Gigantic List of Large Numbers - Large Numbers (2023)

In this section, we will investigate numbers that are reallyreallyREALLYsmall, even by a layman's standards. These numbers are SO small they're actually enormous. How is that possible??? Start reading to find out.

Zero: 0

This is basically the integer representation of nothing. I don't even know if it belongs on this page. It took thousands of years to even be recognized as a number and get its own symbol! In other words, numbers like one and two existed for THOUSANDS of years before someone decided to come up with a name and symbol for zero.

However, it does have some interesting properties: n + 0 = n, n × 0 = 0, and n^0 = 1 for all numbers n. In fact, n^^^...^^^0 = 1, no matter how many up-arrows you have (as stated in the introduction, they are explained at the end of Part 2). But that's boring. So let's look at some numbers that are a tiny bit bigger.

It seems kind of mysterious that zero wasn't considered a number for many thousands of years. But why? Well, no one knows. But, when you think about it, zero is technically the reciprocal of infinity. Infinity isn't a number, so is zero = 1/infinity a number? You and I consider it a number, but maybe there's a problem with that - no one knows!

Googolplexminex: 10^-(10^10^100)

Whoa now!!! This is the reciprocal of agoogolduplex(or sometimes known as googolplexian)! A googolduplex is 1 followed by a GOOGOLPLEX zeroes, and a googolplex is 1 followed by a googol zeroes, and a googol is already a lot bigger than the number of particles in the universe. This is pretty much impossible to picture, and as you can imagine, a googolplexminex is a VERY VERY SMALL number. If we were to write this in decimal form, it would be 0.00000000...00000001 with a googolplex zeroes (counting the zero before the decimal point). And if we can't even write a googol zeroes out, how are we going to even think about writing a googolplex?

Googolminex: 10^-(10^100)

This is a lot bigger than the previous number (almost a googol times bigger) but it is still VERY SMALL. It is the reciprocal of a googolplex and is pretty much useless to the non-googology community. But this can give us a glimpse into how small the previous entry is. If this number is so small that it has no real-world meaning whatsoever, how small would it be if we dwarfed it by a factor of a googol? Incomprehensibly small, right?

Now hang on! Didn't I say this would be a site about LARGE numbers, not incomprehensibly tiny numbers? I did. But to understand large numbers we first must understand smaller numbers. I'm sure that if you scroll down to the entry on Graham's number, you won't understand at all how huge and horrifying it really is. And that's because YOU HAVEN'T READ WHAT'S BEFORE IT YET. So please, don't skip this part.

Googol-minutia: 10^-100

This is the reciprocal of a googol. Now this is still a very small number, but this is a huge step. We can easily write this number out in decimal! It would be exactly 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001.

We can use this to get some real-world meaning. Now, there are less than 10¹⁰⁰particles in the universe, but we could fit a lot more than 10¹⁰⁰particles in the universe if we had them, because the universe is mostly empty space. Thus, a googol-minutia portion of the volume of the universe should be bigger than a particle! Specifically, it would be about the size of a human blood cell.

Gogol-minutia: 10^-50

This is not a typo - the first part of the number's name is actually "gogol". This is clearly a lot bigger than the previous entry (being its square root, and when you are smaller than 1 your square root is bigger than you), and we can get more real-world meaning out of it. This portion of the volume of the universe would be a sphere with a radius of about 10¹⁰or ten billion meters! That's larger than the diameter of Earth and even larger then the distance from the distance from the Earth to the Sun, which is about 93,000,000 miles! And that's just the radius of the sphere - imagine how big its VOLUME is. It would be about the size of a black hole!!! A gogol-minutia doesn't seem like such a small number now, does it?

However, if we take this portion of the number of atoms in the universe, we will obtain about 10^28 atoms, or tenoctillionatoms. That may seem enormously huge at first, but it's actually very close to the number of atoms in your body. But really, this tells is more about how small an atom is rather how small a gogol-minutia is.

But really, I hope I haven't blown your mind yet with all the black holes and stuff. This is still a VERY small number by most people's standards. Imagine if your dick was this many meters long! That would really have to suck. But, on the bright side, at least it wouldn't suck as much as it being a googolplexminex meters long.

Planck time: about 5.391 × 10^-44seconds

This is believed to be the smallest possible length of time. You see, Max Planck theorized that there were fundamental building blocks of time, temperature, and length, and at some point these measurements had to have some limits. If you are a 13-year-old right now, you have lived approximately 2.329 × 10⁵³Planck times, or 232.9sexdecillionPlanck times! Wow! You should be proud of yourself.

However, don't start bragging yet, because you would have to live to be almost 56 years old to reach 10⁵⁴Planck times, and about 558 years to live 10⁵⁵Planck times! I think it's a safe bet to say that no human will ever live to be 10⁵⁵Planck times old. If a human was 10⁵⁵Planck times old right now, they would have been around in the time of William Shakespeare. And remember, you're 2.329 × 10⁵³Planck times old RIGHT NOW, so on this scale, 10⁵⁵seems awfully close! But it's really 42.937 times bigger.

Planck length: about 1.616229 × 10^-35meters

The Planck length is believed to be the smallest possible length measurable. It is believed that beyond this miniscule length, quantum foam exists and the laws of physics simply do not hold true anymore.

To put that into perspective, a hydrogen atom is about 5 × 10^-10 meters in diameter. So that means that if you had a bunch of spheres with the diameter of a Planck length, about 2.9607 × 10^52 of them would fit inside of a hydrogen atom, or 29.607sexdecillion. Holy shit! The Planck length is so tiny it's actually huge!

Guppy-minutia: 10^-20

This number is equal to the reciprocal of 10²⁰, or 100 quintillion. This number seems more comprehensible than most other numbers discussed earlier. This portion of the volume of the universe would be about the size of a large galaxy, and this portion of the atoms in the universe would be about 10^58 atoms, which is greater than the volume of the earth!! Clearly, comparing numbers like this to something as large as the universe is kind of pointless now. For example, you would have to dwarf a hydrogen atom by this factor to obtain a sphere the size of a Planck length as discussed earlier.

So that's it for Part 1. Not very long...

This section contains numbers that are relatively easy for humans to understand. If you're looking for many of the numbers we use in everyday life, you've come to the right place. This is actually one of the longest sections on this page, since there are a lot of numbers that we commonly use!

One hundred millionth: 10^-8

This is a number which is equivalent to 0.00000001. It is very, very easy to write out in decimal (googol-minutia is writable but tedious). It is comparable to the real world in many ways. For example, if you are a middle-class person living in the USA you may own this approximately portion of the world's money. If you are rather wealthy you may own 10^-7of the world's money.

1/998001

This number has a very interesting decimal expansion. We already know that 1/81 is 0.01234567912345679... infinitely repeating without the 8. But this number is 0.000001002003004...996997999000001...! What an interesting decimal expansion!

1/225983

This fraction has the longest decimal expansion of any fraction I have calculated with a program I have made; it has a whopping 225,982 digits.

One hundredth: 0.01

I know this seems like a big jump from one hundred millionth, but there really isn't much in between. This is a number which is encountered very often, both in math and daily life. Most people see this number as pretty small, but when comparing it to the numbers above, is it really that small? And, is a 1% chance really that rare? It's a little more likely than the chance of getting 7 heads from coin flips in a row (the chances of that are 1/128). If you're sending a probe to Mars and the chances of it exploding are 1%, you've got a huge problem on your hands!

One fiftieth: 0.02

Not really much to say here, except that it's twice the previous number. It's encountered just as often as 1% is.

One tenth: 0.1

This is a common decimal number that is very easy to visualize and is encountered often in daily life. Remember, this is still seen as pretty small, and most people won't rely on these types of odds.

For example, let's say you're having a bet with your best friend. There is a 0.1 chance you will win. If you win, you get $50, but if you lose, you have to give your best friend $5. Would you take the bet? Most people won't rely on a 10% chance.

One ninth: 0.1111111...

This is a common example of an non-terminating decimal which is still a rational number. 9 is not a factor/divisor/multiple/etc. of 10, and will not have a terminating decimal. However, if you were working in base 3, 1/9 would have a terminating decimal and would be expressed as 0.01.

Champernowne Constant: 0.123456789101112131415...

This is a rather childish and useless number. You may have once thought of it when you were a child. Well, now you know its name. It is an irrational number (it cannot be expressed as a fraction). Of course the pattern is obvious, but if you were to convert this to another base, it would be a number similar to e and pi, whose digits go on and on with no observable pattern.

One fifth: 0.2

Not really much to say about this one either. It's twice one tenth. By now, these numbers are encountered very often and are distinguishable from their neighbors.

i^i: 0.207879576...

This is an unusual number which is the result of raising the complex number i to its own power, which strangely is a real number. In case you didn't know, i is the square root of -1. (The square root of -1 isn't -1, because -1^2 = -1 × -1 = 1). Then by raising this number to it's own power, how did we get this weird number? As it turns out, it's not a weird number at all - it's equal to e^(-pi/2). We can calculate it by extrapolating from infinite series, but that's a little too complicated for me to explain. If you're curious, you can look it up, but right now, I'd recommend staying on this site.

One fourth: 0.25

This is the next major jump from one half, and it's one of the most common fractions both in daily life and mathematics. It is the exact probability of getting exactly two heads from a coin flip. Pretty common, right? 0.25 is used as an approximation very often for other numbers.

Log 2-10 Constant: 0.301029996...

This is a number equal to log10(2) or 1/log2(10). I call it the Log 2-10 Constant. Of course, you could also have the Log 3-10, Log 4-10, etc. Constants. The reason why I listed this on this site was because it is very useful in computing large numbers.

For example, take Megafugafour, a number which is larger than a googolplex (we'll examine both in more detail later) It is equivalent to 4^^4 or 4^4^4^4. Using a trick with the Log 2-10 Constant we can see that it is approximately 10^(8.02 × (10^157)).

Log 3-10 Constant: 0.47712125...

Not much to say here. It's exactly log10(3) or 1/log3(10).

One half: 0.5

Probably the most common fraction that we as humanity use. It's the chances of getting heads from a coin flip. It's the chances of getting an even number from a die roll. God, there are countless places we can apply this number. It is the smallest number that can be rounded to one.

Three fourths: 0.75

This is now a number that is used to mean "most"; likewise, "most" people might have trust in these odds.

Ninety-nine hundredths: 0.99

This is now a number that is very close to 1, and can be described as "almost all". However, it might not be as accurate as it seems. Let's say we want to check if a person has a very rare virus which only has a one in one million chance of occurring, with a machine with 99% accuracy. If we were to perform that check on the entire population of the USA, nearly all of the people diagnosed with the virus would be perfectly okay! As you can see, this number is actually very far from 1.

One minus a googolplexminex

Now, this is a number so mindbogglingly close to 1 that we can't describe it. It would be 0.99999...99999 with Googolplex-1 9's! These are odds that every human being on earth should wish for.

One: 1

Ultimately one of the most important numbers in mathematics and all of existence. It is the foundation of the natural number system, and the building block of all numbers! With it, we can create practically every other number.

It has very many notable properties. It is the only non-composite and non-prime number. It is the identity of multiplication, exponentiation, etc. (i.d. n × 1 = n, n^1 = n, n^^1 = n, etc. no matter how many arrows you have! and also 1 × n, 1^n, 1^^n, etc. all equal 1 for all values of n)

It's a triangular, square, pentagonal, hexagonal, ..., etc. number. It is the only known odd multiperfect number. It is a default argument in many notations.

One plus a googolplexminex

Because I can.

Twelfth root of 2: 1.059463...

This is the number, that when multiplied by itself 12 times, you get 2. It is notable for being the increase in pitch for every musical note, e.g. C#4 has a frequency 1.059463 times higher than C4 (not the explosive but the musical note mind you!). That's why notes sound the same, just higher pitched, when you go one octave higher, because the frequency is exactly 1.059463^12 = 2 times higher!

1.1

1.1 is a number regarded as being just a tad larger than 1, but when you think about it, it is really much, much, larger. Assume the world population increases by a factor of 1.1 annually (which it doesn't!). In just 7.273 years, the population has doubled, and in 24.159 years, it has increased by a factor of 10! Holy crap!! In a century, two humans could end up having well over 13,000 descendants! This number doesn't strike you as small now... or does it?

The reality is, the world population only grows by a little over 1% each year, and the population growth of the United States is even less, at about 0.8%, and the largest annual population growth of any country, ever, was still less than 4%. A society where there was a 10% annual growth in population would be unimaginable.

e^(1/e): 1.444667...

This may just seem like a random number, but it is the maximum number for which the infinite tetration (we'll get into that later too) n^n^n^n^n^... converges. As you probably already know (or can tell just by looking at it), 2^2^2^2^... does not converge. If you took a number like 1.443, however, you could build a power tower of 1.443's a googolplex 1.443's tall and it would not only be a finite number, but it wouldn't even be that big!! How cool is that?

Sqrt(2): 1.4142135...

This is an example of one of the most common irrational numbers. We see it all the time in algebra, trigonometry, etc. For example, sin(pi/6) is exactly sqrt(2) / 2.

1.5

Again one of the most common numbers in the real world, it is equal to 3/2. You could use it by saying that "X is 50% bigger than Y" and X and Y don't even have to be numbers.

Phi: 1.618033...

You may have learned about this number, also called the Golden Ratio, in school and how it can form the "Golden Rectangle" that you can chop squares off of infinitely, and that 1/phi = phi - 1 and phi^2 = phi + 1. So 1/phi = 0.618033... with the exact same digits after the decimal, and vice versa. You may have also learned that as elements get larger in the Fibonacci sequence, F(n + 1) gets closer to F(n) × phi.

What they don't teach you at school, however, is that this number is exactly equal to (1 + sqrt(5)) / 2. It is a little-known fact that many people have never learned about.

Pi²/6: 1.64493407...

You may have seen this unusual number before; it's the sum of the infinite series 1/1 + 1/4 + 1/9 + 1/16 ...

Note that the harmonic series 1/1 + 1/2 + 1/3 + 1/4 ... does not converge; it goes on up to infinity!

But what is pi doing here, and why is it squared? We don't usually see it squared. You might want to check out this video laterhttps://www.youtube.com/watch?v=d-o3eB9sflsthis guy explained it way better than I ever could. But for now let's move on.

