Back to Ultimate Large Numbers List Part I
ULTIMATE LARGE NUMBERS LIST
PART II
Hyper-E & Extended Hyper-E
Entries: 405
Numbers larger than those expressible with E-Notation but still expressible in E# and xE#.
[10^^100,10^^10^^100)
Entries: 71
Tetronomical is a portmanteau of "tetration + astronomical". These are numbers so large that they can only be expressed using so called "power-towers", which are stacks of exponents many terms high. This Epoch contains power towers of up to "tetronomical" heights, exceeding the limits of power towers that could actually be written and reaching a point where thinking of it as a power tower is no longer helpful since the height is itself a power tower. These numbers can be expressed using power towers of 10s a hundred to a giggol terms high. The limit is considered to be the beginning of Pentational Numbers beginning at around 10^^^3.
The giggol is the first of Jonathan Bower's original extensions to the googol series. It was once the 4th largest number to appear on Robert Munafo's Number list, although he later removed it as not "notable enough". We could say the very large numbers begin here. This is where conventional notations, like power towers with repeated exponentiation start to become impractical. To write out this number in full in conventional notation we would have to write:
10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10
By contrast we can write this succinctly in Knuth's Up-Arrow Notation as 10^^100. In Bowers' Exploding Array Function (BEAF) we can write this compactly as ⟨10,100,2⟩. The giggol can also be written in Extensible-E (ExE) as E1#100. As we continue pas this point the numbers will quickly become to large to explicitly write out in any conventional notation, and thus we will have to rely on notations like BEAF and ExE to go further. Unlike conventional notations BEAF and ExE are "extensible" in that, the notations can be continued as long as spaces can be defined for BEAF and delimiters can be defined for ExE. Welcome to the exciting and terrifying world of googology ...
This might seem like a significant improvement over the giggol, like having a googolplex googolplexes is a lot more than a googolplex. It might seem that (10^10^10^...^10)^2 is the same as 10^10^10^...^10^2, but in fact a giggol squared is much smaller! (See next entry)
This number is still vastly smaller than even E2#100. Simply observe:
(10^^100)^3 = (E1#100)^3 = 10^(3E1#99) < 10^10^(1+E1#98) < ... < E(1+E1#1)#99
= E11#99 < E100#99 = E2#100
Surely this is greater than E2#100? Nope, not even close:
(10^^100)^(10^100) = 10^(E1#99 * E100) = 10^10^(100 + E1#98) < 10^10^10^(1+E1#97)
= E(1+E1#97)#3 < E(1+E1#96)#4 < ... < E(1+E1#1)#99 = E11#99 < E100#99 = E2#100
Believe it or not, this is barely an improvement over the last entry. We simply get:
(10^^100)^(10^10^100) = 10^10^(E100 + E1#98)
< 10^10^10^(1+E1#97) < ... < E11#99 < E100#99 < E2#100
This number is 10^10^10^ ... ^10^10^11 w/99 10s. It's what you would get if you changed the top most exponent in a giggol with 11. Amazingly, despite the fact that this seems like a very minor improvement, it's better than even raising the giggol to the googolplex.
This number is 10^10^10^ ... ^10^10^2 w/100 10s. It looks like giggol^2 but it's actually a lot larger. This serves as a lower-bound of megafuga-hundred. This number is also a very small example of a ternary Hyper-E Expression.
In Alistair Cockburn's number system, megafuga-n = n^^n. So naturally megafuga-hundred = 100^^100. It's obvious that this must be larger than 10^^100 (giggol) but probably not by as much as you might think. In fact this number is less than 10^^101. In fact it's even less than E3#100.
This number is 10^10^10^ ... ^10^10^3 w/100 10s. It looks like giggol^3 but it's actually much much larger. This serves as an upper-bound for megafuga-hundred.
It is a common rookie response to take whatever the largest named number they know is, and simply have 1 followed by that many zeroes. By the time we get up to numbers of this size however things are moving much much faster, so it turns out to be not all that competitive a response. The reason for the prevalence of this kind of response is probably because f(n) = 10^n is the fastest growing function most people know of.
In Hyper-E this number can be represented as E1#101, or E10#100. E10#100 acts as a lower-bound for giggol^giggol (see next entry).
