“My space commander rules the whole world!”

“Yeah, well my space commander rules the whole star and all the planets.”

“Yeah, well my commander rules two stars.”

“Mine rules ten stars.”

Now is the big moment for the five-year-old. Five-year-olds have to learn to count to 100 in kindergarten, so One Hundred is a really big number. It is so big and frightening to five-year-olds that they never name 101. Always 100.

“My space commander rules 100 stars!”

At this point the five-year-old will lose, because the eight-year-old can say,

“Well mine rules 1,000 stars, so there.” And the five-year-old can’t say anything.

But the ten-year-old can, and jumps in with,

“But my commander rules a million stars.”

And now the five-year-old is back in the game,

“Well my space commander rules a million million million million million million…” and keeps going until the other two walk away or mom or dad show up and say, “Quiet down in here and just play.”

Gartwenty = 400

Garhundred = 10,000

Garthousand = 1,000,000

Garmillion = 1,000,000,000,000

Gargoogol = 10²⁰⁰

Now a gargoogol is a pretty cool number. It's 1 followed by 200 zeros, exactly a googol times larger than a googol. While a googol written out 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,

And then there's the eponymous gargoogolplex, a googolplex googolplexes. Evaluating it gives us:

10^{10^100}*10^{10^100} = 10^{10^100+10^100} = 10^{2*10^100}

In the form 10^{2*10^100}, a gargoogolplex doesn't seem much bigger than a googolplex! Hell, it's less than 10^{10*10^100} = 10^{10^101}. Nonetheless it's a pretty cool number.

NOTE: It's a common misconception that "gargoogolplex" refers to 1 followed by a googolplex zeros. That number is most often known as a googolduplex or a googolplexian. There are some large number lists that use that erroneous name, and people have taken the erroneous value one step further by defining a "gargantugoogolplex" as 1 followed by a "gargoogolplex" zeros, which is usually known as a googoltriplex. People have applied other prefixes in the fuga- family to these erroneous numbers.

NOTE 2: "Gargoogolplex" could alternately be interpreted as not gar-googolplex (square of a googolplex) but as gargoogol-plex, which is applying the -plex function to a gargoogol. That would give you the larger 10^{10^200}. To avoid confusion, we can use the name "gargoogol-plexed" for 10^{10^200}.

Since gar-x = x*x, it follows that gar-x = x². The expression x² begs the question: why stop there? Why have 2 as the exponent where you can make the exponent any number? This brings us to the next member of the fuga- family.

Cockburn came up with a prefix to continue gar-, defining fz-x to be x^x. That is a MUCH more powerful prefix than gar-. Once again let's look at some examples of the prefix in action:

Fzone = 1¹ = 1

Fztwo = 2² = 4

Both of those numbers are the same as their gar- counterparts. However those are the only numbers for which that is true, because the fz- prefix soon leaves the power of gar- in the dust. Here are the next few numbers the prefix can name:

Fzthree = 3³ = 27

Fzfour = 4⁴ = 256

Fzfive = 3125

Fzsix = 46,656

Fzseven = 823,543

Fzeight = 16,777,216

Fznine = 987,420,489

Fzten = 10,000,000,000

The tenth member of the sequence is ten billion. That's pretty good so far. However we can easily go much further with something like:

Fztwenty = 104,857,600,000,000,000,000,000,000

This is about 105 septillion! Imagine having this many pennies, and imagine that the sun is solid and you could lay pennies on it. Then all these pennies would cover the sun ... with 103.3 septillion pennies to spare! All the pennies would be able to cover 62 suns total! And that's just the 20th member of the sequence—what follows is far worse.

Fzhundred = 100¹⁰⁰ = 10²⁰⁰

This is the same value as a gargoogol. Note that the fz- function achieves the same value the gar- function does with a much smaller input!

Fzthousand = 1000¹⁰⁰⁰ = 10³⁰⁰⁰

This is pretty close to a millillion (10³⁰⁰³).