Sqrt(Pi): 1.7724539...

Just like with pi², we don't usually see the square root of pi very often. However, it is exactly 2x the factorial of 1/2 (how is that possible)? It was also mentioned once in an episode of Rick and Morty.

Two: 2

This is the second natural number, the third whole number (that is if you count zero to be a number at all), and is again very common in daily life. I don't think I really need to explain that much.

Many people know that the powers of two grow quite fast (2^10 is already 1,024, so if the world population doubled each year, we would have TRILLIONS of humans 10 years from now). However, they grow very slowly compared to other numbers (3^10 is 59,049, and 4^10 is 1,048,576).

2.54

This is the number of centimeters in an inch. Not really much else to say about this number except that most people round it to 2.5, giving them 30 centimeters in a foot rather than the correct 30.48.

e: 2.718281828...

This number is a very important constant in mathematics, and is the root of all logarithms. It can be calculated by the following process: 1/(0!) + 1/(1!) + 1/(2!) + 1/(3!) + 1/(4!) + 1/(5!) + ... which eventually converges to this constant e.

This number is very helpful in estimating values of large numbers. For example, we can use a little trick with this number to calculate the factorial of a googol!

Another way to approximate e is to pick a really big number n, and then compute (1 + 1/n)^n. If I selected one million as my number n, I can get e accurately up to 5 decimal places. That may not seem like much, but if I picked a number like a googolplex, I would be accurate to about a googol digits! Wow!

Three: 3

You may be wondering why there is a rather large jump in between e and 3 compared with all the other numbers I've listed. There really isn't a lot in between 2 and 3, and even less in between 3 and 10.

Three is a number with many interesting properties. It is the first odd prime number. It is a triangular number (obviously, because a triangle has 3 sides). It is the only prime known to be one less than a square number.

In fact, over 2,000 years ago, this was considered an accurate approximation for pi! Nowadays, pi can be calculated to TRILLIONS of digits, but back then they had to rely on estimation, and since the diameter of a circle was about 3 times smaller than its circumference. But we'll get to that in the next entry.

Pi: 3.14159265...

I'm pretty sure that this is a number that everyone knows about, but I'm going to explain it anyway. Pi is the ratio of a circle's circumference to its diameter. It is a very important constant in mathematics that can be found almost anywhere: algebra, trigonometry, geometry, calculus, etc. For some weird reason, people try to struggle to memorize as many digits of this number as they can. Here is how much I know as of 21 June 2018:

3.14159265358979323846264338327950288

Not a lot, right? I memorized 35, but it turns out that the world record holder for most digits of pi memorized over 70,000. WOW! The current world record for most digits of pi computed is 22.459trilliondigits. However, it took thousands of years just to get a few digits of this value.

3.1416

An approximation of Pi given by Ptolemy in 150 AD. For back then, this was pretty good.

22/7: 3.142857142857...

This was an upper bound for Pi given by Archimedes in 250 BC. Amazingly, that value is still used today as a rough approximation.

Sqrt(10): 3.16227766...

Not really much to say here, except that it's the square root of our numeral base. Surprisingly, it actually doesn't come up as often as other irrational square roots such as sqrt(12), sqrt(18), etc.

Four: 4

This is another cool number with many properties. Perhaps its most notable property is that 2+2, 2 × 2, 2², 2^^2, 2^^^2, etc. is always equal to 4, no matter how many arrows you have. This is the only number with this property.

Four is known as bad luck in China because it sounds almost exactly like the Chinese word for death.

Five: 5

This number also has many different properties. It is a Fermat prime, it is a prime number, etc. It is most notable for being the number of fingers on each of our hands. But the reason why most people like it is because it is half of 10, our base, but I will discuss later that 10 isn't really as special as it seems either.

Six: 6

Another very common number in our world.

Tau: 6.2831853...

This is a much lesser-known number than pi, despite being based off of pi. It is exactly equal to 2 × pi, or the ratio of a circle's circumference to its radius.

Some people think that we should actually be using tau instead of pi, that March 14 "Pi Day" is wrong, and that overall using tau is better than pi.

Seven: 7

This is a number which many people see as "odd," or for some reason, people just don't like it. It has been known to even be the name of some people because "seven is a powerful number and nothing can divide it." It's true that seven is a prime number, and has no factors, but why do so many people see it as so strange? No one knows...

Eight: 8

A power of two, the first cube number, etc. This is also a very common number.

Nine: 9

A square, a multiple/power of three, and another number very common in our world. But, it's not even prime, and many people still don't like it that much, similar to seven. But why? We have yet to find out.

Ten: 10

Ten is most notable for being our numeral base. Many people "like" this number because of that property, and see it as a very special number. But in reality, this number is no more special than the number 7 or the number 9. So why do people see it as special?

Our ancient forefathers created a base-10 system because they had 10 fingers on their hands. That's literally it. So it's not a coincidence that you can count to exactly 10 on your fingers. Of course, people have devised crazy ways of multiplying one and two digit numbers on your fingers, but let's face it, does anyone ever do that?

Twenty-seven: 27

This number is exactly 3^^2, 3³, or 3 × 9. It comes up very often when working with threes. It appears in numbers like 7,625,597,484,987, tritri, Graham's Number, ultatri, and many more.

Forty-seven: 47

This number is the most psychologically random number between 1 and 100. In other words, if you were to choose a random number between 1 and 100, you most likely would pick 47.

But why? For example, between 1 and 10 the most psychologically random number is 7, and between 1 and 20 it's 17. Interestingly, 7, 17, and 47 are all prime.

Generally, we don't want a round looking number because that wouldn't seem to be random; so we wouldn't want to pick 25, 50, or 100, or maybe 12, 24, or 60. We also wouldn't want to pick a very small number like 2, or a larger number like 99, for obvious reasons.

Sixty: 60

Instead of using a base-10 system like we do today, the ancient Babylonians used base-60. They also developed a way to count to sixty on your fingers using your knuckles. Sixty may not be as "round" as ten or a hundred, but we still use it EVERY DAY - a minute has sixty seconds, an hour has sixty minutes, etc.

Hundred: 100

We're starting to get bigger now. A hundred is equal to 10², as you probably already know. It is used in the definitions of countless googolisms (really big numbers), such as googol, googolplex, giggol, gaggol, etc., the list goes on and on. Because it's the square of our base, it gets its own unique name in almost all languages.

One hundred eight: 108

This is the third hyperfactorial number, and is also equivalent to 9 × 12.

What is a hyperfactorial number? It's actually much more simple than it may sound. A regular factorial is just 1 × 2 × ... × n. However, a hyperfactorial is 1¹× 2²× ... × nⁿ. Thus, 3 hyperfactorial is 1¹× 2²× 3³= 1 × 4 × 27 = 108. The hyperfactorial function grows much, much faster than the normal factorial.

Gross: 144

Most times tables we are taught in second grade go up to the "legendary" 12 × 12, which is 144. Weirdly enough, its prime factorization contains only 2's and 3's.

It is the largest known Fibonacci number to be a perfect power.

145

This number is notable for having a very interesting property having to do with factorials. When you look at it, 145 = 1! + 4! + 5! Cool, isn't it? Aside from the trivial 1 and 2, the only other number known to have this property is 40,585.

Two hundred: 200

We see this number a lot when working with hyperfactorial array notation. It seems to be Lawrence Hollom's favorite number.

256

This is a number which is relatable to computers. Computers operate in binary, and since 256 is a power of two, it can be associated with computers. Specifically, it is 2^8, which in binary is 100000000. This is the smallest binary number to take up more than one byte.

Devil Number: 666

This number is known as the devil's number, or sometimes Satan's number. It is made up of pure evil.

A few YouTubers such as Don't Turn Around made videos where they call the number 666 and claim to be "talking to the devil".

Thousand: 1,000

This is considered by many to be a number that is relatively large. A thousand dollars is a lot of money, a thousand miles is already a large distance that would take almost a whole day to drive. If you stacked 1,000 random people on top of each other, that tower of people would be about 1.7 kilometers tall. That's taller than the Burj Khalifa!

1,024

Another number commonly associated with computers. It is equal to 2^10. This number would take up 11 bits of memory.

Interestingly enough, the kilo-, mega-, etc. prefixes are not based off of 1,000 when dealing with computers - 1 KB is actually 1,024 bytes, not 1,000. Additionally, 1 MB is 1,024 KB, or 2^20 = 1,048,576 bytes of space.

Piplex: 1385.45573...

This is a funny sounding number that is formed by applying the -plex suffix to pi. n-plex is equal to 10^n (think googolplex). Thus, piplex must be equal to 10^Pi.

Hardy-Ramanujan Number: 1,729

This number was made popular by G.H. Hardy. One day, he pointed out his friend Srinivasa Ramanujan a taxicab, numbered 1729. He stated what a dull number it was. Ramanujan immediately stated that, on the contrary, it was not a dull number at all, it was the smallest number expressible as the sum of two cubes in two different ways (12^3 + 1^3 and 10^3 + 9^3). How was he able to do it so fast? Well, you see, Ramanujan was a man who had received little education, yet still shocked the world with incredible equations he discovered. Sadly, he got sick and died at the age of just 32.

2,000

I really don't think I have to go much in depth about why this number is on this list. For you adults, this was a year that many of you witnessed (I was born after this year unfortunately). It's a nice round number in our base-10 system.

2,018

The year I am writing these words and the year that I published this site.

2,048

Specifically, this is 2^11 and takes up 12 bits of computer memory, but I put it on here for a different reason (and you all probably know why).

2048 is the name of a highly addictive browser game where you play on a 4 × 4 grid and merge identical numbers. The goal is to make it to the legendary golden 2048 tile. However, this can turn out to be a bit of a problem considering you only have 16 tiles of space. I highly recommend you try out the game herehttp://gabrielecirulli.github.io/2048/. I'm proud to say that I've gotten well over the 2,048 tile and have gotten very close to 16,384!

Additionally, this is a year that most of us should hope to live until. An eleventh power year? That's going to be one hell of a New Year's party. The only other time this happened so far was the trivial 1 AD, and it will happen next in the year 177,147. That's in the distant, distant, future.

3,125

This is 5^^2, or 5^5.

4,096

The smallest power of two larger than 2,048. With a bit of skill this shouldn't be hard to achieve in the game 2048.

4,098

This is a lower bound for BB(5), where BB is the busy beaver function. It's a classic example of an uncomputable function. By uncomputable, I mean a computer cannot compute it if it were given all the time and memory in the world; there is no algorithm to compute it.

BB(n) is the maximum number of marks an n-state Turing machine will make before halting. Now, that sounds complicated. What the fuck is a Turing machine?? It's sort of complicated to explain so read this article on Wikipedia (https://en.wikipedia.org/wiki/Busy_beaver) since they explain it way better than I can. BB(5) is highly believed to be equal to 4,098, but we don't yet know BB(6), the lower-bound for that is a whopping 10^18,267. Currently BB(22) is already the smallest value of BB(n) known to be larger than Graham's Number! Since it's equal to G(64), the busy beaver function is even more powerful than Graham's legendary function!

5,000

This is one of the largest numbers that a human being can actually picture in their heads. For example, 5,000 asterisks would look like so:

********************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************

And remember, I changed the text size to small just so it wouldn't take up too much space, so this is actually smaller than it should be.

7,776

This is equivalent to 6^5. It's notable for being nearly... nearly a repunit number (a number whose digits are all the same, e.g. 11, 666, 55,555,555).

8,000

Few people know this, but the original version of the Dragonball Z meme "Over 9000" was originally "Over 8000"!

8,192

The largest tile I have gotten on the game 2048 as of 21 June 2018. It takes skill to get, but it's nowhere near as hard as the next one, 16,384.

9,001

This number is commonly given when trying to state a number "Over 9000". This number has almost become as much of a meme as 9,000 has.

10,000

This number is called "ten thousand" in English, but many languages give it its own name. For example, the ancient Greeks gave it the name "myriad," and even its own special symbol - the letter M. 10,000 objects is pretty hard to visualize in your mind, when given the 5,000 asterisks above, and multiplying that by two!

17,576

This is equal to 26³, and it's the number of possible 3-letter combinations of words, from "AAA" to "ZZZ".

Fugathree: 19,683

This is 3vv3 using the weak hyper-operators. It is equivalent to (3^3)^3 = 27^3 = 3^9 = 19,683. This number is common when working with large numbers having to do with three.

32,768

This number is again seen when working with computers. It is hard to believe that many programming languages, such as Pascal, go only up to this number! That is because they allot 16 bits of space for a number (1 for the sign, so that leaves only 15). All numbers equal to 32,768 or higher cannot be represented in this 16-bit system.

46,656

This is 6 tetrated to 2, or 6 raised to its own power.

50,000

This number is just a tiny bit beyond what we can comprehend in our minds. However, it has multiple uses in everyday life. 50,000 seconds is actually less than just a day, and most houses and even some cars cost more than $50,000.

65,536

This is one of the largest numbers that can be easily represented both with up-arrows and with the decimal system. It is equal to 2^^^3 = 2^^2^^2 = 2^^4 = 2^2^2^2 = 2^2^4 = 2^16 = 65,536. The next tetrational number starting with 2 is a 19,000 digit number (we'll discuss it too later on).

86,400

This is equal to the number of seconds in a day, since 24 × 60 × 60 = 86,400. The first time I calculated this was when I was a small child (about 7 or 8), and it was actually all in my head.

100,000

By now, the numbers are getting horrifying, right? If these numbers overwhelm you, then brace yourself for what will come next. We haven't even scratched the surface yet. Remember the first law of googology - it only gets worse!

This number is sometimes called a "lakh". It is very hard to visualize. For example, the human head contains about 100,000 hairs. 100,000 miles is about halfway from the earth to the moon, and a car with 100,000 miles is considered quite old and not very valuable. A sphere with 100,000 gallons of water would be a little smaller the size of your house.

Still not convinced that this number is huge? There's one YouTuber called MrBeast who counted to 100,000, and it took him a whopping 40 hours of his life!!! And this number has only 5 zeroes!

131,072

This is the largest number possible in the browser game 2048. However, the chances of getting it are so fucking tiny that it's safe to assume that no human being will likely ever get it. For the odds, think of the numbers at the very top of this list, such as the googol-minutia.