This was a number I used to illustrate how much larger a grangol is than a giggol. Even if you raised a giggol to it's own power, you still would come up vastly short from a grangol. giggol^giggol is also "ever so slightly" greater than 10^^101. This implies that giggol^giggol ~ 10^giggol. However it must be remembered that this is only an approximation. In truth "ever so slightly" is really a huge unimaginable gulf. You would need to raise 10^^101 to the 10^^99th power to get giggol^giggol, so it is really much much much larger in the ordinary sense. The following calculations confirm that giggol^giggol is indeed bounded by 10^^101 and grangol:
giggol^giggol = (E1#100)^(E1#100) = (10^E1#99)^(E1#100) = 10^(E1#99 * E1#100) [Law of Exponents]
= 10^(E1#100 * E1#99) = (10^E1#100)^(E1#99) = (E1#101)^(E1#99) = (10^^101)^(10^^99) > 10^^101
: 10^^101 < giggol^giggol
giggol^giggol = (E1#100)^(E1#100) = 10^(E1#99 * E1#100) = 10^10^(E1#98 + E1#99)
< 10^10^(2E1#99) < 10^10^(10E1#99) = 10^10^(10 * 10^(E1#98))
= 10^10^10^(1+E1#98) = E(1+E1#98)#3
< E(1+E1#97)#4 < E(1+E1#96)#5 < ... < E(1+E1#1)#100 = E(1+10)#100 = E11#100 < E100#100 = grangol
As amazing as this number sounds, it isn't even as large as E(1+E1#98)#3, let alone E11#100. In my original proof for giggol^giggol < grangol there was an implicit proof that in fact (giggol^giggol)^giggol < E(1+E1#98)#3. This suggests that grangol is much much larger than a giggol than even my initial proof would suggest!
This number is a relatively exacting upper-bound on giggol^giggol. You can envision it as:
10^10^10^(1+10^10^10^10^ ... ^10^10^10)
where there are 98 10s inside the parenthesis. In other words, this number is a power tower of tens 101 terms high, with a +1 occurring at the 4th position heading upwards. In comparison with E11#100 (see next entry) this seems like a good bound, but in truth even if you raised giggol^giggol to the power of a giggol you still would come up vastly short of this number!
This number acts as an upper-bound on giggol^giggol. It proves conclusively that it is less than a grangol. Even this upper-bound is actually a huge overestimate.
This nifty little entry is a "binary grangol", in analogy with bingol, the binary googol. It is significant in that it is a simple lower-bound for f_3(100). Since f_2(100) = 100*2^100 > 2^100, and f_2(n) > 2^n for n>1, it follows that f_2^2(100) > 2^2^100, and f_2^3(100) > 2^2^2^100, etc. So we know brangol is lower than f_3(100). On the other hand we can show that 2^100#100 is larger than E26#100, making it larger than the previous entry. To do this we note that 10^10^27 < 2^2^100, and that if 10^10^N < 2^100#n, then 10^10^(10^N-1) < 2^100#(n+1). This leads to a power-tower with -1s. Using the "Zipper Technique" we can zip them up and subtract one from the top exponent to obtain E26#100 as a definite lower-bound. One amusing thing about this number is, despite having a hundred 2's in it's power tower, because of that 100 on the top, it's still much much larger than if we simply had a 100 10s with a 1 on top. That 100 on the top makes a big difference despite the small base.
This is f_3(100). This number is approximately E32#100, so it lies between a giggol (E1#100), and grangol (E100#100). To compute this value we expand f_3(100) to f_2^100(100), which is 100 applications of f_2 where f_2(n) = n*2^n. This quickly becomes very complicated in standard mathematical notation, so bounds are created instead. In more standard notation we can create the following power-tower bounds:
2^100#100 < f_3(100) < 10^32#100
[3](100) is an alternative notation I use for f_3(100), where [a](n) = f_a(n). Not to be confused with Stepney's Notation for Steinhaus-Moser Polygons.
This is actually an upperbound on f_3(100). This can be demonstrated by first showing that f_2^2(100) < 10^10^32. f_2^2(100) = 100*2^100*2^(100*2^100). This will be smaller than 2^(102*2^100+108). Since 102 and 108 are multiples of 3 we can divide them by 3 to obtain 8^(34*2^100+36). This will be smaller than 10^(35*2^100) which is smaller than 10^(70*2^99) from here we note that 2^99 < 10^30 which means 10^(70*2^99) is smaller than 10^(100*10^30) = 10^10^32. f_2(n) < 10^n for all n. From this it follows that f_2^3(100) < 10^10^10^32, and f_2^4(100) < 10^10^10^10^32. From this we conclude that f_3(100) = f_2^100(100) < E32#100. This is smaller than a grangol which proves that f_3(100) is smaller than a grangol.