Fzmillion = 10^6,000,000

Cockburn mentioned this number in his article as an example of a number nameable with the fz- prefix.

Fzbillion = 10^{9,000,000,000}

A fzbillion is 1 followed by nine billion zeros!

Fztrillion = 10^{12,000,000,000,000}

A fztrillion is 1 followed by twelve trillion zeros! Now we're heading into some pretty big numbers, but this is much smaller than:

Fzgoogol = (10¹⁰⁰)^{10^100} = 10^{100*10^100} = 10^{10^102}

Written as 10^{10^102} a fzgoogol doesn't seem much bigger than a googolplex. However is really equal to raising a googolplex to the 100th power:

(10^{10^100})¹⁰⁰ = 10^{10^100*100} = 10^{10^100*10^2} = 10^{10^102}

And finally we have a number similar in spirit to Kieran's gargoogolplex:

Fzgoogolplex = (10¹⁰¹⁰⁰)^1010100 = 10^{10100*1010100} = 10^{10(100+10100)}

In that last form a fzgoogolplex might not seem much bigger than a googolduplex. However it's actually equal to raising it to the googolth power! Simply observe:

Of course we can also have a "fzgargoogolplex", whether we use the correct definition or the wrong definition (i.e. 1 followed by a googolplex zeros), and a "fzgargantugoogolplex" as well. Now that's a cool number alright. But Cockburn, in his large number spirit, came up with yet another prefix:

Alistair Cockburn and his family decided to come up with a prefix for x to the xth power x times. Actually that phrase contains ambiguity: does that refer to ( ... ((((x^x)^x)^x ... )^x or x^(x^(x^ ... (x^x)) ... )))? Fortunately, Cockburn gave examples that show that it refers to the first. 

Cockburn's family cam up with the name "fuga-" for this prefix, from a combination of "fugue" plus the gar- prefix. He says it's pronounced /few-ga/, but I myself always read it as /foo-ga/.

Cockburn mentioned in a post-script to his article that ((((x^x)^x)^x ... )^x with x copies of x is "x^{x^2} or a little less". However, it really is equal to x^x^(x-1). It's easy to see why using the laws of exponents (specifically (a^b)^c = a^b*c):

= x^x^(x-1)

Fugaone = 1

Fugatwo = 2^2 = 4

Once again those two are the same as their gar- and fz- equivalents.

Fugathree = (3^3)^3 = 19,683

Fugafour = ((4^4)^4)^4 ~ 3.4028*10^38

Only plugging in four to the fuga- function already outputs a huge number! It's about 340 undecillion and it's also equal to 2¹²⁸.

Fugafive ~ 7.1821*10^436 

Plugging in five to the function not only surpasses a googol but also a centillion!

Fugasix ~ 8.0191*10^6050

This number is larger than a millillion or even the square of a millillion.

Fugaseven ~ 8.6958*10^99,424

This number has almost 100,000 digits.

Fugaeight ~ 10^1,813,917

Fuganine ~ 10^41,077,011

Fugaten = 10^1,000,000,000

I assume you get the pattern. Even better would be:

Fugahundred = 10^{100^(100-1)} = 100^{100^99} = 10^{2*10^198}

This is larger than a googolplex or a gargoogolplex. In fact it's the 2*10⁹⁸th power of a googolplex!

Let's skip to two awesome numbers:

Fugagoogol = 10^{10^(10^102-98)} ~ 10^10^10^102

A fugagoogol is somewhat larger than a googolduplex. However the top exponent is deceiving. Actually the number of digits in this number is approximately:

The number of digits in this number is equal to about 10^{10^(10^100+100)} = 10^{(10^10^100)*(10^100)} = (10^10^10^100)^(10^100) = googolduplex^googol

which is:

a googolduplex times a googolduplex times a googolduplex times a googolduplex times a googolduplex times a googolduplex times a googolduplex times a googolduplex times a googolduplex times a googolduplex times a googolduplex times a googolduplex times a googolduplex times a googolduplex times a googolduplex times a googolduplex times a googolduplex times a googolduplex times a googolduplex times a googolduplex times a googolduplex times a googolduplex times a googolduplex times a googolduplex ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