200,000

Now this is beginning to get scary. Imagine that sphere with 100,000 gallons of water that is the size of your house. Now imagine a sphere with twice as much water. It would be larger than most buildings. By now the numbers are getting really hard for us to comprehend.

500,000

We are making very huge jumps now... This number is half a million!!!

It is already getting difficult to find real-world examples of numbers like these. For example, a large city would have about 500,000 people. 500,000 gallons of water in a sphere would be about as tall as a three-story building. This many blood cells would most likely be visible to the naked eye at last.

Legendary Slither.io Number: 733,094

This number has no googological significance, but is personally important to me. 733,094 is the highest score I have ever gotten onthis game(I am still trying to get a million though) and amazingly, I have video evidence of it right here: https://www.youtube.com/watch?v=-P3JhEk2OXgI suggest you click on the video right now, take a break from all these mind-blowing numbers, and leave a like. Note that my score is the WORLD RECORD for Slither.io on mobile! I mean, just think about how big that ginormous number is. Even just 100,000 is already so big, most people have not achieved it, and 200,000 takes about an hour, which means that this score took well over 3 HOURS!

My previous record was just 434,804, which means I completely destroyed it with this new 733K! The 434K video is herehttps://www.youtube.com/watch?v=mg_dYKpstEM.

Ugh... I really don't have a life. Not only does no one play Slither.io anymore except for me, but it turns out that that score is one of my greatest achievements in life. But that doesn't matter now, so let's keep going.

823,543

I know this seems like a seemingly random number, but when I explain tetration, it will make sense. 823,543 is equal to 7^^2 = 7^7 = 823,543. It is one of the largest notable numbers smaller than a million.

You really have to understand that between this and a million is a HUGE difference. Just to represent the vastness of the continuum from here to a million, we will select a few more random numbers.

823,544

877,913

900,000

901,232

907,115

934,889

979,007

992,668

999,999

This number is sometimes used to describe numbers that are ever-so-close to a million, but not quite there yet. That's the point of this number, to signify being "that close".

Million: 1,000,000

We are finally here. The amazing and legendary one million. This is considered by many people to be a very large number, since it has much fewer real-world applications and is much harder to comprehend than previous numbers. I mean, amillion. That's pretty big. It is the boundary between Class 1 and Class 2 numbers by Robert Munafo's idea of classes (class 0 = 0 - 6, class 1 = 7 - 1,000,000, class 2 = 1,000,001 - 10^1,000,000, etc.)

I have an insight into how big this number is from playing Slither.io. A score of 10,000 points is usually already enough to get you on the leaderboard, and it's tiny - it only takes a minute or two to get. It's 1% of a million. But amillionis something that I have NEVER gotten and for the record, I'm not sure if ANYONE EVER HAS without cheating or using bots!

So if you are lucky enough to have a million subscribers on YouTube, or a million followers on Instagram or Twitter, or a million friends on Snapchat (is that even possible??), take it as a huge honor. That's a HUGE NUMBER, PEOPLE. For every little dot that you need to eat to get 1 million points on Slither.io, a person has subscribed to your channel/followed you/etc. That's insane. If a person subscribed to your channel every SECOND, it would take 12 days to hit a million subs. THAT'S TOTALLY INCREDIBLE.

As a comparison, I only have 11 subscribers on YouTube and 97 followers on Instagram (and that number is dropping fast). For a person to have 90,909 TIMES the number of subscribers that I have, strikes me as just amazing. And to think that we've just scratched the surface... No wait. We haven't even reached the surface. We're in the sky right now, and once we reach the surface it will still take some time to even leave a scratch on it.

So that's it for Part 2. But before beginning Part 3, I think I need to take a break to explain what tetration and up-arrows are, because I probably have you confused by now.

What is Tetration and What are the Hyper-Operators?

We have all learned in school that:

• Addition and subtraction are first-level, basic operations.
• Multiplication is repeated addition, and division is the opposite of subtraction. Thus, multiplication and division are second-level operations.
• Exponentiation is repeated multiplication. Roots are the opposite of exponentiation. Thus, exponentiation and roots are third-level operations.

What comes next? What is the mysterious fourth-level operation? It has a name, and it is calledtetration. It is repeated exponentiation. Numbers with tetration grow incredibly quickly, and as we know even exponential numbers grow pretty fast.

I once had an idea for tetration when I was 7 or 8 years old. At the time I knew a lot for my age; I was very good with second-level operations and knew what third-level operations were and how to use them. I was told that there was no fourth-level operation, but I knew it was possible. You would draw a triangle for the notation, so 2 tetrated to 3 would be 2 triangle 3. Here are some examples I thought of as a small child:

• 2 tetrated to 2 = 2²= 4.
• 2 tetrated to 3 = 2^2^2 = 2^4 = 16.
• 2 tetrated to 4 = 2^2^2^2 = 2^2^4 = 2^16 = 65,536
• 2 tetrated to 5 = 2^2^2^2^2 = 2^2^2^4 = 2^2^16 = 2^65,536 = a 19,729 digit number (we'll get into it later)

At the time, I thought 2 tetrated to 5 was impossible to compute. Now I know that there are things like logarithms and Python and stuff which can compute that. But since this number was already too gigantic for me to imagine, I wondered how horrifying 2 tetrated to 6 would be (it's a lot larger than a googolplex!), and realized my notation was pretty much useless. I never told anyone about my idea, almost forgot about it, and kept living my life.

But recently I discovered that my notation was real, and it had a name: tetration. Tetration is notated with two up-arrows: ^^ Since its growth rate is enormous, it is usually referred to a "hyper-operator" instead of just an "operator."

So you could have 2^^2, 2^^3, etc. Let's look at more examples which I didn't think of as a child. Also note that you have to compute the power tower from TOP TO BOTTOM. Since exponentiation does not have the associative property hold true anymore (i.d. x^(x^x) is not the same thing as (x^x)^x).

3^^2 = 3³= 27. You may notice that every number tetrated to 2 is equal to it raised to its own power.

3^^3 = 3^3^3 = 3^27 = 7,625,597,484,987

3^^4 = 3^3^3^3 = 3^3^27 = 3^7,625,597,484,987 = a 3.6 trillion digit number.

6^^2 = 6⁶= 46,656.

6^^3 = 6^6^6 = 6^46,656 = a 36,306 digit number.

7^^2 = 7⁷= 823,543

7^^3 = 7^7^7 = 7^823,543 = a 695,975 digit number. Remember, this is a bit too big for me to compute on a computer instantly, but I can use things like the Log 7-10 Constant (remember the Log 2-10 Constant?)

10^^3 = 10^10^10 = 10^10,000,000,000 = a 1 followed by 10,000,000,000 zeroes.

10^^4 = 10^10^10^10 = a 1 followed by 10^^3 zeroes. We have gotten past a GOOGOLPLEX!

I think you guys are getting the point. But there's more. If you thought this was crazy, brace yourselves for pentation - the 5th hyper-operator. It is notated with three up-arrows: ^^^ Let's look at some examples:

• 2^^^2 = 2^^2 = 4. Just as I stated earlier, 2^^^...^^^2 = 4 no matter how many arrows you have.
• 2^^^3 = 2^^2^^2 = 2^^4 = 65,536.
• 2^^^4 = 2^^2^^2^^2 = 2^^65,536 = 2^2^...^2^2 with 65,536 2's

Holy shit! With just 2^^^4, we have already created a number that is so gigantically unfathomable that no one can hope to ever describe it! If you want a taste of how big it is, then:

2^2^...^2^2 with 65,536 2's = 2^2^...^2^4 with 65,534 2's = 2^2^...^2^16 with 65,533 2's = 2^2^...^2^65,536 with 65,532 2's = 2^2^...^2^(that legendary 19,729 digit number) with 65,531 2's

With just five of the twos, we've already created that legendary 19,729 digit number! And we've still got 65,531 layers to go! As you can see, if we can't comprehend the tip of the iceberg how are we going to understand the whole tower??

3^^^2 = 3^^3 = 7,625,597,484,987. This number crops up pretty often when working with 3's and up-arrows.

3^^^3 = 3^^3^^3 = 3^^7,625,597,484,987 = 3^3^...^3^3 with 7,625,597,484,987 3's.

As you can now see this number is even more insane than the previous 2^^^4! We have no hope of ever understanding these numbers... or do we? After pentation comes hexation, which makes pentation look like shit:

• 2^^^^2 = 2^^^2 = 4
• 2^^^^3 = 2^^^2^^^2 = 2^^^4 = that big huge power tower of 2's 65,536 2's high

It seems totally hopeless now right? Actually, 2^^^^4 is equal to a power tower of twos2^^^^3 twos high. The number of twos in here is equal to the power tower of twos that is already gigantic.

• 3^^^^2 = 3^^^3 = that big huge power tower of 3's 7 trillion 3's high
• 3^^^^3 = 3^^^3^^^3 = 3^^3^^...^^3^^3 with 3^^^3 3's

Now this is pretty much hopeless. This is not a power tower anymore, this is a TETRATIONAL TOWER. And it's still impossible to write out completely. So how big is it? We will never know... until we reach Part 7!

We haven't quite gotten into the tetrational range yet, but we will, and if these numbers scare you, I suggest you close out of this window right now and stop looking. If you are scared right now or feel uncomfortable, you have googolophobia. If you keep scrolling down and you have googolophobia, you might literally do permanent damage to your brain, you might have a seizure, or you might even DIE. You have been warned...

This section contains numbers that are between a million and a googol.

1,048,576

This is the number of bytes in a Megabyte. If each letter takes up one byte, this would be enough space to store something like the U.S. Constitution.

Two million: 2,000,000

What could be better than one million of something?? Easy! Two million. Hey, two is better than one!

10!: 3,628,800

This is the equivalent of 10 factorial, and it is the largest factorial that I have memorized. As you can see, the factorial function is very fast-growing, and it's one of the fastest-growing functions that kids are taught in school. But it's nowhere near as fast as the busy beaver function, Graham's function, and so many more.

Number of arrangements on a 2x2 Rubik's Cube: 3,674,160

This is the number of ways you can arrange a 2x2 Rubik's Cube. It's really amazing to think that that tiny little baby version of a Rubik's Cube has almost 4 million combinations. By comparison, the 3x3 Rubik's Cube has a whopping 43quintillioncombinations. 3 million may seem small now, but remember - this many years ago, ancient hominids were still learning how fire worked and living in caves and all that. So this is still a really big number.

Pi^^e: about 5,328,483.61

Now you may be wondering: How do you tetrate a number to something that isn't an integer? In other words, how do you have a power tower of pi's e things tall? It doesn't make sense!!! Or does it? We use a process similar to that to calculate decimal exponents: x^0.1= a number y such that y^10 = x. To calculate x^2/7, for example, you take the seventh root of x first, then you square the result. The process is very similar for tetration. But, when the number of arrows is a decimal, things become a lot harder.

Ten million: 10,000,000

This number is also sometimes called a "crore". It's a pretty big number. It's close to the number of people that live in New York City. If you had a sphere with ten million gallons of water in it, it would be about ten stories tall. Imagine how long it would take to drink all that water! And yet all of humanity consumes almost a HUNDRED times this a day (but that's only because there's 7.6 billion humans).

16,777,216

This is 2²⁴, 8⁸, or 8^^2. From 7^^2 to 8^^2 we made a pretty big jump. But the jump to 9^^2 is even more horrifying.

This is another number which has to do with computers. It's equivalent to the number of bytes in 16 MB. Kind of surprising how that mere 4.9% increase (1MB is actually 1.049 million bytes remember) can add up to almost a million extra bytes. That's because the Log 2-10 Constant is not exactly 0.3, but about 0.301. That extra 0.001 can do horrifying things on a large scale.

33,554,432

This number is another power of two, at 2²⁵. It's a pretty interesting number, as it is composed of strings of two identical digits! Of course that two at the end ruins everything. But it's a pretty cool number.

Approximate circumference of the earth in meters: 40,000,000

This number may not seem that special to you, but there is one very little-known fact about it: It is the length of the meridian which passes through Paris. This was the original definition of a meter. Then, they used one centimeter cubed of water to get one gram. And that one gram of water was exactly one milliliter. And now you know where they all originated!

Of course, now that definition has changed. Light travels at almost exactly 299,792,458 meters per second, and now the official definition of a meter is the distance that light travels in a 299,792,458th of a second.

PewDiePie's number of subscribers on YouTube as of 22 June 2018: 63,533,349

As I mentioned earlier, 1 million subscribers is already fucking huge. But this guy has 63 MILLION!! That's a world record! Holy shit! This is unimaginable! That's almost one-fifth of the entire US population and almost one-hundredth of the world's population!!! But since not all the world has computers, more than 1% of people you probably know have subscribed to him. You may have even subscribed to him yourself. If someone subscribed to him every second, it would take 2 YEARS for him to get this many subs. He gets an estimated 30,000 a day, so that's about 1 subscriber every 2-3 seconds. As you have read this paragraph, he's gained an extra 10-20 subscribers (depending on how fast you read of course). But that is still crazy!! You are getting subscribers literally all the time!

73,939,133

This is a prime number with a very cool property. You can shave one digit off of the end and get a prime, and keep doing that until you get just 7, and you get a prime each time! It is the largest number to have this property. So 73,939,133 is prime, 7,393,913 is prime, 739,391 is prime, etc.

Myllion: 100,000,000

This is a modified version of million. It along with the -yllion naming system were developed by a mathematician named Donald Knuth, the same Knuth that developed the up-arrow notation described earlier. He invented numbers like byllion, tryllion, etc. which grow much faster than normal -illions. A myllion is equal to 10^8, or one hundred million. But abyllionis a lot bigger.

Selena Gomez's number of followers on Instagram as of 22 June 2018: 135,458,041

Now, this is completely and utterly insane. A million subs on YouTube is already a lot, and PewDiePie has a shitload of subs, way more than a million. But, he is dwarfed by Selena Gomez, who has more than twice as many followers on Instagram! That's about 2% of the world population. But then again, not everyone in the world owns a phone, but there are also people with multiple accounts. There's approximately 800 million accounts worldwide, and that means that 1/5 of everyone who uses Instagram has followed her. If you're reading this right now, you probably have Instagram yourself, and that means there's a 1/5 chance that you have followed her yourself!! God, this number is bigger than 2²⁷(that is 134,217,728). That means that if you started with one Instagram follower and doubled the number of followers every day, you could wait 27 days and you still wouldn't have as many followers as she did!

