lot easier to test for primality than other prime numbers for primality.Largest known perfect number (as of 2016)2^{74,207,281}*(2^{74,207,281}-1)~ 4.51*10^44,677,234When 2
^{74,207,281}-1 (currently the largest known prime number) was proven to be prime, it quickly followed that this number was a perfect number. It's the largest known perfect number, since all even perfect numbers connect to Mersenne primes, and it is not known if there are any odd perfect numbers (see 496 for more). This number has 44 million digits, so it's about the square of the largest known prime.Fuganine9^9^8~ 10^41,077,011This is the ninth member of the fuga- series, a number with about 41 million digits which falls somewhere between the 2014 record holder for largest known perfect number and a googolbong.
Googolbong 10^100,000,000 A googolbong is 1 followed by 100 million zeros. It's a continuation of the googolgong recently coined by Sbiis Saibian. It's expressible as E100,000,000 or as E8#2 in Hyper-E notation, and it's the thousandth power of a googolgong and the millionth power of a googol. Aarex (?) gave this number an alternative name, googolmine. Some users of Googology Wiki pointed out that the number's name is a little weird since "bong" is also the name of a device used to smoke marijuana. Sbiis Saibian said that he was aware of this, and altough the marijuana device was not the main point of the name, it matches the number's theme since he says that one would have to be high to think they are able to understand this enormous number. Besides, as Sbiis Saibian said, Bowers already beat him with a "bongulus" :)
f_{3}(3)~ 6.8591*10^121,210,694 This is a number equal to f _{3}(3) or f_{w}(3) that shows up often when working with threes in the fast-growing hierarchy. It it a very large number with over 100 million digits that falls between a googolbong and 9^9^9. The number, however, can still be written out in full - since 9^9^9 has been written in full, this number, with a third as many digits, can definitely be written out.
9^9^9 / Ulysses Number4281247731..........2627177289 with 369,693,110 digits 9^^3 (nine tetrated to three) or 9^9^9, a 369-million-digit number. You could store this on a flash drive but it would take up over a gigabyte of space. 9^9^9 is one of the largest numbers to appear in fiction. It appeared in the novel "Ulysses" by James Joyce, in this paragraph: “Because some years previously in 1886 when occupied with the problem of the quadrature of the circle he had learned of the existence of a number computed to a relative degree of accuracy to be of such magnitude and of so many places, e.g. the 9th power of the 9th power of 9, that, the result having been obtained, 33 closely printed volumes of 1000 pages each of innumerable quires and reams of India paper would have to be requisitioned in order to contain the complete tale of its printed integers of units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions, hundreds of millions, billions, the nucleus of the nebula of every digit of every series containing succinctly the potentiality of being raised to the utmost kinetic elaboration of any power of any of its powers” "The 9th power of the 9th power of 9" would normally be interpreted as (9^9)^9 = 9^81, which is much smaller, but from describing the large amunt of space that would be needed to hold its decimal expansion, he was clearly describing the hyper-exponential number 9^9^9. There is a website which has the full decimal expansion of 9^9^9 in spirit of this passage, with the digits stored in 33 webpages with 11 to 12 million digits each. This number shows up sometimes in large number discussions, since it's the largest number you can make with just three digits and mathematical operators. Though three digits can name more different numbers than four fours, the largest number formable with three digits is vanishingly small compared to the largest such number for four fours (see 4^^4).
Billionplex / Fugaten 10^1,000,000,000 This is 1 followed by a billion zeros, a number shown here mostly for comparison's sake. Also, this number can be named "fugaten" with the fuga- prefix.
Googolpleiv 4^4^16 ~ 10^2,585,897,973 This is another cool googolple- number, about 2.6 billion digits long - certainly too large to store efficiently on this website, but still small enough to realistically store. It's equal to 4^4,294,967,296 - the exponent, 4,294,967,296, also happens to be a member of the googo- series (googoviji). Interesting coincidence in my opinion.