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:                                        :                                        :                                        :

:                                        :                                        :                                        :

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These are some pretty awesome numbers. The growth rate of the fuga- prefix is roughly double exponential (comparable to functions of the form a^{a^x}). As I said two articles ago, in double exponential functions, the number of digits in their outputs grow at exponential rates. But the number of digits in numbers formed with the fuga- prefix grows a bit faster than exponential. Despite how fast growth that is, these are still nowhere near numbers we examined earlier in section 2! The fuga- prefix is the most powerful of the prefixes Cockburn introduced, but some people have come up with even better prefixes.

Alistair Cockburn shared the prefixes he and his family came up with on the "Really Big Numbers" forum page. He described the fuga- prefix like so:

However, Stephan Houben, a friend of Cockburn's, noticed an ambiguity in how Cockburn described the fuga- prefix. He said:

Cockburn then said that he had the former in mind (fuga-3 = (3^3)^3), and said that a "coffee buddy" of his suggested the name megafuga- for the higher prefix. With that, a fourth still more powerful member of the fuga- family of prefixes was born.

Now, since megafuga-x = x^(x^(x^( ... (x^x))) ... ))) with x copies of x, it can be expressed more compactly using Knuth's up-arrows as x^^x; it doesn't collapse into an exponential expression like fuga- does. That said we can look at some examples, largely recapping what we looked at in the up-arrows article:

megafugaone = 1^^1 = 1

megafugatwo = 2^^2 = 4

Once again the first two members are 1 and 4. However after this the sequence grows much more quickly.

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

This is much larger than fugathree = 19,683. It's a prevalent number in googology as we saw when we examined Knuth's up-arrows.

megafugafour = 4^^4 = 4^4^256 ~ 10^(8.0723*10^153)

Megafugafour is a pretty awesome number. It's bigger than a googolplex or even a gargoogolplex, and therefore it's a number we can safely say we can never hope to write out.

However, we can compute the first and last digits of megafugafour. It begins:

23610226714597313206............

and ends:

Now we're heading into some REALLY big numbers. This surpasses a googolduplex, and it's just the fifth member of the sequence.

Megafugaten, which Jonathan Bowers calls "decker", is another pretty cool number.

As you can see, megafuga- is not a little more powerful than fuga-, it's MUCH more powerful. An even better number would be "megafugahundred", which is a power tower of 100 100s.

A megafugagoogol is a power tower of a googol googols! If it were written as googol^googol^googol^ ... ^googol and each googol was 1/2 cm high, then the tower would stretch 5.68*10⁷⁰ times higher than the diameter of the observable universe!

And why stop at a megafugagoogolplex when you can have a megafugagargoogolplex? It's a power tower of a gargoogolplex gargoogolplexes, a googolplex times taller than the tower used to represent a megafugagoogolplex! In addition we can have things like a megafugagoogolduplex, or a megafugagoogoltriplex, and so on through the googol series. An alternate name for "megafugagoogoltriplex" is megafugagargantugoogolplex, which is the largest number on some online large number lists. Despite how epic a "megafugagargantugoogolplex" sounds, we can go much further by continuing the googol series with Latin prefixes, for example a megafugagoogoldeciplex or a megafugagoogolcentiplex.

Sbiis Saibian, in his article on the fuga- family of prefixes, came up with a prefix booga-, which takes n and turns it into n n-ated to n, i.e. n^^...^^n with n-2 ^s. The sequence of numbers formed with that prefix is very similar to the Ackermann numbers which we learned about when we examined Knuth's up-arrows. 

Here are some examples of the booga- prefix:

boogatwo = 2*2 = 4

Note that the first two members of the sequence are not 1 and 4, but instead 2 and 4.

boogathree = 3^3 = 27.