387,420,489

This is 9^^2 = 9⁹- I told you it would be a big jump from 8^^2.

Billion: 1,000,000,000

This is the second number in the -illion system, and is equivalent to 10^9. The rule of the -illion system is that each -illion is 10^(3x + 3). So if x = 59, for example, the 59th -illion will be equal to 10^(3(59) + 3) = 10^180. That number is known as anovemquinquagintillion.

Now, everyone take a deep breath and consider just how huge this number is. A billion seconds is about 27 years. If you were to count from 1 to a billion, it would take about 125 years, assuming you would never sleep. But you wouldn't get even close to 125 years old if you never slept, and no one has ever gotten to that age in the history of our world! It's unlikely it will happen soon. So I think it's safe to say that a billion is just a number that no human will ever count to or be able to count to.

1,073,741,824

This is the number of bytes in a gigabyte. This is really huge. Only large games like GTA and Hearthstone take up more space than a gigabyte. With a gigabyte of space, I could store numbers up to 2^1,074,741,824 which is a 323,228,497 digit number!! And this huge number has only 10 digits!

2,147,483,647

The eighth Mersenne prime. It is equal to 2^31 - 1. This is the maximum 32-bit integer possible. You need one bit for the sign, and that leaves 31 bits to play with. Thus, every number smaller than 2^31 and larger than -2^31 is possible to make.

2,147,483,648

This is a variant of the popular browser game 2048, developed by CyberZHG. Instead of playing on a 4x4 board, you have an 8x8 board, and you need to get 2^31 instead of 2^11. There are multiple modes and cool features as well! However, getting 2,147,483,648 would take years and years of gameplay though.

Views on the YouTube video "Despacito" as of 23 June 2018: 5,254,815,996

This is the most viewed video on YouTube, and has over five billion views! That's more than half the world has seen this video (of course, some people see it twice, etc., but still a large portion of the world has seen this video). This is a number that puts Selena Gomez's follower count on Instagram to shame. The video is about a year and a half old as of 23 June 2018, and that means that it must have received more than 20 views every second! As you have read this paragraph, HUNDREDS of people have clicked on the video! This is a more fast-growing number than the population of the earth. THAT'S INSANE!

World population as of 23 June 2018: 7,631,177,134

Every minute, there are about 300 births and about 200 deaths. There's about a population increase of about 200,000 every day. The world population is one of the most well-known numbers larger than a billion. The thing about it, though, is that it is constantly changing. However, there are more bacteria on your body than there are people on the planet, so the bacterial population on Earth is WAY BIGGER!!!

Ten billion: 10,000,000,000

This number can be called 10^^2 or 10^10. It can be called dialogue by a system which extends to trialogue, tetralogue, etc. we'll talk about it later.

e^^Pi: about 19,337,456,547

Since I included Pi^^e in this list, it should make sense that I'm including e^^Pi as well. It's an example of a much larger number.

108,000,000,000

This is a rough estimate for how many people have ever lived on Earth. According to a video by RealLifeLore (https://www.youtube.com/watch?v=l-S34gZ3xwg), if every human who ever lived on our planet came back to life, most of them would die within a few months. I suggest you click on RealLifeLore's channel right now (the video that I provided a link to is the first one of his videos I saw), his videos are very cool and fun to watch, and are intriguing even for topics that seem relatively boring.

Trillion: 1,000,000,000,000

This number is equal to 10¹², and it's the third -illion. This is the largest -illion many people know, as the question "What comes after a trillion?" is Googled very often. The answer to that is a quadrillion, but I think I should explain what a trillion is first.

A trillion is already a number that is unimaginably huge. The SI prefix for this number is "tera", which means "monster." While it may seem tiny, it's true that this number is really monstrous! I mean, a trillion seconds is over 32,000 years, a trillion dollars would probably fill up the Empire State Building (even if it were $100 bills), and a trillion miles would be the distance from the Earth to the Sun... 10,753 times!! Now, 10,753 is already pretty big, but a TRILLION!! It is very difficult to wrap your mind around...

Megafugathree: 7,625,597,484,987

This is a famous number which can be expressed in many ways. It is 3^^^2, 3^^3, 3^3^3, 3²⁷, or 7,625,597,484,987. It comes up very often when working with threes and power towers and is an example of a tetrational number that is not too big, but not too small. This is the first step toward computing the great Graham's number: To compute Graham's number (G64) you first need to know G1, and for that you need to know 3^^^3, and for that you need to know the value of this number! Cool, right?

Megafugafour, however, is a much larger number. It's already larger than a googolplex!

Quadrillion: 1,000,000,000,000,000

This number is equal to 10¹⁵. It is encountered rarely except in areas such as cosmology. However, it is important not to neglect such numbers, as they can actually be very useful.

It's kind of shameful to see that this is the largest number that Google puts into its dictionary (the actual English dictionary goes up to vigintillion, the 20th -illion = 10^63). Google based its own name off of the infamous googol, a super big number, and it doesn't even bother to put some large numbers into its dictionary? In fact, the word "googol" itself is not in Google's dictionary! How pathetic is that? Microsoft, however, puts "googol", "googolplex", and even "googolplexian" in its dictionary. Kudos to you Bill Gates!

Gogolbit: 1,125,899,906,842,624

This is 2^50, the number of bytes in a petabyte. It's about 1.13 quadrillion. By now, an amount of storage this large is probably useless; even a terabyte (which is 2⁴⁰= about 1.1 trillion bytes) is big enough to support almost anything.

9,007,199,254,740,992

This is the largest and one of the most famed variants of the browser game 2048 (link: https://www.csie.ntu.edu.tw/~b01902112/9007199254740992/). Just like in 2147483648, you are on an 8x8 board. However, there are no cool mods, but there is a robot which can play for you very fast. However, even with this AI, it would still take longer than the universe has existed to beat the game.

A YouTuber by the name of "bigmacdontcare" once uploaded a video where he beats the game (he stated that he uses hacks in the description). The video received over two million views.

Byllion: 10,000,000,000,000,000

This is the second member of Knuth's -yllion series and is equal to 10¹⁶. But how exactly does his series work?

Basically, he changes all the -illions to -yllions, and goes from here:

• Myllion = 10⁸
• Byllion = 10¹⁶
• Tryllion = 10³²
• ...
• Decyllion = 10^4,096
• N-yllion = 10^(2^(N + 2))

As you can see this is an EXTREMELY fast growing sequence by the layman's standards, but very, very slow by googological standards.

Quintillion: 1,000,000,000,000,000,000

This seems to be a turning point for most people in large numbers. Numbers beyond this are used vary rarely. Even a cosmologist probably wouldn't use numbers like this because he would be afraid that people wouldn't know how big the numbers were!

9,223,372,036,854,775,807

This is the largest number that can be expressed in the 64-bit format. You probably already know why from reading all the other entries (32,767, 2,147,483,647, etc.). You may notice that all of these numbers end in 7. That's because the last digits of powers of two always follow a pattern: 2, 4, 8, 6, 2, 4, 8, 6, ... and in this case, 2^63 ends in an 8. Using this technique we can find many of the terminating digits of large powers of numbers. A common AMC/Math Kangaroo problem would be to find the last digit of a power of two, say 2^2016. In this case it would be 6 since 6 is the 2016th element in the 2, 4, 8, 6, ... pattern.

18,446,744,073,709,551,616

The number of possible 64-bit integers.

Number of combinations on a Rubik's Cube: 43,252,003,274,489,856,000

This is the number of combinations on a 3x3 Rubik's Cube - it's about 43.3quintillion. It's really hard to believe that a tiny little toy like the Rubik's Cube could have a number of combinations this big! But that's not it - if you're allowed to "cheat", or take the Rubik's Cube apart, you can produce 12x more combinations! How insane is that?

However, if you have a 4x4 Cube, the number is WAY bigger. It also has its own entry on this list.

Guppy: 100,000,000,000,000,000,000

This number is equal to 100 quintillion, or 10²⁰. It is another of Sbiis Saibian's numbers. The reason it is called a "guppy" is because, horrifyingly, it is only a guppy compared to what will come next, despite looking enormous...

Sextillion: 10²¹

This is the 6th -illion, and it is 1 followed by 21 zeroes. It's already incredibly enormous and numbers this big are only encountered in astronomy and cosmology! To get an insight into how big this is, a big mound of a sextillion would be about the size of a flea, or big enough to see with the naked eye! Of course, if you put all those atoms into a big line, it would be longer than the earth's diameter, but that line would be way too thin to see. This really tells us about how small an atom is rather than how big a sextillion is, though.

Septillion: 10²⁴

The 7th -illion. It's larger than the diameter of our universe in miles. It's the largest -illion to receive a name in the SI prefixes (yotta- for 10²⁴and yocto- for 10^-24), but considering that a "yottameter" is already about the size of our universe, this prefix is pretty much useless.

Octillion: 10²⁷

The eighth -illion. It's the first not to receive an official name in the SI prefixes.

Seven octillion: 7 × 10²⁷

This number is close to the number of atoms in the average human body! As you can see, there are a LOT of atoms in your body, way more than you may have previously thought.

Nonillion: 10³⁰

The ninth -illion. It's a 1 followed by 30 zeroes!!

Tryllion: 10³²

This is the 3rd -yllion, and it's equal to 10^32. It's equal to 100nonillion.

Planck temperature in Kelvins: 1.417 × 10³⁰

Remember when I mentioned the for Planck time, length, and temperature? Well, I never included a Planck temperature. But now I did!

So, this is the hottest temperature possible. It is about 141.7nonilliondegrees. This temperature is the hottest temperature possible because the radiation that it emits has wavelengths of a Planck length, and as you guys already know, that is the smallest possible length.

If you know someone who is this hot, good for you.

Decillion: 10³³

This is the 10th -illion, and one of the most famous larger ones. It is incredibly enormous and it is hard to think of any good examples! For example, the Sun weighs about 1.989 decillion grams. People, that is simply gigantic and mind-boggling!

Undecillion: 10³⁶

This is a much lesser-known -illion than a decillion. It is 1,000 times bigger. It would be about the volume of a red supergiant star in grams. Now, an object of that size is already very, very difficult to imagine - but in terms of grams it is almost imossible.

Two undecillion: 2 × 10³⁶

This number appeared in a lawsuit. Someone tried to sue the company Au Bon Pain for two undecillion dollars once, and, unsurprisingly, lost. But now that lawsuit has become sort of a joke.

Imagine if the plaintiff trying to sue Au Bon Pain had won, though. If the company owned the earth's weight in diamond, it would still not be enough!

Duodecillion: 10³⁹

Tredecillion: 10⁴²

Quattuordecillion: 10⁴⁵

Quindecillion: 10⁴⁸

Gogol: 10⁵⁰

This number was named by Sbiis Saibian, and it was used to define "gogol-minutia" for 10^-50. Just think about how big it is. Take the goby - that's 10³⁵. It's a quadrillion times larger than a guppy. This gigantic number we refer to as the gogol is a quadrillion times larger than a goby - the goby doesn't seem much bigger than a minnow or even a guppy now!

Sexdecillion: 10⁵¹

Septendecillion: 10⁵⁴

Octodecillion: 10⁵⁷

Novemdecillion: 10⁶⁰

Vigintillion: 10⁶³

This is the largest -illion number in the English dictionary besides centillion. It is much more well-known than all of the previous -illions larger than a decillion. It is possible to name numbers unvigintillion, duovigintillion, etc. but these numbers are not in the dictionary. Hopefully you understand the pattern. By now this figure is so large, it is almost impossible to think of a real-world comparison, but I'll try my best. A red supergiant is made of about a vigintillion atoms.

Quadryllion: 10⁶⁴

This is the 4th -yllion, equal to 10^(2^(4+2)) = 10^64. Hopefully you are getting the pattern by now. It may seem only a tad bigger than a vigintillion, which on this scale it is, but in reality it is 10 TIMES BIGGER!!

Hundred quinvigintillion: 10⁸⁰

This is a 1 followed by 80 zeroes, and it is also called an "ogol" by Sbiis Saibian. It is a rough estimate of the number of particles in the universe, and it is believed that at the time of the Big Bang, all those 100 quinvigintillion particles were squished in a very very tiny space smaller than a proton, at a temperature nearing the Planck temperature. The question is, how did those particles get squished in a space that small, and even more importantly, where did 100 quinvigintillion particles even come from??

Thirty septenvigintillion: 3 × 10⁸⁵

This is a three followed by 85 zeroes, and it was the largest number to ever appear in a Calvin and Hobbes comic.

Trigintillion: 10⁹³

This is 1 followed by 93 zeroes. We're getting very close to that mighty googol!

Of course, you could have untrigintillion, duotrigintillion, etc.

69!

This is sixty-nine factorial, and it is the largest factorial expressible on calculators that overflow at a googol. It's about 171.1untrigintillion.

Its full decimal representation is: 171122452428141311372468338881272839092270544893520369393648040923257279754140647424000000000000000.

Googol: 10¹⁰⁰

We're finally here. The legendary and amazingGOOGOL!!

Googol is a very well-known large number; it is so well-known that is has made its way into the English dictionary. But where did it come from? In 1920, a mathematician named Edward Kasner wrote down a 1 followed by 100 zeroes, and wondered what this huge number should be called. He turned to his nine-year-old nephew, who thought that it should be called something silly, so he said, "a Googol!" and a googol it was. It became popular both in pop culture and mathematics. Just so you get a feel for how big this number is, it is about 100 quintillion times bigger than the number of atoms in the universe. 100 quintillion is itself a number very hard to grasp in your mind. But it's not bigger than the volume of the universe, in, say, nanometers! Therefore, we can still find real-world examples of this number.

The googol is called 10 duotrigintillion by the -illion naming system. Its full decimal representation is: 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.

What's shameful is that the company Google didn't even put this word in its dictionary, and its name was actually based off of this word (Actually, it was a typo, the name of the company was actually supposed to be Googol)! Wait, why am I building this website with Google Sites, then?

I think I've written enough, so I'll stop here.

That's it for Part 3, and here comes Part 4 with some numbers to truly scare the shit out of your ass!