A "grangol" is a number I myself coined in 2011. The name is derived from combining the words "grand" and "googol", thus a grangol is short for "grand googol". It's an example of a number expressible using my Hyper-E notation. Hyper-E notation is a modern equivalent of a notation I devised as a kid. For simple expressions like above let:
Ea#b = 10^10^10^ ... ^10^a w/b 10s
A grangol is therefore 10^10^10^...^10^10^100 w/100 10s. One way to think of a grangol is as a continuation of the googol tradition. We begin by creating a "googol series". The first member of this series is the eponymous googol, or 10^100. The 2nd member of the series is a googolplex, or 10^10^100. The 3rd member is a googolduplex or 10^10^10^100, and so on. A grangol is defined as the 100th member of the googol series. A grangol turns out to be "just a little" larger than Jonathan Bower's "giggol". In fact, it can be shown that:
giggolgiggol < grangol
That is, a giggol raised to a giggol is still smaller than a grangol. A giggol in Hyper-E is equal to E1#100. We can work it out as follows:
giggol^giggol = (E1#100)^(E1#100) = (10^E1#99)^(E1#100) =
10^(E1#99*E1#100) < 10^(E1#100*E1#100) = 10^(E1#100)^2 =
10^(10^E1#99)^2 < 10^(10^E1#99)^10 = 10^10^(10*E1#99) =
10^10^(10*10^E1#98) = 10^10^10^(1+E1#98)
= E(1+E1#98)#3 < E(1+E1#97)#4 < E(1+E1#96)#5 < ... etc. ... < E(1+E1#1)#100 =
E(1+10)#100 = E11#100 < E100#100 = grangol
Thus we conclude that giggol^giggol < grangol.
This is the smallest power tower of 10s larger than a grangol. As such it acts as an upper-bound, allowing the grangol to be compared to other numbers. To prove grangol < 10^^102 observe:
grangol = E100#100 < E10,000,000,000#100 = E(10^10)#100 = E1#102 = 10^^102.
This number is more imporant than it initially looks, as this forms a basis for upper-bounding Hyper-E Numbers.
This was a number I coined to illustrate the inadequacy of the -plex prefix to capture higher level recursions. It is true that Bower's has used the -plex prefix to refer to any type of recursion, however based on the googolplex many have concluded that n-plex always means 10^n. If that is so than a grangolplex is an inadequate name for E100#100#2 (see grangoldex). Following the above definition it follows that grangol-plex = 10^grangol = E(E100#100) = E100#101.
This number is greater than 10^^102, but less than grangol^grangol. First observe:
E100#101 > E10#101 = E1#102 = 10^^102
Since grangolplex = 10^grangol it follows that it is less than grangol^grangol.
As usual, this number isn't "much larger" than a grangol, at least in terms of power tower height. It must be greater than 10^^102 since grangol^grangol > 10^grangol = E100#101 > E10#101 = E1#102 = 10^^102. However it must be less than 10^^103. This is a little more involved, but can easily be established as follows:
(E100#100)^(E100#100) = 10^(E100#99 * E100#100) = 10^10^(E100#98 + E100#99)
< 10^10^10^(1+E100#98)
= E(1+E100#98)#3 < E(1+E100#97)#4 < E(1+E100#96)#5 < ... < E(1+E100#1)#100
= E(1+E100)#100 < E(E101)#100 = E101#101 < E(10^10)#101 = E1#103 = 10^^103
An upper-bound for grangol^grangol.
This is a lower-bound that is commonly used for a Mega since it is the largest integral power tower of 10s which is less than a Mega. This lower-bound is still much much bigger than a grangol, proving that a Mega is also larger.
It can be written in Hyper-E as E1#257 or E10#256.
This is a more accurate lower bound on the Mega which attempts to narrow down the top most exponent. Written in full it looks like:
10^10^10^10^ ... ... ^10^10^10^10^619 w/256 10s
This number goes by various names, "Mega", "Zelda", "two in a circle" or "two in a pentagon". It is among the "classic" large numbers along with a googolplex, and Graham's Number. It was first defined by Hugo Steinhaus using his own custom operator notation...(READ MORE). This number is notable for being the largest number on Munafo's Large Number list for a long time. Currently it is still the 3rd largest entry on the list.
This is a more accurate upper-bound on the Mega. It looks like:
10^10^10^10^ ... ... ^10^10^10^10^620 w/256 10s
This is a common upper-bound on the Mega. It is the smallest expression in the form of 10^^N , where N is a positive integer, that is greater than the Mega.
It can be written in Hyper-E as E1#258 or E10#257 or E10,000,000,000#256.