This is smaller than megafugathree or even fugathree. It's the same value as fzthree. As you can see this sequence is humble so far.

boogafour = 4^^4 ~ 10^10^154

This is the same value as megafugafour. It's around the same level of numbers as a googolplex, so it's a very big number, but still small enough to have some real-world meaning if you think back to the article on large numbers in probability.

But what about boogafive = 5^^^5? That number would be represented like so:

5^5^5^5^5.......^5

6^6^6....^6 w/ 6^6^6....^6 sixes w/ 6^6^6....^6 sixes ......... w/ 6^6^6....^6 sixes w/ 6 sixes

w/ 6^6^6....^6 w/ 6^6^6....^6 sixes w/ 6^6^6....^6 sixes ......... w/ 6^6^6....^6 sixes w/ 6 sixes power towers

w/ 6^6^6....^6 w/ 6^6^6....^6 sixes w/ 6^6^6....^6 sixes ......... w/ 6^6^6....^6 sixes w/ 6 sixes power towers

w/ 6^6^6....^6 w/ 6^6^6....^6 sixes w/ 6^6^6....^6 sixes ......... w/ 6^6^6....^6 sixes w/ 6 sixes power towers

w/ 6^6^6....^6 w/ 6^6^6....^6 sixes w/ 6^6^6....^6 sixes ......... w/ 6^6^6....^6 sixes w/ 6 sixes power towers

w/ 6 power towers

which is very difficult to even understand!

As you can see, this is an awesome sequence so far. It starts with the trivial numbers 2 and 4, then with 27, then jumps to a decent tetrational number, but all of a sudden a pentational number that leaves all that in the dust, but then a super-unfathomable hexational number that leaves THAT in the dust.

Of course we can go all the way up to something like a "boogagoogolplex", a googolplex "googolplexated" to the googolplex. That would be represented as googolplex^^googolplex-2googolplex, and it's a truly unfathomable value.

If we can have a boogagoogolplex, we can have a boogagargoogolplex, a boogafzgoogolplex, a boogamegafugafzgargoogolplex, or how about a boogaboogagoogolplex? That would be applying the booga- prefix to a boogagoogolplex. That is the largest number we have encountered yet, bigger than even Bowers' boogolplex!

Of course we can transcend all that with:

boogaboogaboogaboogaboogaboogaboogaboogaboogaboogaboogaboogaboogaboogaboogaboogagoogolplex

or however many boogas as we please ... a boogagoogolplex boogas, X boogas where X is the booga ... boogagoogolplex number shown above, and so on ...

The gag- family of prefixes

Someone else on the "Really Big Numbers" forum page suggested another large number prefix, gag-, which is A(x,x) with the Ackermann function. As we learned earlier, weith Knuth's up-arrows this translates to 2^^x-2(x+3)-3. Let's try the first few values:

gagzero = 0+1 = 1

gagone = 1+2 = 3

gagtwo = 2*2+3 = 4+3 = 7

gagthree = 2^(3+3)-3 = 2^6-3 = 64-3 = 61

gagfour = 2^^(4+3)-3 = 2^2^2^65,536-3 — a really big number bigger than a googolduplex

Hold on now. We have 1, 3, 7, 61, but all of a sudden a number that transcends a googolduplex? What kind of world is this?! Welcome to the world of googology. Next, of course, we can have gagfive:

gagfive = 2^^^(5+3)-3 = 2^^^8-3 = 2^^2^^2^^2^^2^^65,536-3, can be thought of like so:

2^2^2^2.......^2 - 3

with 2^2^2^2.......^2 2's

with 2^2^2^2.......^2 2's

with 2^2^2^2.......^2 2's

with 2^2^2^2.......^2 2's

with 2^2^2^2.......^2 2's

with 65,536 2's

Note how much this is looking like the booga- sequence. The gag- sequence and the booga- sequence are roughly on par with each other in terms of how fast they grow. Gagsix is 2^^^^9-3; I'll leave you to imagine that number, but it's in about the same realm of numbers as boogasix.

