Introduction
Now that we've examined a few large number notations, let's take a break from those and examine prefixes and suffixes to name numbers. Those are some popular ways to make large numbers, and the most popular of those prefixes is Cockburn's fuga- family of prefixes. Let's begin.
The Fuga- Family of Prefixes
A popular family of large number prefixes what I like to call the fuga- family of prefixes. How did those prefixes come to be? It began with Alistair Cockburn (pronounced /co-burn/) in his 2001 article A Fuga Really Big Numbers. In this article, he first talks about how kids "like to hurl big numbers at each other", and as an example he makes a story of kids arguing:
“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.”
Saying "million million million million ... " would usually be how the game ended. Surprisingly, the kids would not just end with "a zillion" or "infinity" or "the largest number anyone can think of plus 1". It seems that Cockburn's kids knew the spirit of googology: to not just stop with infinity or some cheating nonsense like "the largest number anyone can think of plus one" or a fake number like a zillion or a gazillion, but to continually come up with larger and larger well-defined numbers, just for the fun of it!
Repeatedly saying "million" was the best way to name large numbers for Cockburn's children, until one day, Alistair introduced them to the googol and googolplex, which I went over back in section 1. Now the kids could go way way further than a million. If you repeatedly said "million" for 24 hours, saying "million" three times per second, you'd get to 101,555,200. That's pretty big, but hardly scratching the surface of a googolplex. But now that the kids knew what the googol and googolplex were, they could continue:
"My commander rules a googol stars!"
"My commander rules a googolplex stars!"
Is a googolplex the end for the kids? Not quite. Kieran Cockburn, Alistair's youngest son (6 years old as of 2001), had a flash of imagination. He said "my commander rules a gargoogolplex stars", and said that a gargoogolplex is a googolplex googolplexes!
After that moment, the fuga- family of prefixes was born. Cockburn decided to generalize the name "gargoogolplex" to a prefix gar-, which multiplies the number it's applied to by itself. That is the first member of the fuga- family.
The Gar- Prefix
There are many numbers we can name with the gar- prefix. You can apply it to any number to multiply it by itself. Here are some examples:
Garone = 1*1 = 1
Gartwo = 2*2 = 4
Garthree = 3*3 = 9
Garfour = 16
Garfive = 25
Garsix = 36
Garseven = 49
Gareight = 64
Garnine = 81
Garten = 100
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,
000,000,000,000,000
a gargoogol 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,
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,
000,000,000,000,000,000,000,000,000
And then there's the eponymous gargoogolplex, a googolplex googolplexes. Evaluating it gives us:
1010^100*1010^100 = 1010^100+10^100 = 102*10^100
In the form 102*10^100, a gargoogolplex doesn't seem much bigger than a googolplex! Hell, it's less than 1010*10^100 = 1010^101. 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 1010^200. To avoid confusion, we can use the name "gargoogol-plexed" for 1010^200.
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.
The Fz- Prefix
Cockburn came up with a prefix to continue gar-, defining fz-x to be xx. 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 = 106,000,000
Cockburn mentioned this number in his article as an example of a number nameable with the fz- prefix.
Fzbillion = 109,000,000,000
A fzbillion is 1 followed by nine billion zeros!
Fztrillion = 1012,000,000,000,000
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 = 10100*10^100 = 1010^102
Written as 1010^102 a fzgoogol doesn't seem much bigger than a googolplex. However is really equal to raising a googolplex to the 100th power:
(1010^100)100 = 1010^100*100 = 1010^100*10^2 = 1010^102
And finally we have a number similar in spirit to Kieran's gargoogolplex:
Fzgoogolplex = (1010100)1010100 = 1010100*1010100 = 1010(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:
(101010100)10100 = 10(1010100*10100) = 1010(10100+100)
The fz prefix is a pretty cool prefix because it can name some numbers pretty compactly. For example, take the number name:
sixteen million seven hundred seventy seven thousand two hundred sixteen
That's a pretty unwieldy name. However, the fz- prefix can give it a much shorter name:
fzeight
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:
The Fuga- 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 "xx^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 (ab)c = ab*c):
((((x^x)^x)^x ... )^x with x x's
= x^(x*x*x ... *x) with x-1 x's in the exponent
= x^x^(x-1)
Here are some examples of the fuga- prefix:
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
A sizable third value. This number shows up a lot with threes using the weak hyper-operators, analogous to 7,625,597,484,987 with the normal hyper-operators.