Nanillion 10^3,000,000,003 The billionth -illion, exactly equal to one followed by three-billion-and-three zeros. Its decimal expansion is a little longer than that of a googolpleiv. Robert Munafo's number of possible human beings 3.62*10^3,576,838,408 Robert Munafo mentioned this number on his number list, as his estimate for the total number of possible combinations for the human genetic code. He calculated this number from taking 4 (number of DNA bases) to the power of 5,941,000,000 (estimate of the number of bases in a human's DNA). It's a HUGE number with 3.576 billion digits; although not each of these combinations would produce a distinct human being (due to the majority of DNA being noncoding DNA), but it's so big that, as Sbiis Saibian said on his large number site: unless you're a member of twins, triplets, etc, you really don't need to worry about bumping into someone with the same DNA as you. For more about this and related numbers see here.
Trialogue / School House Rock Number10^10^10 A trialogue is a googolism coined by Sbiis Saibian, equal to 10 tetrated to the third or 1 followed by ten billion zeros. It's also notable because it appears in a School House Rock song ( ) READ MORE
Gartrialogue (10^10^10)^2 = 10^20,000,000,000 On Andre Joyce's googology page, this number was formerly listed as a (erroneous) value for the googolplex. This demonstrates a misunderstanding of exponential laws - instead of being a googol digits long, this number is a mere 20 billion digits long. Simply observe: 10^10^10^2 = 10^10^100 = googolplex
(10^10^10)^2 = 10^10^10*10^10^10 = 10^(10^10+10^10) = 10^(2*10^10) ≠ googolplex
Googolthrong 10^100,000,000,000 A gogolthrong (coined by Sbiis Saibian) is one followed by a hundred billion zeros. It's the millionth power of a googolgong or the tenth power of a trialogue. It's expressible as 10^10^11, or as E11#2 in Hyper-E, and it falls somewhere between a trialogue squared and Ballium's number.
Ballium's Number(794,843,294,078,147,843,293.7+1/30)*e^π^e^π ~ 2.040427*10^138,732,019,349
A number that appears in a joke YouTube video by a channel called Meerkats Anonymous. In the video, a fictional physicist named Samuel Ballium says he was studying relativistic space theory while his wife was reciting Shakespeare, and because of his wife reciting Shakespeare he inadvertently added Hamlet to an equation. Ballium claims that the Hamlet operator caused the equation to return the highest number, equal to 794 quintillion 843 quadrillion 294 trillion 78 billion 147 million 843 thousand 293 point 7 3 recurring multiplied by e to the power of pi to the power of e to the power of pi. The narrator of the video says that one of the reasons why this number has not yet been discovered is because it's so "phenomenally big". However, the narrator then says that most people didn't believe that Ballium's discovery was true, and a fictional mathematician named Dr. Nilda Christobal Haverman commented that she refuses to believe that there is actually a largest number, and proclaims that by adding 1 to the number she's made a larger number. But then, Ballium says that Haverman doesn't understand how the Hamlet operator works, and says that adding 1 to the number he discovered doesn't make it any higher, because after the number he discovered, numbers just stop growing. The narrator then interviews a man on the street about his opinion on the number, and the man on the street has no words. The video ends with the narrator saying that whether or not the largest number really exists, it will have interesting ramifications for us all. For obvious reasons, there is no largest number and the entire video is complete bullshit, but hey, jokes are jokes. Sbiis Saibian gave this number a name, Ballium's Number. But how big is this "largest number" really? It's about 138 billion digits long and begins 2040427..., so it falls between a googolthrong and 11^11^11. Although it represents an ordinary person's idea of a really big number, since it's only 138 billion digits long, that means it is vastly smaller than a googolplex, so it turns out not to be all that big. It's not even close to Graham's number, which is itself vanishingly small to the unspeakable numbers googologists have studied! Therefore, it would be hugely disappointing to googologists if this was indeed the largest number.