This section contains numbers that are in between a googol and a googolplex. There are still a few numbers in here which can relate to the real world in some primordial way, but soon we will leave the real world far, far, far behind us.

Quadragintillion: 10¹²³

The 40th -illion. After this would be unquadragintillion, duoquadragintillion, ..., octoquadragintillion, novemquadragintillion. Hopefully you are getting the pattern. If you aren't, though, feel free to look it up.

Quintyllion: 10¹²⁸

The 5th -yllion.

Quinquagintillion: 10¹⁵³

The 50th -illion. There really isn't much to say about numbers on this scale.

Quattuorquinquagintillion: 10¹⁶⁵

This is the longest -illion name smaller than a centillion.

Quinquinquagintillion: 10¹⁶⁸

Not really much to say about this one, except that it has a nice ring to it. Because of this, a lot of people cite this as their favorite number, and this number receives more Google searches than usual.

Sexagintillion: 10¹⁸³

This is the 60th -illion. Getting closer to a centillion.

Three hundred sexagintillion: 3 × 10¹⁸⁵

This is very close to the number of Planck volumes in the universe. A Planck volume is a cube with a Planck length as its side length. Now, this may only seem about 125% bigger than the number of particles in the universe, but this is where exponents can begin to get tricky - you could square the number of particles in the universe and it would still be smaller than this. Of course, 185 is is about 125% bigger than 80, but that's where exponents start to trick you. 10¹⁸⁵is 10¹⁰⁵times as big as 10⁸⁵.

Gargoogol: 10²⁰⁰

This is equal to googol × googol - that's what the gar- prefix is. You probably have never heard of this prefix before, because it isn't an official prefix. This is a prefix invented by Alistair Cockburn. In fact, I'm going to take a break to explain it right now and his other prefixes.

What are the Gar-, Fz-, Fuga-, and Megafuga- prefixes?

The gar-, fz-, fuga-, and megafuga- are prefixes invented by Alistair Cockburn. So basically:

• Gar-n = n × n = n². Nothing big. Thus,
• Garten = 100
• Garthousand = 1,000,000
• Gargoogol = 10^200

Next we have fz-, the next most powerful prefix:

• Fz-n = nⁿ= n^^2. Getting stronger. So,
• Fzeight = 16,777,216
• Fztwenty = 1.048576 × 10²⁶(exact)
• Fzthousand = 10^3,000

Fuga- is a much more powerful prefix:

• Fuga-n = n vv n = n vvv 2. These are down-arrows instead of up-arrows. This is just like tetration, but computing the power tower from bottom to top instead of from top to bottom, so it is weaker. Therefore,
• Fugatwo = 4
• Fugathree = 19,683
• Fugafour = 2^512 = about 1.34 ×10^154

Megafuga-, however, is the most powerful of all of these, and megafugafour is already larger than a googolplex.

• Megafuga-n = n^^n = n^^^2. These are up-arrows, unlike the fuga- prefix. Hence,
• Megafugatwo = 4
• Megafugathree = 7,625,597,484,987
• Megafugafour = 2^2^513 = about 10^10^157

These prefixes are already very powerful as they are, but Cockburn invented another prefix that is far more powerful than all of these combined: booga-. It achieves a growth rate of ω in the fast-growing hierarchy, while all the others achieve growth rates of just 2 to 3. ω is an ordinal, ordinals are the last section on this list, and ordinals are sort of like infinity. So:

• Booga-n = n n-ated to n^^^...^^^n with n - 2 arrows. We can get,
• Boogaone = 1 + 1 = 2
• Boogatwo = 2 × 2 = 4
• Boogathree = 3^3 = 27
• Boogafour = 4^^4 = about 10^10^157
• Boogafive = 5^^^5 = ???

I mean, imagine how big boogafive would be! That would be 5^^5^^5^^5^^5, and 5^^5 is already WAY too big for us to compute. Let's assume we can. Then we have to compute a power tower of 5's THAT MANY 5's tall, and then compute a power tower of 5's THAT MANY tall, and do that 2 more times! Holy shit!

Septuagintillion: 10²¹³

Octogintillion: 10²⁴³

WEED₄₂₀(420)

This number was part of a Brilliant.org problem I posted on 4/20 Day 2018 (https://brilliant.org/problems/happy-420-day/?ref_id=1492899). WEED(x) = 4x + 20 and WEED_y(x) = WEED(WEED(WEED(...(WEED(x)...))) with y nested functions. In this case, our number is equal to WEED(WEED(WEED(...(WEED(420)...))) with 420 nested functions. The result is about 3.128132 × 10^255.

Also see (10^63)!!.

Nonagintillion: 10²⁷³

Novemnonagintillion: 10³⁰⁰

The 99th -illion. This is the largest -illion that is smaller than a centillion, and often confused with it. But a centillion actually has 303 zeroes, not 300, and a number with 300 zeroes is called a novemnonagintillion.

Centillion: 10³⁰³

Finally, we have reached the infamous centillion. The 100th -illion. This number dates back to as early as the 1850s, when someone combined the cent- prefix from Latin with the -illion root. This is the largest -illion that most people have heard of, and some people confuse it with a googol and googolplex; they aren't sure if a googol is bigger (which, of course, it isn't). However, it is way smaller than a googolplex - we will get to that one later.

Uncentillion: 10³⁰⁶

This is the 101st -illion, and much rarer than centillion. It is the largest -illion which can be represented in the double-precision floating point - we will get to that next entry.

Googlek: 2^1,024

This is the limit for the floating-point class in many programming languages (although some languages like Python have a separate class that can go further). It's about 1.79709 × 10³⁰⁸, or 179.719 uncentillion.

Duocentillion: 10³⁰⁹

The first -illion inexpressible in the floating-point. This looks very similar to ducentillion (which is a lot bigger), but is not the same thing.

Faxul: 200!

This is a number equal to 200 factorial, which is about 7.886579 × 10³⁷⁴. That's a 375 digit number! However, don't let that deceive you, as all of Alistair Cockburn's prefixes are WAY more powerful than the factorial function.

Octodragondilliatillion: 10⁵⁰⁰

This number was coined in a strange way. An article was created on Googology Wiki titled "Octodragondilliatillion," which apparently was 10 raised to the 500th power. The creator of the page, J200pd, stated that his friend had come up with the number and provided no other sources. The article was quickly deleted because, well, one can't just put an article on Googology Wiki and say that their friend is the source; the policy is that you need to have a source like a website, YouTube video, etc. but since 10⁵⁰⁰has no other name, I thought I would list it on here and "bring it back to life."

Googocci: 402²⁰¹

This is the 201st number in Andre Joyce's system. His system uses Roman numerals at the end (CCI = 201). His system, is as you can imagine,very weird. You could have numbers like a googoviji (you need to use a "j" because a googoviii wouldn't be distinguishable from googovi, etc). But this number is notable because it sounds Italian-like, as stated by Joyce himself. It's about 2.814729 × 10^523.

Ducentillion: 10⁶⁰³

This is the 200th -illion. Not to be confused with the duocentillion stated earlier.

Trecentillion: 10⁹⁰³

The 300th -illion.

10^1,000- 1

This is a long string of 1,000 9's and is a common response to "the largest number one can write out." Suppose there is a competition to whoever can write the largest number on a notecard, and suppose you can fit 1,000 characters on that notecard. A naive googologist (or in other words, a FUCKING NOOB) would probably write this number. A slightly more skilled googologist would write, say, 995 9's with a 99,999 exponent. But that's still noob level.

Googolchime: 10^1,000

This number was another one of those coined by Sbiis Saibian, and it is one followed by a thousand zeroes.

Quadringentillion: 10^1,203

The 400th -illion.

Quattuorquinquagintaquadringentillion: 10^1,365

The longest -illion that can be formed with the conventional -illion system - it is 37 letters long. After that Jonathan Bowers and Sbiis Saibian have developed their own -illion systems, but they are not official. But when you consider that only the first 20 -illions are in the dictionary, is this official either?

Quingentillion: 10^1,503

Sescentillion: 10^1,803

Sexsexagintasescentillion: 10^2,001

The 666th -illion. Just try to say it out loud and you'll say the word "sex" a total of 3 times.

Septingentillion: 10^2,103

Octingentillion: 10^2,403

Nongentillion: 10^2,703

Novemnonagintanongentillion: 10^3,000

The 999th -illion. With the fz- prefix it can be called fzthousand.

Millillion: 10^3,003

The 1,000th -illion. Some people call it a millinillion, but millillion is more preferred among googologists. Now, people, take a moment to appreciate how huge this number is (I haven't asked you to do that in a while, but I will now). It's greater than the number of particles in the universe... raised to the 37th power!!! So to understand what I'm saying, imagine a "first-order sphere" as one with as many particles as there are in the universe. Now a second-order sphere is one with as many first-order spheres as the first-order sphere has particles. A third-order sphere has as many second-order spheres as the first-order sphere has particles. Keep going until you reach the 37th order sphere. That sphere has roughly a millillion particles. People, that is insane!!

Millimillion: 10^3,006

The 1,001th -illion; it has a sort of funny name. After this you continue with millibillion, millitrillion, ..., millidecillion, ..., millicentillion, ..., millinovemnonagintanongentillion, etc.

Decyllion: 10^4,096

This is the 10th -yllion in Knuth's series. As you can see this series is growing EXTREMELY quickly.

Googolbell: 10^5,000

This is another of Sbiis Saibian's numbers, and is a 1 followed by 5,000 zeroes.

Dumillillion: 10^6,003

The 2,000th -illion - it comes after that "millinovemnonagintanongentillion" I mentioned earlier.

Tremillillion: 10^9,003

This is the 3,000th -illion; it is 1 followed by OVER 9,000 ZEROES!! As you guys can see, even the NUMBER OF DIGITS in the number is growing fast.

Googoltoll: 10^10,000

Another of Sbiis Saibian's googolisms.

Quadrimillillion: 10^12,003

Quinmillillion: 10^15,003

Sexmillillion: 10^18,003

10^18,726

A lower-bound for BB(6) - as you guys can see this is WAY bigger than the 4,098 lower-bound for BB(5).

2^^5

This is about 2.0035 × 10^19,728, and is the only integer non-trivial form of n^^5 which can be written out. Its full decimal expansion is a whopping 19,729 digits:

20035299304068464649790723515602557504478254755697514192650169737108940595563114530895061308809333481010382343429072631818229493821188126688695063647615470291650418719163515879663472194429309279820843091048559905701593189596395248633723672030029169695921561087649488892540908059114570376752085002066715637023661263597471448071117748158809141357427209671901518362825606180914588526998261414250301233911082736038437678764490432059603791244909057075603140350761625624760318637931264847037437829549756137709816046144133086921181024859591523801953310302921628001605686701056516467505680387415294638422448452925373614425336143737290883037946012747249584148649159306472520151556939226281806916507963810641322753072671439981585088112926289011342377827055674210800700652839633221550778312142885516755540733451072131124273995629827197691500548839052238043570458481979563931578535100189920000241419637068135598404640394721940160695176901561197269823378900176415171900511334663068981402193834814354263873065395529696913880241581618595611006403621197961018595348027871672001226046424923851113934004643516238675670787452594646709038865477434832178970127644555294090920219595857516229733335761595523948852975799540284719435299135437637059869289137571537400019863943324648900525431066296691652434191746913896324765602894151997754777031380647813423095961909606545913008901888875880847336259560654448885014473357060588170901621084997145295683440619796905654698136311620535793697914032363284962330464210661362002201757878518574091620504897117818204001872829399434461862243280098373237649318147898481194527130074402207656809103762039992034920239066262644919091679854615157788390603977207592793788522412943010174580868622633692847258514030396155585643303854506886522131148136384083847782637904596071868767285097634712719888906804782432303947186505256609781507298611414303058169279249714091610594171853522758875044775922183011587807019755357222414000195481020056617735897814995323252085897534635470077866904064290167638081617405504051176700936732028045493390279924918673065399316407204922384748152806191669009338057321208163507076343516698696250209690231628593500718741905791612415368975148082619048479465717366010058924766554458408383347905441448176842553272073155863493476051374197795251903650321980201087647383686825310251833775339088614261848003740080822381040764688784716475529453269476617004244610633112380211345886945322001165640763270230742924260515828110703870183453245676356259514300320374327407808790562836634069650308442258559670392718694611585137933864756997485686700798239606043934788508616492603049450617434123658283521448067266768418070837548622114082365798029612000274413244384324023312574035450193524287764308802328508558860899627744581646808578751158070147437638679769550499916439982843572904153781434388473034842619033888414940313661398542576355771053355802066221855770600825512888933322264362819848386132395706761914096385338323743437588308592337222846442879962456054769324289984326526773783731732880632107532112386806046747084280511664887090847702912081611049125555983223662448685566514026846412096949825905655192161881043412268389962830716548685255369148502995396755039549383718534059000961874894739928804324963731657538036735867101757839948184717984982469480605320819960661834340124760966395197780214411997525467040806084993441782562850927265237098986515394621930046073645079262129759176982938923670151709920915315678144397912484757062378046000099182933213068805700465914583872080880168874458355579262584651247630871485663135289341661174906175266714926721761283308452739364692445828925713888778390563004824837998396920292222154861459023734782226825216399574408017271441461795592261750838890200741699262383002822862492841826712434057514241885699942723316069987129868827718206172144531425749440150661394631691976291815065797455262361912248480638900336690743659892263495641146655030629659601997206362026035219177767406687774635493753188995878662821254697971020657472327213729181446666594218720034745089428309115351892711142871083761592223802766053278233516615551493693757784666701457179719012271178127804502400263847587883393968179629506907988171216906869295382485298300234760684541141781391106485602365497542274972310076151318700240539105109138178437217914225285874320985249578780346837033378184214440171386881242499844186181292711985333153825673218704215306311977485352146709553346263366108646673322924098798492566911095161436186015489097402419135096230436121961281659505186660220307156136847323646608689050142639139065150639081993788523183650598972991254044794434251667742996598118492331515552728832740283526884424087528112832899806259126736995462473415433335001472314306127503903073971352520693381738433229507010490618675394331307847980156551303847581556852362180104196502555961819349863159132330360964619059902361126811960234418433633345949276319461017166529138237171823942992162725384617760656945422978770713831988170369645886898118632109769003557358846244648357062914530527571012788720279653644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Terrifying, right? Just so you know, we're just getting started with large numbers!