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 = 10100^(100-1) = 100100^99 = 102*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 = 1010^(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:
a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex times a googolplex
If that was a headspin, look at a fugagoogolplex now:
Fugagoogolplex = 1010(1010100+100 - 10100 + 100) ~ 10101010100
This is roughly comparable to a googoltriplex, 10101010100.
The number of digits in this number is equal to about 1010^(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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... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 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
where you say "googolduplex" a googol times! How are we even supposed to comprehend that?!?
Additionally we can trump all of that with a fugagargoogolplex, a number described by Cockburn as so large that it boggles the ears in the speaking! And some have extended the somewhat erroneous "gargantugoogolplex" to get a "fugagargantugoogolplex", which is roughly equal to a googolquintiplex (10^10^10^10^10^10^100).
These are some pretty awesome numbers. The growth rate of the fuga- prefix is roughly double exponential (comparable to functions of the form aa^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.
The Megafuga- Prefix
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:
Had fun coming up with the brand-new function fuga(-) to get a faster growing function than any I knew of, so we could say "Fuga(googolplex)." Fuga(N) is N to the Nth power N times, so fuga(2) is 2^2, fuga (3) is 3^3^3, etc. You can see how this grows.
However, Stephan Houben, a friend of Cockburn's, noticed an ambiguity in how Cockburn described the fuga- prefix. He said:
Since the ^ operation is not associative, i.e. (x^y)^z != x^(y^z), this begs the question whether Fuga(3) means (3^3)^3 = 19683 or 3^(3^3) = 7625597484987. Probably the latter, since the goal is to get big numbers.
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:
............36860456095261392896
The leading digits are computable using logarithms, but the terminating digits are a little trickier: you need to use modular arithmetic, the same way the last digits of Graham's number have been found (more on that when we learn about Graham's number).
Anyway, let's continue:
megafugafive = 5^^5 ~ 10^10^10^2184
Now we're heading into some REALLY big numbers. This surpasses a googolduplex, and it's just the fifth member of the sequence.
megafugasix = 6^^6 ~ 10^10^10^10^36,305
Just the SIXTH MEMBER of the series transcends a fugagoogolplex!
megafugaseven = 7^^7 ~ 10^10^10^10^10^695,974
megafugaeight = 8^^8 ~ 10^10^10^10^10^10^10^15,151,335
megafuganine = 9^^9 ~ 10^10^10^10^10^10^10^10^369,693,100
megafugaten = 10^^10 ~ 10^10^10^10^10^10^10^10^10^10
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.
Now let's skip to two unfathomable numbers:
megafugagoogol = googol^^googol
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!
megafugagoogolplex = googolplex^^googolplex
This is a power tower of a googolplex googolplexes! It's a commonly cited example of a large number an ordinary person might come up with to name the largest number they can, perhaps as a retort to the infamous Graham's number (we'll examine that number a little later, and it's MUCH larger than a megafugagoogolplex).
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.
If we can have a prefix for n tetrated to n, why not a prefix for n pentated to n, n hexated to n ... or n n-ated to n? This is where a fifth member of the fuga- family comes in:
The Booga- Prefix
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:
boogaone = 1+1 = 2
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
with 5^5^5^5^5.......^5 5's
with 5^5^5^5^5.......^5 5's
with 5^5^5^5^5 5's
Holy shit!! Only the fifth member of the booga sequence leaves a megafugagoogolplex in the dust! Hell, it even transcends a megafugagoogolduplex, or a megafugagoogoltriplex, or anything like that! It's just such a new kind of pentational number.
How about boogasix = 6^^^^6? It can be imagined as:
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!
Other Prefixes and Suffixes
Prefixes and Suffixes from the English Language
Prefixes and suffixes are everywhere in the English language, even in the words "prefix" and "suffix" themselves. So it shouldn't be surprising that numbers are no exception. There are three suffixes in the English language used to name numbers:
-teen adds ten to a number
-ty multiplies the number by ten
-illion takes n and comes out with 10^(3n+3)
Then we can name some humorous numbers:
googolteen = 10^100 + 10 = 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,010
googolty = 10^100 * 10 = 10^101 = 100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
And googolillion = 10^(3*10^100+3) = one followed by three-googol-and-three zeros, a thousand times the cube of a googolplex.
Though the numbers are amusing, they aren't really that much new since they're really constructible from English.
The -plex suffix, however, is something pretty interesting. Since googol is 10^100 and googolplex is 10^10^100, x-plex can be thought to mean 10^x. For more on -plex refer back to my article on the googol and googolplex.
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 = 10x
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 109.9566*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.