11^11^11 10^297,121,486,765 This is 11 tetrated to the third, a 300-billion-digit number and the first integer of the form x^x^x to pass Ballium's Number. Estimated size of inflationary universe 10^10^12 This is a rough estimate of the size of the whole universe in cubic meters based on inflationary models - not just the observable universe, but the whole entire universe beyond just what we can observe. It's a notably large number, managing to surpass the unimaginably huge trialogue. It’s much bigger than a googol and centillion, and even the googolgong!
Picillion 10^(3*trillion+3) The trillionth -illion, named so by Bowers. It's also the largest number Harry Foundalis describes as nameable in typical modern Greek on his page on Greek numerals (see here and here).
3^3^3^3 10^(3.64*10^12) 3^^4 or 3^3^3^3. A few trillion digits long! As you can see, in x^^y, y matters much more than x - in part 3 of this list you’ll see even more incredible examples of this.
12^12^12 10^(9.62*10^12) This is 12^12^12 or 12^^3. It's the smallest x^^3 number bigger than 3^^4. It’s about the cube of 3^^4.
10^20^10 10^(1.02*10^13) Andre Joyce once said on his infamous googology page that googolple-x is x^(2x)^x when x is written in Roman numerals. This formula is incorrect; it should be x^x^x^2. When you plug in googolplex to that system, you get a number vastly smaller than a googolplex, being merely 10 trillion digits long. 3^(7,625,597,484,987^7,625,597,484,987)~ 10^(9.823*10^13)In the entry for Graham's number (which is the last entry) in David G. Wells' book "The Penguin Dictionary of Curious and Interesting Numbers", he erroneously claimed that 3^^^3 is equal to this value. The real 3^^^3 is equal to 3^3^3^3 ... ^3, with 7,625,597,484,987 3's - that's a FAR FAR FAR larger number.
I suspect that Wells made the mistake of 3^^7,625,597,484,987 = 3^(7,625,597,484,987^7,625,597,484,987) based on the fact that 3^^3 = 3^(3^3); perhaps he incorrectly concluded that since 3^^3 = 3^(3^3), a^^b = a^(b^b).
Googolgandingan
10^10^14 A googolgandingan is 1 followed by 100 trillion zeros, a name coined by Sbiis Saibian. It's yet another number coined on analogy to the googolgong. It's the trillionth power of a googol, or the millionth power of a googolbong. The name comes from "gandingan", an instrument consisting of four chimes in a series.
13^13^13 10^(3.37*10^14) 13 tetrated to the 3rd. This number has 337 trillion digits!
Femtillion 10^(3*quadrillion+3) The quadrillionth -illion. 10^10^16This is Andrei Linde and Vilaty Vanchurin's estimate on the number of distinguishable parallel universes, based on an estimate of humans being able to absorb around 10^16 (ten quadrillion) bits of information in their lifetime. By applying this estimate, they give this figure as an upper bound for how many parallel universes a human can distinguish.
14^14^14 10^(1.27*10^16) 14 tetrated to the third. This number is about 13 quadrillion digits long, so it falls between a femtillion and Archimedes' number.
Archimedes' Number 10 ^{8*10^16} This is 1 followed by 80 quadrillion zeroes. It's the largest number Archimedes named in The Sand-Reckoner, a book of his (ca. 287-212 BC) which discussed large numbers and attempted to estimate the number of grains of sand in the universe. In a section of his book, he named some large numbers. This one was defined as 100 million (known as a myriad myriad in Ancient Greece) to the power of 100 million, to the power of 100 million, or:
((100,000,000) ^{100,000,000})^{100,000,000}The number was the largest number in an extension to the Greek numeral system, which used Greek letters to name numbers up to 10,000. He extended it to name some very huge numbers! In some sense this is the first example of googology: naming very large numbers just for largeness's sake. Googology Wiki lists this as the first event in its large number timeline. This makes Archimedes' work pretty notable in the world of googology in itself!