Septimillillion: 10^21,003

Octimillillion: 10^24,003

Nonimillillion: 10^27,003

Decimillillion: 10^30,003

The 10,000th -illion - wow!

Myriabang: 10,000!

This is 10,000 factorial - that's what the -bang suffix is. This number is about 2.84625 × 10^35,659- that's 35,660 digits.

6^^3

Six tetrated to the third - this number is about 2.65911 × 10^36,305. 7^^3, however, is WAY bigger.

Hitchhiker's Number: 2^276,709

This was jokingly coined inThe Hitchhiker's Guide to the Galaxyas the reciprocal of the probability that a spaceship would pick someone in the cosmos up in 30 seconds. It's about 5.11764 × 10^83,298.

Googolgong: 10^100,000

Another one of Sbiis Saibian's googolisms. It's supposed to evoke the image of a giant gong that will sound for a googolgong years. The name, however, was coined in an unusual way.

So, basically, a young Sbiis Saibian was very interested in large numbers, and was telling his friend's dad about all of the large numbers he was learning about, the googol, the large -illions, etc. The dad then proceeded to tell Sbiis that "scientists had come up with a number called a googolgong, and it is a 1 followed by 100,000 zeroes" and then Sbiis thought, "If 'scientists' had come up with this number, I WILL COME UP WITH EVEN MORE INSANE NUMBERS!!"

But it turns out, there was no such thing as a googolgong. Somehow, the dad managed to screw up both the name, definition, and origin! Three mess-ups in one! First of all, the number he was referring to was probably a googolplex, and it is a followed by a GOOGOL, not 100,000, zeroes. And a single mathematician and his nephew came up with this number, not "scientists." But I've been mentioning this Sbiis Saibian figure a lot recently, without explaining who exactly he is. So I'll do that now.

Who is Sbiis Saibian?

Sbiis Saibian (the name is a pseudonym) is a googologist who has been fascinated with large numbers from a very young age. As a second-grader, he was always struggling to reach infinity. To him, infinity was real and reachable, and he kept trying to devise ways to create bigger and bigger numbers which no one had ever concieved before, with notations he has come up with for large numbers, such as Poly-Cell Notation and Hyper-E Notation, and managed to surpass Graham's Number without even realizing it until later! People, he found a way to trump Graham's Number in just SECOND GRADE without any naive extensions or nooby stuff. If you've heard anything about that number, you've probably heard that it's the "largest number ever coined", it's "indescribably huge", and "impossible to visualize." Needless to say, it is NOT the largest number ever coined, and if a SEVEN YEAR OLD CHILD can think of ways to make bigger numbers, I think Graham's number is actually pretty small, not to mention puny.

Saibian has literally named over 15,000 googolisms. They're on his website (https://sites.google.com/site/largenumbers/home). You should check it out!

Centimillillion: 10^300,303

The 100,000th -illion.

Hamlet Monkey Number: 48^182,831

This is about 1.0391 × 10^307,383, and it is one of those numbers that seems like it's smaller than actually is, until I describe it. Let's assume you have a monkey sitting at a 48-key typewriter, with equal odds of typing each key. This number is the reciprocal of the odds of the monkey typing the entire Shakespearean play Hamlet - perfectly - on its first try!! How insane is that! It only has a 1 in 48 chance of typing the FIRST CHARACTER right, and a 1 in 2,304 chance of typing the first TWO right, let alone all 182,831 characters that are in the play (letters, not fictional characters, mind you)!! This probably wouldn't work with a real monkey, of course, but this is just one example of the crazy numbers you can create very easily!

Lakhbang: 100,000!

This is about 2.824229 × 10^456,573, and it's equal to the Lakh (100,000) factorial.

7^^3

This is about 3.75982 × 10^695,974. That's a 695,975 digit number! It comes close to limits of numbers that an ordinary computer can compute without crashing. I tried to put its entire decimal expansion on this page, and it resulted in the page being very unresponsive and I almost had to delete my site! But if numbers this tiny cause this much trouble, what sort of chaos can we expect from numbers that are BIGGER?

Milliplexion: 10^1,000,000

The boundary between Class 2 and Class 3 numbers; it is 1 followed by a MILLION zeroes! It could be called "millionplex", but milliplexion sounds cooler, don't you think?

Micrillion: 10^3,000,003

The millionth -illion. You may notice that centi- = 1/100, milli- = 1/1,000, micro- = 1/1,000,000, etc. That's how people have extended the -illion system!

Micromillion: 10^3,000,006

No, this is not a millionth of a million - it's the 1,000,001th -illion. After this you can have crazy names like microbillion, microtrillion, etc. just like with millillion.

Vigintyllion: 10^4,194,304

The 20th -yllion.

10^6,000,000

A million raised to the millionth power. Can also be called fzmillion.

8^^3

This is 8^16,777,216, or about 6.01452 × 10^15,151,335. Wow!

Quinvigintyllion: 10^134,217,728

The 25th -yllion.

9^^3

9^387,420,489, or about 4.28145 × 10^369,693,099.

Nanillion: 10^3,000,000,003

The billionth -illion.

Trialogue: 10^^3

This is equivalent to 10^10^10, or 10^10,000,000,000. A 1 followed by ten billion zeroes!

Picillion: 10^3,000,000,000,003

Femtillion: 10^(3 × (10^15) + 3)

10^(8 × 10^16)

This number was the largest number mentioned in Archimedes' bookThe Sand Reckonerin 250 BC, where it was described as ((10^8)^(10^8))^(10^8). This number is a 1 followed by 80 quadrillion zeroes. For those times, this number was quite enormous!

Attillion: 10^(3 × (10^18) + 3)

Zeptillion: 10^(3 × (10^21) + 3)

Yoctillion: 10^(3 × (10^24) + 3)

Notice how fast we're progressing? We're already at the septillionth -illion! A yoctillion is the approximate result of raising the previous number to the 1,000TH POWER!

Sexsexagintasescentiyoctosexsexagintasescentizeptosexsexagintasescentiattosexsexagintasescentifemtosexsexagintasescentipicosexsexagintasescentinanosexsexagintasescentimicrosexsexagintasescentimillisexsexagintasescentillion: 10^(2 × (10^27) + 1)

This is the 666,666,666,666,666,666,666,666,666th -illion. If you say it out loud, you'll say the word "sex" a total of 27 times (please don't try to pronounce it). Notice how long the names can get with -illions this big.

Xonillion: 10^(3 × (10^27) + 3)

The octillionth -illion. Notice how xon- isn't an official SI prefix anymore.

Vecillion: 10^(3 × (10^30) + 3)

Mecillion: 10^(3 × (10^33) + 3)

The decillionth -illion!! We're not far from the legendary GOOGOLPLEX!!

Centyllion: 10^(2^102)

The 100th -yllion. It's about 10^(5.071 × 10^30). As you can see, the -yllion series is BLASTING into the stars... or is it?

Icosillion: 10^(3 × (10^63) + 3)

The vigintillionth -illion. You may notice now that in Jonathan Bowers' system, he starts using Greek prefixes instead of Latin ones.

Triacontillion: 10^(3 × (10^93) + 3)

Googolplex: 10^10^100

We have finally made it! Another great step - the GOOGOLPLEX!

This is a 1 followed by a googol zeroes, which is a 1 followed by a hundred zeroes, which is a 1 followed by 2 zeroes. Want me to go even further? A 2 is technically a 1 followed by about 0.30129995 zeroes (because of the Log 2-10 Constant). But we're getting off-topic.

A googolplex is the largest number most people have heard of, and for many it is accepted as the largest named number!! However, the way it was coined was a bit different than the googol.

Right after Milton Sirotta had come up with the name "googol", he immediately told Kasner, "a googolplex should be a 1 followed by writing zeroes until you get tired." Kasner didn't like that definition because it was too subjective, so he changed it to a 1 followed by agoogolzeroes. I mean, if you write 1 zero and you're already tired, you've written a 10, would you call that a googolplex? Besides, the vast majority of us would get tired before writing 100 zeroes, so would a googolplex be less than a googol by Sirotta's definition? We shall never know...

The crazy thing is, we have surpassed the point where n is distinguishable from 2n. In other words, if you multiplied a googolplex by two it would make absolutely no difference! But that's not it. You could multiply a googolplex by ten, and it would be 10^(10^100 + 1). Since adding 1 to a googol is a pretty negligible effect, multiplying a googolplex by ten is just as negligible. Crazy! But there's more. You could multiply a googolplex by a hundred, a thousand, a million, and it still wouldn't make a difference; hell, even a GOOGOLPLEX GOOGOLPLEXES wouldn't make a very big impact!

So that's it for Part 4... if those numbers don't scare you yet, brace yourselves, because the next ones are going to make your heart skip a beat.

Here are numbers in between a googolplex and a dekalogue. They leave numbers like the googol IN THE DUST, and are so big that the layman has trouble even understanding what they are.

Gargoogolplex: (10^10^100)^2

This is a googolplex squared, or 10 to the power of 2 googols. It was coined by Kieran Cockburn, when he said that he rules "a gargoogolplex stars", where a gargoogolplex is "a googolplex googolplexes." That was where the gar- prefix was coined and where his brother Alistair got ideas for the fz-, fuga-, etc. prefixes.

10^10^101

This may seem only a tiny bit bigger than a googolplex, but in fact, it is a googolplex raised to the 10th power. Why is that? Every student who has been in a basic algebra course will understand that (a^b)^c = a^(b × c). So if 10^101 = 10^100 × 10, then (10^googol)^10 = 10^(googol × 10) = 10^(10^100 × 10) = 10^10^101. Sort of complicated-looking, but we can immediately solve it with basic algebraic laws.

10^10^102

This is equivalent to googol^googol, and can be named fzgoogol with the fz- prefix. This is googolplex raised to the 100th power, or a 1 followed by 100 googol zeroes!!! Hopefully, by now, you guys understand that looks can be deceiving on a scale this large.

Megafugafour: 4^^4

This is the 4th megafuga- number, and I've been mentioning it a lot. It's personally one of my favorite large numbers bigger than a googolplex. It's not too small, but not too large (you'll see why later). This can also be called boogafour. We can solve:

4^^4 = 4^4^4^4 = 4^4^256 = 4^2^512 = 2^2^513 = 2^10^(Log 2-10 Constant × 513) = 10^(Log 2-10 Constant × 10^(Log 2 Constant × 513)) = about 10^(8.0723 × 10^157).

This is the largest solution to the four fours puzzle, where someone uses four fours to try to create a number. It's very cool how with just four fours, you can already surpass a googolplex!

This used to be called "Tritet" by Jonathan Bowers since it was {4, 4, 4} in his array notation, but it is now {4, 4, 2} and Tritet is now the MUCH larger 4^^^^4. So, he calls this "Tritet Jr."

Hectillion: 10^((3 × 10^300) + 3)

The centillionth -illion.

Millyllion: 10^2^1,002

The 1,000th -yllion, and it's about 4.286 × 10^301 digits long. Wow.

"Centillionillion": 10^10^303

This number was among the only ones coined by Sbiis Saibian as a kid. It's 1 followed by a CENTILLION zeroes; so he called it "centillionillion." This is technically incorrect, as this isn't the centillionth -illion - it isn't even an -illion at all since it isn't a power of 1,000. Sbiis now calls this number an "ecetonplex", since centillion is sometimes called an eceton.

Centillionillion (corrected): 10^(3 × (10^303) + 3)

Killillion: 10^(3 × (10^3,000) + 3)

Wow - there's about 3,000 digits IN THE NUMBER OF DIGITS OF THIS NUMBER!! Even the number of digits is becoming enormous now (I don't want to put in a 3,000 digit number in here)

Vecekillillion: 10^(3 × (10^30,000) + 3)

Hectekillillion: 10^(3 × (10^300,000) + 3)

We're increasing very, very quickly now - that's because there really isn't a whole lot in between.

Micryllion: 10^2^1,000,002

About 10^(3.9603 × 10^301,030). Wow, the number of digits in this number is almost the same as the HAMLET MONKEY NUMBER!

Millionduplex: 10^10^1,000,000

There are 1,000,000 digits in the number of digits in this number. It's the boundary between Class 3 and Class 4 numbers.

10^10^1,000,006

Could be called "fzmilliplexion" with the fz- prefix. This is the effect of raising the previous number to the MILLIONTH power, but on this scale, the previous number is practically identical to this, since adding 6 to a million is a negligible effect.

Megillion: 10^(3 × (10^3,000,003) + 3)

You may recognize the mega- suffix put onto this number - that's how Bowers names his numbers from this point on. This is the largest of his -illions I'm going to list on here.

Nanyllion: 10^2^1,000,000,002

The billionth -yllion; it's about 10^(1.84519 × 10^301,029,996). I'm going to skip a few -yllions.

Tetralogue: 10^^4

Equal to 10^10^10^10.

Yoctyllion: 10^2^((10^24) + 2)

About 10^10^10^23.

First Skewes' Number: e^e^e^79

This was the original Skewes' Number, and it's about 10^10^10^33.947. It's an upper bound to the smallest number where π(n) (the prime counting function, denoted by the π symbol, the function is unrelated to Pi) > li(n) (the logarithmic integral).

But what exactly is the prime counting function? It's a function which returns the number of prime numbers less than or equal to n, so so π(2) = 1, π(3) = 2, π(7) = 4, etc. But π(10) is still 4 since there are only 4 prime numbers smaller than 10: 2, 3, 5, and 7. Only π(11) will be bigger than 4; it will be 5.

So Skewes' problem is to find the smallest number where π(n) actually surpasses li(n). For smaller inputs, π(n) is always very short of li(n), but Skewes managed to prove that li(n) eventually falls behind π(n).

But let's take a moment to appreciate how big this is. First, imagine a number, such that IT'S OVER 9 DECILLION! That's a pretty big number alright. Now, you have to take a gigantic piece of paper, and write down a 1, then write down that many zeroes!!! Just imagine how long that would take to write and how much paper it would take up.

Now, take your over-9-decillion-digit-number, and do what you just did again. This gigantically huge piece of paper - is the amount of ZEROES in the next number! This behemoth is too big for anyone to comprehend and to put to any practical use!