But that's not the end of the story. We can have gagten, gaggoogol, or something like a gagboogafugafzgargoogolplex, but there's a further way to extend the gag- prefix as well.

Someone on the site had an idea for an additional prefix: mag-x = gag-gag-gag-gag-gag-gag ... -gag-x with x gag's. Let's try out some small values:

magzero = 0 (no gags)

magone = gagone = 3

magtwo = gaggagtwo = gagseven = 2^^^^^10-3 — only the second member of the sequence is a HUGE number!

magthree = gag-gag-gag-three = gag-gag-61 = gag(2^⁵⁹64-3)

which is equal to about:

2^^^...(X ^s)...^^^3

where X is 2^^^...(Y ^s)...^^^(gag-61)-3

where Y is 2^⁵⁹64-5

That just transcends anything previously encountered! But magfour is an even bigger number equal to roughly:

2^^^...^^^3

with 2^^^...^^^3 ^s

with 2^^^...^^^3 ^s

with 2^^^...^^^3 ^s

with gagfour ^s

I think you get the pattern. Magfive is about:

2^^^...^^^3

with 2^^^...^^^3 ^s

with 2^^^...^^^3 ^s

with 2^^^...^^^3 ^s

with 2^^^...^^^3 ^s

with gagfive ^s

Of course, we can have a maggoogolplex and the like. The mag- prefix is roughly on par with repeatedly applying booga- to a number.

Andre Joyce (we'll learn more about him in the next article) decided to extrapolate from THAT and go a step further with bag-, trag-, quadrag-, etc. bag-x = mag-mag-mag ... (x mags) ... mag-x, trag-x = bag-bag-bag ... (x bags) ... bag-x, and continue with quadrag-, quintag-, etc,. using the Latin prefixes, the same ones used to name the -illions. Imagine something like a centaggoogolplex that way. Centag- is the 100th member of the sequence mag, bag, trag, quadrag, etc. That is the very largest number we have envisioned as of yet!

Cantor's Attic Prefixes

Cantor's Attic is a small wiki that focuses mainly on infinite numbers, with some info on large finite numbers. One thing it has is the so-called "plex bang stack hierarchy". It gives a trio of prefixes:

x-plex = 10^x

x-bang = x! (x factorial)

x-stack = 10^^x

The first is just backformed from the googolplex, but the other two are original inventions. Several googolisms are invented with that system. The two main ones are:

googolbang = (10¹⁰⁰)!

This number can be approximated 10^{9.9566*10^101}, using the approximations for the factorial. Therefore, relatively speaking it's not that much bigger than a googolplex, even though it's really roughly raising it to the 100th power. For more on the googolbang, check out Sbiis Saibian's article on the googolbang.

googolstack = 10^^(10¹⁰⁰)

That's a power tower of a googol tens, a pretty awesome number alright. That tower would be MUCH larger than the observable universe!

While the googolbang is roughly comparable to the googolplex, the googolstack is far larger than either.

The "plex bang stack hierarchy" is that set of prefixes, and how you can append the googol with any combination of these. For example, we could make a googolplexbang, the factorial of a googolplex which is roughly equal to a googolduplex. Or a googolbangplexstack, equal to 10^^(10^{(10^100)!}), roughly equal to 10 tetrated to a googoltriplex, which is the largest number you can define with a googol plus one each of the plex/bang/stack prefixes.

Conclusion

Large number prefixes and suffixes are a fun way for amateurs to come up with large numbers. However, most of the prefixes I discussed aren't all that powerful, with the exceptions of megafuga-, booga-, the gag- family, and -stack. If we really want to go further, large number prefixes alone can't take you that far. To go further we'll need to devise large number notations, which we'll further examine in not too long, but not before we look at a weird system devised by Andre Joyce.

2.5B. Joyce's googol-based naming systems