Googolplev 5^5^25 ~ 10^2.08*10^17 The fifth member of the googo- series. This one would take up an exabyte of space, so it's pretty difficult to store its decimal expansion.
Attillion 10^(3*quintillion+3) The quintillionth -illion.
Duquomevalka 2[2,4] ~ 10^(2*10^21) This is another copy notation (see 22) number by SpongeTechX, equal to 222......222 with a dutrimevalka twos. Amongst Bowers' -illions this falls just under zeptillion. The next number is this many digits long - the du-x-mevalka function is a good example of tetrational growth.
Zeptillion 10^(3*sextillion+3) The sextillionth -illion.
Yoctillion 10^(3*septillion+3) The septillionth -illion, and the last of Bowers’ -illions to use a SI prefix. The next ones are all Bowers' own names.
20^^3 10^(1.36*10^26) Let’s skip through the numbers of the form x^^3 to this number, equal to 20 tetrated to the third. It has 136 septillion digits. little foot100000000000000000000000000000000^10000000000000000000000000 = 10^{3.2*10^26}This number is incredibly huge ... it's almost as big as the current largest named number, BIG FOOT -
not!It was given by a user new to Googology Wiki by the name Vilius2001 as a guess of how big BIG FOOT might be. The name "little foot" was given by Sbiis Saibian, and this number is not even bigger than a
googolplex, which is itself far smaller than Graham's number, which is itself much smaller than a tethrathoth which is itself not even close to as big as the largest known numbers really are! If anything it represents an ordinary person's idea of an incredibly huge number!
Xonillion 10^(3*octillion+3) The first of Bowers' -illions to use an extended SI prefix. Also the octillionth -illion. Lower bound for SSCG(2)3*2^{3*2^95}-8~ 10^{3.775*10^28}This is a lower bound for SSCG(2) with the SSCG (Simple Subcubic Graph) function, a sibling of Harvey Friedman's SCG function (see SCG(13) for details). SSCG(0) is 1, SSCG(1) is 5, and SSCG(2) is known to be at least this big, a number with 37 octillion digits. SSCG(2) is believed to be the largest output of the SSCG(x) function that can be expressed with conventional methods, since all larger outputs require using the fast-growing hierarchy or similar.
Vecillion 10^(3*nonillion+3) The nonillionth -illion.
10^{5,070,602,400,912,917,605,986,812,821,504} ~ 10^{5.071*10^30}In Knuth's -yllion system a
centyllion is equal to exactly 10^2^102, a number with about 5.071 nonillion digits. It's a pretty awesome number in my opinion. If you were to stack enough sheets of paper to store all the digits of this number (assuming each sheet can hold 20,000 digits, 10,000 on each side, then the stack would reach 1.34 million light-years high, which would be about 13 times larger than the Milky Way galaxy!
Mecillion 10^(3*decillion+3) The decillionth -illion. At this point Bowers begins to do something interesting with his -illion names - he gives a new system of names unlike the SI prefixes.
Estimated odds of a person living 1000 years or more 10^10^36 Some inscurance companies have tables that calculate the odds that a person would live past a certain age, and after a certain point they give formulas to estimate the odds of living past even older ages. Robert Munafo used one such formula to calculate this value, the odds that someone would live 1000 years or more. Even the number of digits of this number is ridiculously huge now, and we still haven’t reached a googolplex!