10^10^10^34

This seems like a pretty good approximation for Skewes' Number, since it's 10^10^10^33.947, but in reality, it is Skewes' number raised to approximately the 10^10^33RD POWER!! That's a number with a decillion digits!!

(10^63)!!

I mentioned this number on one of my problems on Brilliant.org (https://brilliant.org/problems/googology-is-always-fun/?ref_id=1519579). This number is about 10^10^10^65.

Also see WEED₄₂₀(420).

Googolduplex: 10^10^10^100

This is another big step from a googolplex; it is a one followed by a GOOGOLPLEX zeroes. A googolplex is already an incomprehensibly large number that can't even be written out, and now imagine a number with that many digits!!

Ecetonduplex: 10^10^10^303

Extending the centillion (or eceton) just like we are the googol and million.

Megafaxul: 200!!!

This is 200 factorial factorial factorial. To compute this, you first have to compute the factorial of 200 (a faxul), then take the factorial of THAT (the result is larger than a googolplex) then take the factorial of this mega huge big number!

Second Skewes' Number: e^e^e^e^7.705

This number is actually a lot larger than the previous Skewes' number. It was calculated by Skewes a long time after he calculated the original Skewes' number, but this time, he assumed the Riemann hypothesis to not hold true. That actually made the problem harder to solve, as this new value is about 10^10^10^963 compared to just 10^10^10^33.947 before.

Note: This was the largest number that I knew of for a very long time. It was larger than a googolplex, which many people considered the largest named number. Hell, it was larger than agoogolduplex, which is a lot bigger than the measly googolplex, and back then I didn't even know that a googolduplex existed. I couldn't think of any real-world meaning for this number. I now know that there's the Poincare recurrence time, which is bigger, but let's face it, that's kind of useless too, don't you think?

Milliontriplex: 10^10^10^1,000,000

The boundary between Class 4 and Class 5 numbers. With numbers this big, we could start using Hyper-E Notation; this would be E1,000,000#3 or E6#4.

Pentalogue: 10^^5

A power tower of 5 10s. In Hyper-E this would be E1#5.

Poincare Recurrence Time: 10^10^10^10^10^1.1

This AMAZINGLY HUGE number is the number of years it will take for the universe to return to the state it is now (it could be Planck times or millennia, it really doesn't make a difference on this scale). This is the longest time calculated by any physicist. In Hyper-E it would be E1.1#5.

But what exactly is the Poincare Recurrence Theorem? Pretend you have a deck of 52 cards, and you start shuffling it. After shuffling for a really long time, the cards will eventually return to the way they were when you originally started shuffling, right? That's what the theorem says, but on a much larger scale. The Poincare Recurrence Time is the amount of time it will take all 10⁸⁰particles in the universe to return to the state they are in now (assuming the smallest possible length is the Planck length).

You really have to keep in mind that this is averypoor upper-bound, however, and after doing a calculation myself, I came at 10^10^194 as the upper-bound, by taking the factorial of the number of Planck volumes in the universe, and even that isn't the best possible upper-bound. That means that sometime in less than 10^10^194 years, you're going to be in the exact same spot you are now, reading this!

Googoltriplex: 10^10^10^10^100

The third element in the googolplex, googolduplex, etc. series. Would be E100#4 or E2#5 in Hyper-E.

Hexalogue: 10^^6

A power tower of 6 tens. This would be E1#6 in Hyper-E.

Now, take a moment to realize how huge the numbers are getting. So the first two layers of the 6-layer tower is 10^10 = 10,000,000,000. Nothing big.

Now, for the next level, we need to do 10^10,000,000,000 = a 1 followed by 10,000,000,000 zeroes!! That number approaches the limits of numbers we can store! And we've STILL GOT 3 MORE LAYERS TO GO, and just the first one out of those 3 already takes up more space than the universe has!!

Googolquadriplex: 10^10^10^10^10^100

Heptalogue: 10^^7

Googolquintiplex: E100#6

Octalogue: 10^^8

Googolsextiplex: E100#7

Ennalogue: 10^^9

Googolseptiplex: E100#8

Dekalogue: 10^^10

Finally. We have made it. A power tower of 10 10s. This could be called E1#10 in Hyper-E. Just TRY to understand how huge this number is. You have to take 10 to the power of 10, then take 10 to that power, then take 10 to THAT power, then repeat 6 times!

Remember how I said a hexalogue was horrifyingly huge? Well a dekalogue is 10^10^10^10^hexalogue. That's incomprehensibly huge! Not only can we not think of a real-world application for this, we can't even understand what the fuck we just created! With just two tens and two up-arrows, we've already created this monster of a number. And to think that 10^^11 is a 1 followed by this many zeroes is crazy. While that may seem like a HUGE improvement, it really isn't doing that much on this scale - does that seem possible? And yet, this number is TINY compared to what we're going to encounter in the future.

Well, that's it for Part 5. If these numbers haven't scared you at least a little bit, then I don't know what will. But I do! In Part 6, we're going to get into some REALLY BIG numbers.

In this section, we will examine numbers in between the puny dekalogue and the still puny tritri.

10^^11

This isn't too big a step from a dekalogue, but remember, it's 1 followed by a DEKALOGUE zeroes.

Icosalogue: 10^^20

Not only is this a power tower of 10 10's, it's a power tower of 20 10's. It may only seem twice as big, but in reality, it's incomprehensibly larger!

Penantalogue: 10^^50

Ogologue: 10^^80

Giggol: 10^^100

This is a power tower of 100 tens. It should be called "hectalogue", but the name "giggol" was coined by Jonathan Bowers. I like that name better because it makes me giggol.

Now, if this number is really hard to comprehend, which it should be, let's think in terms of stages:

• Stage 1 = 10
• Stage 2 = 100...000 with Stage 1 zeroes
• Stage 3 = 100...000 with Stage 2 zeroes
• Stage 4 = 100...000 with Stage 3 zeroes
• ...
• Stage 100 = Giggol

If we were to write a giggol out as a power tower, it would look like this:

That's already a power tower so big that I can barely fit it on my page, and it's so big that most of us probably wouldn't write it down.

10^^101

If you think that this is a rather small step from a giggol, you're right. Just remember that this is 1 followed by a GIGGOL zeroes, and a giggol is 10^^100, so on this scale large numbers can be very, very deceiving!

Grangol: E100#100

Grangol, short for "grand googol," is another of Saibian's googolisms which is a power tower of 100 tens topped off with a 100, and it's E100#100 in Hyper-E Notation. This is NOT equal to a giggol raised to the hundredth power; that is a way smaller number that is practically indistinguishable from the giggol itself.

Expofaxul: 200!1

This is the exponential factorial of 200, and it's about E5.263#197 in Hyper-E Notation. I think you guys are getting the point with Hyper-E; this would be about a power tower of 197 10's topped off with a 5.263.

But what exactly is an exponential factorial? It's something much more powerful than even the hyperfactorial. The expofactorial of n would be n^(n-1^(n-2^(...(2^1)...))). So an expofaxul must be about 200^199^198^197^...^2^1. Of course you have to start backward because if you didn't, it would just decay into 1 (it would be 1 to the power of a super big number).

The Mega: 2[5]

This is "2 in a pentagon" in Steinhaus-Moser Notation, and it's about E619#256. But what exactly is Steinhaus-Moser Notation?

So, we define N in a triangle as N^N. That's just it. So 2 in a triangle is 4, 5 in a triangle is 3,125, 1,000 in a triangle is a novemnonagintanongentillion, etc. We'll denote that as a function triangle(n).

Now, N in a square is triangle(triangle(...(triangle(N))...)) with N triangles. When you think about it, even the triangle function is powerful enough, but the square function is incredible! 2 in a square is triangle(triangle(2)) = triangle(4) = 256. 3 in a square is triangle(triangle(triangle(3))) = triangle(triangle(27)) = triangle(27^27) = triangle(10^38.647) = about 10^10^39

So, you guys can already see that the square function is INSANE and just 3 in a square is a gigantic number with about aduodecilliondigits, and a duodecillion is already super big by a normal person's standards!

Now, N in apentagonis square(square(...(square(N))...)) with N square. That's INSANE. So if the Mega is 2 in a pentagon, we get:

pentagon(2) = square(square(2)) = square(triangle(triangle(2)) = square(triangle(4)) = square(256) = triangle(triangle(...(triangle(256))...)) with 256 triangles

HOLY COW! That's insane. We already saw that 3 in a square is horrifying, and the Mega is 256 in a square! If we solve the first triangle, we get 256^256, which is equal to 2^2048. That's already a SUPER BIG NUMBER. If we were to solve another one of the triangles, we get (2^2048)^(2^2048) which is already WAY TOO BIG for us to even compute! This number would probably be larger than a googolplex, and we still have 254 triangles to go!

So if anyone would be daring enough to try to solve all 256 triangles, they would get about E619#256, as I stated earlier, or about a power tower of 256 tens topped off with a 619. N in a square is approximately a power tower of N tens on this scale.

Giggolchime: 10^^1,000

A power tower of 1,000 10's - a clever combination of Bowers' giggol with Saibian's -chime suffix.

Grangolchime: E1,000#1,000

Another number you can form with Saibian's -chime suffix - this would be 1,000 tens topped off with a 1,000. Just like the grangol, but with 1,000 instead of 100.

Giggoltoll: 10^^10,000

Yet another number you can form with Saibian's suffixes.

Grangoltoll: E10,000#10,000

2^^^^3

This is a cool number which actually isn't as big as it seems, but it's still REALLY BIG. I mean, two HEXATED to three? As I already discussed in the hyper-operator section, hexation makes pentation look like crap, and pentation is WAY more powerful than tetration, and tetration already blows the minds of most people! But we can simplify this crazy hexation thing down to something that is only a moderately sized tetrational number:

2^^^^3 = 2^^^2^^^2

Giggolgong: 10^^100,000

Remember that -gong suffix that was created by Saibian's friend's dad? We can combine it, too to form this cool number that's a power tower of 100,000 tens.

Grangolgong: E100,000#100,000

E1,000,000#1,000,000

Just for the heck of it, here's a power tower of a million tens topped off with a million.

Grangolbong: E100,000,000#100,000,000

10^^(10^10)

This is the limit of Perl Hypercalc; it's a power tower of 10 billion tens. Hypercalc is a cool calculator which "never overflows," but it does at this point. In other words, the maximum number it can store is this, which makes it wide-ranged enough to store numbers like a giggol, or even worse, a giggolgong. Because of this, it can compute things like 27^(86!) very easily.

Grangolthrong: E100,000,000,000#100,000,000,000

Tritri: 3^^^3

Well, you guys asked for it, here it is! The legendary and amazing tritri, which is 3 pentated to 3 or 3 hexated to 2. Just so you guys (and girls!) get an idea of how big this is:

3^^^3 = 3^^3^^3 = 3^^(3^3^3) = 3^^(3^27) = 3^^7,625,597,484,987 = 3^3^3^...^3^3^3 with 7,625,597,484,987 3's

With just three up-arrows we've already created this legendary number. It's cool how this is another step toward Graham's number: to compute it you first need to know G1, and to know it you have to know the value of this number!!

Ronald Graham himself, the creator of Graham's number, incorrectly described this number as "a 3.6 trillion digit number" on a Numberphile video. Tritri is not a 3.6 trillion digit number; it is so much more horrifying than that. The number Graham was referring to when he said "a 3.6 trillion digit number" was probably 3^^4, and that's nowhere near 3^^7,625,597,484,987. In fact, if tritri would be a power tower of tens instead of threes, it would literally only be about two or three levels shorter!! If we were to write out this tower, it would reach from the Earth to the Sun if each 3 was an inch tall. Just the first five threes in that tower already surpass a GOOGOLPLEX, so how could this number be "only" 3.6 trillion digits long?

Fun fact: I wrote an Urban Dictionary definition for this number where I said it was "the length of my dick") and it actually got published! You can read it herehttps://www.urbandictionary.com/define.php?term=Tritri

That's it for Part 6; up next is Part 7 with some even scarier numbers.

Here, we will examine numbers in between tritri and boogol.

Googol-stack: 10^^(10^100)

This is a power tower of a googol tens. I know this is a HUGE improvement from tritri, but there's literally almost nothing in between. Some people call this number "googolgoogolplex," but that's incorrect because googolplex is 10^googol, googolduplex is 10^10^googol, googoltriplex is 10^10^10^googol, etc. A friend of mine was the first to realize this mistake. Googol-n-plex is basically a power tower of n tens topped off with a googol.

Googolgoogolplex: E(googol+1)#100

A power tower of a googol tens topped off with a googol, or a power tower of a googol+1 tens topped off with a 100. Remember, it's a misconception that a googol-stack and a googolgoogolplex are the same thing.

Zootzootplex: googolplex!1

This is the exponential factorial of a googolplex, and it was coined by a guy named Andrew Schilling at the age offour. It would be (googolplex)^(googolplex-1)^...^2^1. Just like with expofaxul, you have to start backward, so it doesn't decay into 1.

Schilling also coined the numbers "sillion" which is supposed to be "very silly" and "sillyillion" which is "like sillion, but bigger."

Googolgoogolplexplex: E(googolplex-1)#100

A power tower of a googolplex tens topped off with a googol. A friend of mine coined this number (the same one that realized the mistake for googolgoogolplex).

Giggolplex: 10^^10^^100

A power tower of a giggol tens. Bowers uses the -plex suffix to perform any type of recursion, which is kind of cool.

Kiloexpofaxul: (200!1)!1

Just like expofaxul, but starting with an expofaxul instead of 200, so it would be (expofaxul)^(expofaxul-1)^...^2^1.

Giggolduplex: 10^^10^^10^^100

A power tower of a giggolplex tens. By now, power towers are becoming obsolete.

Megaexpofaxul: ((200!1)!1)!1

Just like kiloexpofaxul, but starting with a kiloexpofaxul instead of an expofaxul. After this you could have a gigaexpofaxul, teraexpofaxul, ..., yottaexpofaxul, etc.

Googolgoogolgoogolgoogolgoogolplexplexplexplexplex

Just for the sake of it, I'm going to put this stupid, pointless number on here. I think you guys get what this means, it's the googolgoogolgoogolgoogolplexplexplexplexth element in the googolplex, googolduplex, etc. sequence. I don't think I really need to explain.