Duecillion 10^(3*10^36+3)
Trecillion 10^(3*10^39+3) e^(1.59*10^40)~ 10^(6.8952*10^39)This number is the current upper-bound for the smallest counterexample of the
Mertens conjecture, a conjecture in mathematics which turns out to be incorrect. The conjecture states that |M(x)| (| | are absolute value bars and M(x) is the Mertens function) will always be less than the square root of x.The Mertens function M(x) is defined as the sum of the values of the Mobius function for integers 1 through x. The Mobius function (denoted with the Greek letter mu (µ)) is itself defined for positive integer outputs:
µ(x) =:
1 if x is squarefree (is not divisible by any non-trivial square number) and has an even number of prime factors
-1 if x is squarefree and has an odd number of prime factors factors
0 if x is not squarefree
It has been proven that there exists a number x such that the absolute value of M(x) is not bounded by the square root of x, but it is not known exactly what the smallest such x is. The current best upper-bound for the smallest such x is e
^{1.59*10^40}, a number with about 6.89 duodecillion digits.An interesting fact about the Mertens conjecture is that if it were true, that would imply that the Riemann hypothesis is also true.
See also Skewes' number, an well-known old upper-bound for a somewhat similar problem.
Tetrecillion 10^(3*10^42+3)
Pentecillion 10^(3*10^45+3)
Icosillion 10^(3*10^60+3) This number is the point where Bowers begins using Greek roots for numbers in his -illion names - the next major gear shift.
Lynz at the time of my birth10^(63*2^198) ~ 10^10^61.40 The lynz (pronounced lines) is an ever-growing number which is calculated as follows: On February 26, 1998, it's equal to 100. Every day after that, the value doubles, so on 2/27/1998 it's 200, on 2/28/1998 it's 400, on 3/1/1998 it's 800, etc. But that's not the end of the story - on September 17, 1998 the value is rounded to one vigintillion and squares every day fro m there on, making it grow much faster - on 9/18/1998, it blew past the googol and soon after a centillion, quickly reaching the class 3 range of numbers. On April 3, 1999 (my birth), it was equal to around 10^10^61, approximately an icosillion in terms of Bowers' illions. See also the late 2008 and mid-2014 values of the lynz.
Lower bound for the size of Linde's Grand Universe10^10^64 There is a theory of chaotic inflation devised by Andrei Linde that states that our universe is part of a Grand Universe which consists of many universes, with local Big Bangs constantly going off. Unlike our plain old universe, such a Grand Universe would inflate eternally with all the local big bangs. Not a lot is known about how that theory would really work or how large or old the Grand universe is, but we can come up with a lower-bound for how big the Grand Universe is.Sbiis Saibian devised such an estimate in his article "
Surveying the Cosmos". He assumed that the size of the Grand Universe multiplies by at least 10^10^12 each Planck time, and let it run for 13.7 billion years (since that's the age of our universe, we know the Grand Universe must be AT LEAST this old). There he got 10^10^64 as a lower-bound for the size of the inflationary universe in meters, light years, or whatever. We can't come up with a reasonably certain upper-bound for the size though, since we have no way of upper-bounding how old Linde's Grand Universe might be - we have no reason to believe our local universe we necessary the first of the universes to have existed!
Triacontillion 10^(3*10^90+3) 2^10^100~ 10^10^99.4786This number is two to the power of a googol. It was once erroneously listed as a way to express the googolplex by Andre Joyce - he used his g-function, expressing it as g(2,g(2,100,10),2). This is rather inexplicable, since one a simple correction gives us googolplex = g(2,g(2,100,10),10).
This number appears to be almost nothing less than the googolplex, but actually you need to raise it to the fourth power to get it past a googolplex. This tells us that we're moving through the numbers very quickly far faster than you think! But hold on, you know what they say,
it only gets worse ...See also log(log(2)).
The Googladigit Range10^10^100 ~ 10^10^999,999Entries: 42
Googolplex10^10^100 Main article: Googol and GoogolplexA googolplex is a well-known large equal to 1 followed by a googol (10^100) zeros. For many people, it is the largest number with a name. It's gotten a lot of attention due to its simple explanation and vast size, and it has been referenced many times in culture. It is the origin of some notable ideas in googology, mainly the -plex suffix (most often defined as 10 ^{n}). However, as far as large numbers go it really isn't that big!To get an idea of how big a googolplex really is, consider the difficulty of just writing it out in full as 10000 ... ... 0000 with a googol zeros. Imagine that everyone alive today were to write zeros continuously, writing a zero every
Planck time. At that insane way-beyond-superhuman rate you could write about 10^{43} zeros in a second! But even at this rate, it would take about 2.38 duodecillion (2.38*10^{39}) years for all of us to write out a googolplex. For more about a googolplex, read my googol and googolplex article (link above).
Googolplex plus one10^10^100 + 1 No, this number isn't just here to parody those who add one to the largest number they know of; it's actually significant as a number outside of that. Why is that? People have calculated factors of numbers just past a googolplex. That sounds impossible, but with some clever tricks relating to the modulo function that can be done without too much difficulty. This number's smallest known factor is 31,691,265,005,705,730,374,175,801,344,000,001, or 31.7 decillion.
Googolplex plus thirty-seven 10^10^100 + 37 This is the smallest number after a googolplex with no known factors. ~ 10^(10^100 + 267)This is a googolism defined in the inactive blog "A googol is a tiny dot" defined as the smallest prime larger than 10
^{gooprol} (gooprol = 10^100+267 is the smallest prime number larger than a googol). It's quite trivial to see that this number must lie somewhere between 10^(10^100+267) and 10^(10^100+268) (closer to the former) based on what is known about the distribution of primes. And yet, it's not possible with currently known methods to find exactly how big this number is, more specifically how much larger it is than 10^(10^100+267), and therefore the number doesn't have an article on Googology Wiki. Nonetheless this is still a completely well-defined number.
50^100^50 10^(1.70*10^100) The result of plugging "googolplel" into Joyce's verbatim googolple- system. It's the closest the system gets to a googolplex. This value falls just between the cracks of a googolplex and a gargoogolplex. Googolplel could logically be interpreted as this if we forget the whole googolple- system and instead use the -ple- infix: x-ple-y is y^x where y is written in Roman numerals. So googolplel in this case would be 50^googol since L is Roman numerals for 50. However, that wouldn't be as interesting in my opinion.
Gargoogolplex (10^10^100)^2 = 10^(2*10^100) This number is the square of a googolplex. Its name has an interesting story. The children of a man named Allstair Cockburn played a game where the goal was to name a number larger than the previous person's number by saying "My space commander rules x stars". It would usually end at a million until Allstair introduced his kids to the googol and googolplex. Then, Allstair's youngest son Kieran said "My space commander rules a gargoogolplex stars", and said that a gargoogolplex was a googolplex googolplexes. That was the origin of the gar- prefix, and the inspiration for its more powerful brothers fz-, fuga-, and megafuga-. Even though a googolplex googolplexes sounds impressive, wiith numbers this big this isn’t a good extension. This is 1 followed by only twice as many zeros as a googolplex, less than even 10^10^101. Therefore, not a great attempt at extending a googolplex.
Googolillion10^(3*10^100+3)This is one followed by three-googol-and-three zeros, a humorous number which is just an -illionized version of the googol. It's a thousand times the cube of a googolplex.
10^10^101
This number looks a tad bigger than a googolplex, but in reality this is raising the googolplex to the 10th power. Don’t believe me? Simply observe: (10^10^100)^10 = 10^(10^100*10) = 10^(10^100*10^1) = 10^(10^(100+1)) = 10^10^101 With numbers this big, looks can be deceiving, but this is a relatively tame example of that kind of thing! Also, this number is a lower bound to the googolbang.
Googolbang(10^100)! A googolbang is the factorial of a googol. The number that came from the website "Cantor's Attic", a wiki that focuses mostly on infinite numbers. It is betwen 10^10^101 and 10^10^102, and it’s only “a little bigger” than a googolplex if by "a little bigger" you mean that you need to raise a googolplex to the 100th power to get a number bigger than a googolbang. It has about 9.96*10^101 digits, so logarithmically it’s closer to 10^10^102 than to 10^10^101. It starts 162940433246.... and has exactly 2.5*10^99-18 trailing zeros and its last nonzero digit is 6. A lot of the information on the number has been found using methods to estimate large factorials.