Gaggol: 10^^^100

This is really insane now. This is 10^^10^^10^^...^^10^^10^^10 with 100 tens. This isn't even a power tower anymore; it's a TETRATIONAL TOWER. That's crazy. So just the first two tens is already a dekalogue, and the first three is a power tower of tens, a DEKALOGUE tens tall! And we still have 97 layers to go!!! Pentation really does leave tetration in the dust.

Greagol: E100#100#100

Greagol (short for "great googol") is another one of Saibian's googolisms and is comparable to the gaggol, at about 100^^^101. It's sort of hard to explain exactly what it is, but I'll try to describe it using stages:

• Stage 1 = E100#100 = grangol
• Stage 2 = E100#grangol. That number is something called a grangoldex.
• Stage 3 = E100#grangoldex
• ...
• Stage 100 = greagol

If you want clarification on how Hyper-E works, then clickhere. It can be seen that Hyper-E notation isveryfast-growing and with just three 100's, we've already created this "great" googol.

Greagolchime: E1,000#1,000#1,000

Just another number you can form with the -chime suffix.

Greagoltoll: E10,000#10,000#10,000

Greagolgong: E100,000#100,000#100,000

Megiston: 2[6]

This is 2 in ahexagonin Steinhaus-Moser Notation, and it's WAY bigger than the humble mega. You may be thinking, 2 in a hexagon can't be that much worse than 2 in a pentagon, right?

Remember how 2 in a square was just 256, and 2 in a pentagon was already an incomprehensibly huge number called the mega? 2 in a hexagon is equivalent to the gigantic mega inside a pentagon!! That is because hexagon(2) = pentagon(pentagon(2)) = pentagon(mega). This number is about 10^^^mega.

We are now moving out of the tetrational and pentational ranges. On to the up-arrow level.

Grahal: G(1)

This is the first member of Graham's sequence, and it's already way bigger than we can imagine. It's not 0, 1, or 3; it is already the gigantic 3^^^^3. With four up-arrows, the numbers get a LOT crazier:

3^^^^3 = 3^^^3^^^3 = 3^^^Tritri = 3^^3^^...^^3^^3 with Tritri 3's

Now THIS is a number that leaves tritri in the dust. It might be easiest to visualize with stages:

• Stage 1 = 3
• Stage 2 = 3^3^3 = 7,625,597,484,987
• Stage 3 = 3^3^...^3^3 with Stage 2 3's = Tritri
• Stage 4 = 3^3^...^3^3 with Stage 3 3's
• Grahal is Stage Stage 3 = Stage Tritri

This is almost impossible to even understand!! But, as you will later learn, G(2) is a number that leaves everything far, far behind it.

Gaggolplex: 10^^^10^^^100

The gaggolth stage in the stages used to calculate a gaggol. (What?)

Okay, so if you go back to the entry for Gaggol, you will see that it is only stage 100 in a series of incomprehensibly huge giants. This is stage GAGGOL. This is 10^^10^^...^^10^^10 with a gaggol tens.

Tritet: 4^^^^4

This is four hexated to the fourth, which can be visualized with the HUGE diagram below. It's very hard to even understand. The rightmost part is how many layers are on the middle part, and the middle part is how many layers there are on the leftmost part.

6^^^^6

Can be called boogasix with the booga- prefix. If we try to solve:

6^^^^6 = 6^^^6^^^6^^^6^^^6^^^6 = 6^^^6^^^6^^^6^^^(6^^6^^6^^6^^6^^6) = ???

By now, eventetrationaltowers are useless and can't even come close to expressing numbers this big.

Geegol: 10^^^^100

Another huge step from giggol and gaggol. If we use the same steps we did for the gaggol:

• Stage 1 = 10
• Stage 2 = 10^10^...^10^10 with Stage 1 10's = dekalogue
• Stage 3 = 10^10^...^10^10 with Stage 2 10's
• Geegol is Stage Stage Stage ... Stage Stage 1 with 100 copies of the word "Stage"

That's pretty mind-blowing, but wait until we get to the gigol.

"Great Graham": 3^^^^^3

This is 3 heptated to 3, or even worse, 3 octated to 2. It's equal to 3^^^3^^^...^^^3^^^3 with Grahal 3's. The name was coined when a random person created an article on Googology Wiki with this number, but they stated, "It's larger than Graham's Number!" but it is much, much smaller than Graham's number. They probably mistook Graham's number for G(1) = 3^^^^3.

Tripent: 5^^^^5

It's called tripent because it's {5, 5, 5} in Bowers' array notation. It follows the same pattern as tritri and tritet. But I think I should explain a little bit about Bowers' arrays before I continue.

What are these arrays and how do they work?

The Bowers Exploding Array Function (or BEAF for short) is a way to create arrays that can quickly "explode" into very huge numbers, hence the name. So, starting:

• The easiest array is the empty one: {} = 1. Why is the empty array equal to 1? One of the notation's rules states that if the last number of the array is a 1, it can be removed.
• Arrays with a single entry work like so: {a} = a. Pretty boring. So {5} = 5, {100} = 100, {10^10^10} = 10^10^10, etc.
• Two-entry arrays are a tiny bit more complicated, but still simple: {a, b} = a^b. So {3, 4} = 81, {2, 20} = 1,048,576, {5, 5} = 3,125, etc.
• Three-entry arrays are way cooler: {a, b, c} = a^^^...^^^b with c arrows. So {3, 3, 2} = 7,625,597,484,987, {5, 5, 5} = the tripent number you just read about, and {10, 10, 100} is a cool number with a hundred up-arrows called a boogol, and that is the last number on this section. We can also use a{c}b to represent {a, b, c}, so 3{10}3 = 3^^^^^^^^^^3. That notation is more efficient than writing out 10 arrows and easier to understand than {3, 3, 10}, so you'll be seeing it a lot in the next few entries.

It can be seen that this notation grows just as quickly as Knuth's arrows do, and we only have three entries! With four entries, we can already surpass Graham's number!!! And once the arrays have millions and millions of entries, we can start using multidimensional arrays. That's why BEAF is such a preferred notation among googologists; it's simple and can easily create amazingly huge numbers. I'm going to stop here for now, but I will describe the larger arrays later.

Gigol: 10^^^^^100

This number is not to be confused with the giggol. It's 10heptatedto 100.

Goggol: 10{6}100

Or 10^^^^^^100, which is 10octatedto 100. That's truly impossible to even try to understand. Right now, even trying to use the crazy stages mentioned earlier will result in complete and utter failure!

Gagol: 10{7}100

Or 10^^^^^^^100, which is 10ennatedto 100. When you think about it, it's a HUGE step from goggol and gigol, since as we've seen, adding just ONE up-arrow to an expression results in complete and utter chaos. But in the multi-arrow range, there really isn't much in between these kinds of numbers.

10{8}100

This googolism actually doesn't have a name.

Gaxoogol: 10{9}100

Unlike the previous entries, this was not coined by Jonathan Bowers, it was coined by Andre Joyce. It's equal to 10^^^^^^^^^100.

Tridecal: 10{10}10

Or three 10's in a linear array, which is {10, 10, 10}. This is 10dodecatedto 10. By now we have lost literally all hope of even trying to make visual representations using stages, layers, or anything else.

2{12}3

This was the the first step in calculating the original Graham's Number, which is much smaller than the Graham's Number we know today. Thus, the one we know today was never the upper-bound to Graham's problem!

3{27}3

This mostly crops up when working with arrays of threes.

100{98}100

Can be called "boogahundred." It's "a little" less than a boogol. It's 100hectatedto 100.

Boogol: 10{100}10

This is 10 "102-ated" to 100 or {10, 10, 100}. Written out in full up-arrow form, it's:

10^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^10

As you can see, up-arrows are ALREADY becoming obsolete! Aren't up-arrows supposed to be the most powerful googological tool on earth that can express ANYTHING comprehensible by humans? I mean, just two arrows already blew your mind, and three arrows made that look like crap! But as we will discover in the next section, up-arrows are actually very weak.

That's it for Part 7. On to Part 8.

This section quickly blasts off into the stars with numbers in between a boogol and Graham's Number.

Gugold: E100##100

Another of Saibian's googolisms, this is short for "golden googol." It's equal to E100#100#100#...#100#100 with 100 100's.

Hyperfaxul: 200![1]

This gigantic number is equal to 200!200, which is 200{200}199{200}198...2{200}1. That may sound gigantic, but it's still smaller than 200{201}200.

200{201}200

Also equal to 200{202}2. It's a rough upper-bound to the hyperfaxul.

Gamoogol: 10{999}100

A continuation from gaxoogol. Sort of weird, I guess.

Boogolchime: 10{1,000}10

Gugoldchime: E1,000##1,000

Boogoltoll: 10{10,000}10

Gugoldtoll: E10,000##10,000

Boogolgong: 10{100,000}10

Gugoldgong: E100,000##100,000

10{10¹⁰⁰}10

Yep, the insane number known as a googol is the number ofarrowsin this. Writing it would be like writing a googolplex, but with arrows intead of zeroes. Just imagine how crazy and gigantic it is!

Great googol: E100##1#2

Now THIS really is an amazing number. It's equal to E100#100#100#...#100#100 with a googol 100s. It's got about 1 or 2 arrows more than the previous entry, which is nothing on this scale, but as we saw earlier, adding just ONE arrow causes complete and utter devastation and chaos.

10{10^10^100}10

Just like 10{10^100}10, but with a googolplex arrows instead of a googol.

The Moser: 2[2[5]]

This is another Steinhaus-Moser notation number, except that this one was coined by Moser, not Steinhaus. This HUGE number is equal to 2 in a mega-gon (a polygon with mega sides).

It may be hard to believe, but we can actually approximate this number accurately: it is about 3^^^...^^^3 with mega-2 arrows. Wow!

Tritriplex: 3{3{3}3}3

This is another number that pops up a lot when working with arrays of threes; it's 3^^^...^^^3 with tritri arrows.

G(2)

This is the second element in Graham's series, and it's equal to 3{3{4}3}3, or 3^^^...^^^3 with G(1) arrows. The crazy, mind-boggling, and insane sequence is as follows:

• G(1) = 3^^^^3
• G(2) = 3^^^...^^^3 with G(1) arrows
• G(3) = 3^^^...^^^3 with G(2) arrows
• ...
• G(64) = Graham's Number

It might help to think of it this way:

• G(1) = an insane number
• G(2) = an insaaa...aaane number with G(1) a's
• G(3) = an insaaa...aaane number with G(2) a's
• etc.

So this number is already INSANE!!! I can't even really think of anything else to say about it.

Boogolplex: 10{10{100}10}10

Or 10^^^...^^^10 with a boogol arrows.

Moserplex: 2[2[2[5]]]

A number I made up myself. It's 2 in a Moser-gon.

G(3)

The next step in calculating Graham's number; it's 3^^^...^^^3 with G(2) 3's. In BEAF it would be 3{3{3{4}3}3}3. Holy shit.

Boogolduplex: 10{10{10{100}10}10}10

10^^^...^^^10 with a boogolplex arrows. You can extend this to boogoltriplex, boogolquadriplex, ..., boogolcentiplex, ..., boogolgoogolplex, etc.

Moserduplex: 2[2[2[2[5]]]]

I don't think I really need to explain much about this either, it's 2 in a Moserplex-gon. You could extend this to Mosertriplex, etc.

Kilohyperfaxul: (200![1])[1]

Or (200![1])!200![1]. So it's 200{hyperfaxul}199{hyperfaxul}198...2{hyperfaxul}1. After this comes megahyperfaxul, gigahyperfaxul, etc.

G(3)

Yet another step toward computing Graham's Number.

Graham-Rothschild Number: 2{2{2{2{2{2{2{12}3}3}3}3}3}3}3

This was the original Graham's number, in between G(7) and G(8). It's much less-known than the Graham's Number known today, as even a Numberphile video states that it is the current upper-bound, but it NEVER EVEN WAS! The paper about the Graham-Rothschild number was published in 1970, yet the paper about the larger Graham's Number we know today was only published in 1977. Then, Martin Gardener wrote about it, never mentioning this much smaller number, and that is why the number received so much attention.

Graham's Number: G(64)

We have finally made it. The legendary, gigantic, crazy Graham's number. This number even made it to Guinness World Records for being the biggest number used in any serious mathematics. In case I need to define again:

• G(1) = 3^^^^3
• G(2) = 3^^^...^^^3 with G(1) arrows
• G(3) = 3^^^...^^^3 with G(2) arrows
• ...
• G(64) = Graham's Number

Graham's Number is just called G sometimes. But what exactly is it and why was it created? What is Graham's problem? It has to do with Ramsey theory. The question is: what is the minimum number of dimensions N of a hypercube such that all 2-colorings of all vertex pairs of the hypercube have a coplanar complete graph of 4 vertices? That may sound super complicated at first, but it's not really that hard to understand.

The last 500 digits of Graham's Number are:

02425950695064738395657479136519351798334535362521430035401260267716226721604198106522631693551887803881448314065252616878509555264605107117200099709291249544378887496062882911725063001303622934916080254594614945788714278323508292421020918258967535604308699380168924988926809951016905591995119502788717830837018340236474548882222161573228010132974509273445945043433009010969280253527518332898844615089404248265018193851562535796399618993967905496638003222348723967018485186439059104575627262464195387

But, the cool thing is that these digits don't just end Graham's number, they end G(63), G(62), ..., G(1), and even tritri. They end any power tower of 500 3s. They would also end G(65), G(66), ..., and so on.

It's really insane how a number this big is considered to be an upper-bound to a problem. Currently, the upper-bound is 2^^2^^2^^9. That number is still REALLY big; it's larger than tritri. But the lower-bound is 13 (this is one of the few facts that Numberphile actually gets right). So it could just be 13, and not some gigantic, crazy number.

And the funny thing is that Graham's function is actually puny, and pales in comparison to larger functions! While it may make the booga function, which has growth rate ω in the fast-growing hierarchy, look like crap, Graham's function actually only has growth rate ω + 1. And if we had a function H(x) that iterated G(G(G(...(G(G(x))...))) x times, that would have growth rate ω + 2. And if we had a function that iterated THAT, it would have growth rate ω + 3. As you could imagine, ω^2 and ω^ω are WAY more powerful than what you might think - and even they only scratch the surface of what googology has to offer!

That's it for Part 8, but beware - Part 9 is a lot longer.

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