Fzgoogol (10^100)^(10^100) = 10^10^102 Fz- is a prefix related to the gar- prefix that takes a number to the power of itself. With numbers this big, it still is pretty powerful. Here’s why it’s 10^10^102: (10^100)^(10^100) = 10^(100*10^100) = 10^(10^100*10^2) = 10^10^102. This number is also an upper-bound for a googolbang.
Tetracontillion 10^(3*10^120+3) The first of Bowers' milestone -illions to transcend a googolplex.
Pentacontillion 10^(3*10^150+3) Tritet Jr. / Megafuga-four / Booga-four4^^4 ~ 10^(8.0723*10^153) This is a large number equal to 4^4^4^4, 4^^4 (four tetrated to four), or 4^^^2 (four pentated to two). It's a tiny pentational number but a mid-sized tetrational number, and it has several different names. It's named "megafugafour" using the megafuga- prefix which tetrates a number to itself, or booga-four with the booga- prefix which turns a number n into n^
^{n-2}n (see here for more on these prefixes). Jonathan Bowers used to call this number "tritet" because it was at the time equal to {4,4,4} in his array notation ({4,4,4} now equals the much larger 4^^^^4 instead), so Sbiis Saibian calls this number Tritet Jr. A Tritet Jr. or whatever you want to call it is a number on roughly the same scale as a
googolplex. It's my favorite example of a class 3 number which we can't write out but can approximate pretty well, with both the number of digits and starting digits. It’s amazing that both the leading and ending digits can be computed quite easily, but that since the number of digits is so large we will never compute all of them. For more on this sort of stuff, see 10^1,000,000.Here is how this number starts and ends: 23610226714597313206............36860456095261392896 It's exactly 8072304726028225379382630397085399030071367921738743031867082828418414481568309 149198911814701229483451981557574771156496457238535299087481244990261351117 digits long. 4^^4 is the highest solution to the four fours puzzle, where your goal is to name as many numbers as you can with four fours and standard mathematical operators. For example, 9 can be expressed as 4+4+4/4, and 172 can be expressed as 4*44 - 4. 4^^4 can be expressed as 4^(4^(4^4)). It's interesting that people usually think of forming smallish numbers with this puzzle, but in fact it can form a number that is bigger than a googolplex!
81^^3 ~10^(7.3786*10^154) The smallest x^^3 number greater than Tritet Jr.
Hexacontillion 10^(3*10^180+3)
Fugahundred (....(((100^100)^100)^100).....)^100) with 100 100’s = 100^100^99 = 10^(2*10^198) A fugahundred is formed by applying the fuga- prefix which takes n and turns it into (...(((n^n)^n)^n ... )^n with n copies of n, which is equal to n^n^(n-1). It is larger than a googolplex, and in fact it's the 2*10 ^{98}th power of a googolplex.Gargoogol-plexed10^10^200This is a nonstandard way to interpret "gargoogolplex", applying "-plex" to "gargoogol" instead of "gar-" to "googolplex". It's the googolth power of a googolplex, meaning that it's a googolplex googolplex googolplex ... ... ... googolplexes (say a googol times).
Heptacontillion 10^(3*10^210+3)
Octacontillion 10^(3*10^240+3)
Ennacontillion 10^(3*10^270+3)
Hectillion 10^(3*10^300+3) Notable for being a little less (in hyper-exponential terms) than an ecetonplex.
Ecetonplex / Centillionillion10^10^303 or E303#2 This is a googolism created by Sbiis Saibian as a "plexed" version of the centillion - it's one followed by a centillion zeros. He names it "ecetonplex" rather than the awkward "centillionplex". Sbiis Saibian called this number “centillionillion” as a kid after "learning" about the googolgong - it's the only number he named as a kid, and he believes that it's the first large number he ever devised. However, despite the name "centillionillion", this is not the centillionth -illion, so the name is erroneous (see next entry).
Centillionillion (corrected version) 10^(3*10^303+3) This number is the centillionth -illion, and "centilionillion" is a silly name that can be logically used for this number. It's called a henhectillion in Bowers' illion system. It's a thousand times the cube of the previous number.
Promaxima 10^10^343 The promaxima is Sbiis Saibian's alternate universe count. It is the most recent revision of a calculation he posted on the infamous "Really Big Numbers" forum, where he calculated the number of possible histories of the universe from its birth to its death, assuming that sub-Planck units aren’t meaningful. He ended up with this crazy number, notable for being bigger than a googolplex. Promaxima means probability maximum, meaning that Sbiis Saibian considers it the largest number probability can meaningfully generate. Although there are larger numbers probability can give us, it's debatable whether they are physically meaninful. See this article for more.Now literally anything you can think of, even if it’s ridiculous, is part of one of these histories. You were born a Planck time later? Whole new history.You were born two Planck times later? New history. You were born a Planck time earlier? Another new history. You never actually read this list? Another new history. This text is in a different color? Yet another new history. You sneeze as you’re reading this? Another history. Even crazy things like the universe being created by your favorite celebrity or your pet cat turning into a mutant alien for one yoctosecond are among those histories.
Kilofaxul (200!)! About 10^10^379, the factorial of a faxul. If we listed every way 200 books could be arranged, those arrangements could be arranged a kilofaxul ways. We could arrange those arrangements of arrangements, but that becomes meaningless quickly.
Dohectillion 10^(3*10^600+3)
Trihectillion 10^(3*10^900+3)
Great googolplex / googolplexichime 10^10^1000 Along with the great googol, Joyce defines the great googolplex. Saibian calls this number googolplexichime with the -chime suffix similar to the -gong suffix. It's equal to 1 followed by a googolchime zeros.
Lynz at the time Googology Wiki was founded10^(63*2^3732) ~ 10^10^1125 Googology Wiki was founded on December 5, 2008; at that time, the lynz (for the definition, see the entry "Lynz at the time of my birth") recently passed a great googolplex. In terms of Bowers' -illions, it was between a trihectillion and a tetrahectillion.
Tetrahectillion 10^(3*10^1200+3)
Pentahectillion 10^(3*10^1500+3)
Lynz at the time this website was created10^(63*2^5732) ~ 10^10^1727 This website was founded May 28, 2014 - at that time the lynz (see here for more about it) was equal to about 10^10^1727, between a pentahectillion and a hexahectillion. It's gradually drifting away from this value, and eventually will pass 10^10^2000.
Hexahectillion
10^(3*10^1800+3)
Heptahectillion 10^(3*10^2100+3)
5^5^5^5 5^^4, ~10^10^2184 Five tetrated to four, a moderately sized tetrational number. This number has somewhere between a googolchime and a millillion digits.
Octahectillion 10^(3*10^2400+3)
Ennahectillion10^(3*10^2700+3)
Killillion 10^(3*10^3000+3) Another milestone in Bowers' -illions - this is where he starts using the large SI prefixes. It's the last of his -illions I'm listing here.
Great great googolplex / googolplexitoll 10^10^10,000 Sbiis Saibian calls this number googolplexitoll, analogizing "googolplex" with "googolitoll".
6^6^6^6 6^^4, ~10^10^36,305 These numbers are growing horrendously fast - this one has a 36,000-digit number of digits. Googolplexigong 10^10^100,000 This is the combination of googolplex with the -gong suffix. It sounds better than googolgongplex.
7^7^7^7 7^^4, ~10^10^695,974 This number is 7 tetrated to the fourth - there are nearly a million digits in the number of digits in this number. Now that we're done with these little babies, how about some bigger numbers? Welcome to part 3 of this list. |