In the infamous "My Number is Bigger" thread, Gmalivuk pointed out that D^^D was vague and could be interpreted as 13^^13 using hexadecimal or 500^^500 using roman numerals. This latter interpretation is much much larger, though still smaller than Twasbrillig's Power Tower, the 12th competitive entry.
This is the 12th valid entry in the "My Number is Bigger" competition. The original posting of this number was deleted unfortunately but traces of it remain in the responses. This was the first entry to officially break the Tetrational Epoch barrier. This is also the last in an uninterrupted string of larger and larger entries. After this some smaller entries are entered in the competition before this "Large Number in play" is overcome. (See Crazyjimbo's Factorial power tower for entry 13 ).
A weak upper-bound on the Mega based on the Left Associative Tetrates Lemma. Note that we can use the Knuth Arrow Theorem to bound this from above: 256^^512 < (10^^2)^^512 < 10^^514 < 10^^1000. Also 256=2^8 < 2^16 = 2^2^2^2 = 2^^4. Therefore: 256^^512 < (2^^4)^^512 < 2^^516 (Knuth Arrow Theorem). So it is much less than 2^^1000.
This is the second largest entry on Robert Munafo's Numbers List. According to Harvey Friedman this is a lower bound on the number of symbols that would be required to prove the finiteness of TREE(3) using finite arithmetic. Suffice it to say no man nor machine is writing out this proof in this universe. Thankfully ... we are not restricted to proving things in finite arithmetic, because by Kruskal's Tree Theorem we know that TREE(n) is finite
Chilialogue is a number that you must take the common logarithm of 1000 times to reduce it to 1. It may also be called the giggolchime, using the chime modifier on giggol. Chilialogue / giggolchime can be written as E1#1000 in Hyper-E Notation. Expressed as a power tower, this number is about twice as tall as the previous entry.
E3#1#2 is a simple Ternery Hyper-E expression which does not have a standard name. Solving it we have E3#1#2 = E3#(E3#1#1) = E3#1000. This makes it a little larger than chilialogue. Note however that this value arises naturally from Hyper-E using single digit arguments. This number can be visualized as 10^10^10^10^10^ ... ... ^10^10^10^10^10^3 w/1000 10s. It can also be written in Hyper-E as E1000#999. By the Hyper-E Theorem this makes it a little less than grangolchime.
Grangol with the chime modifier. This number is larger than giggolchime, as well as the expression E3#1#2. It is smaller than giggoltoll:
E1000#1000 = E(10^3)#1000 = E3#1001 < E10#1001 = E1#1002 << E1#10,000
myrialogue is a number so large you need to apply the common logarithm 10,000 times to reduce it to 1. It may also be called the giggoltoll following another naming convention. This entry is a power tower approximately 10 times taller than the last entry, yet we are only getting started with the tetronomical range, let alone numbers described by pentation.
The next simplest Ternery Hyper-E expression. E4#1#2 = E4#(E4#1#1) = E4#(E4) = E4#10,000. Thus its bigger than E1#10,000 (giggoltoll) and smaller than E10,000#10,000 (grangoltoll).
The grangol modified by toll. The toll modifier takes all instances of 100 and replaces them with 10,000. Despite how large the first element is becoming it still makes no difference to the second element.
We can easily show E10,000#10,000 = E(10^4)#10,000 < E(10^10)#10,000 = E1#10,002 << E1#100,000.
This number is the result of combining one of Bower's numbers with my -gong suffix. If a googolism, call it g, can be defined as f(100), then g-gong is defined as f(100,000). A giggol = 10^^100, and therefore the giggolgong is 10^^100,000. A giggolgong is a power tower of 10s 100,000 terms high! This number is way way bigger than a Mega, yet it's vanishingly small compare to 256^^(2^256), a naive upper-bound on the Mega.
In Hyper-E this number can be written as E1#100,000.
E5#1#2 = E5#(E5#1#1) = E5#100,000 > E1#100,000.
A lower bound on a grangolgong. This can be shown to fall between E5#1#2 and E100,000#100,000. First E5#1#2 = E5#100,000 < E10#100,000 = E1#100,001. On the other hand E1#100,001 = E10#100,000 < E100,000#100,000.
The grangolgong is equal to 10^10^10^ ... ^10^10^100,000 w/100,000 10s. It lies between E10#100,000 = E1#100,001, and E10,000,000,000#100,000 = E1#100,002.
A upper bound on a grangolgong. We can show this is greater than a grangolgong. E100,000#100,000 = E(10^5)#100,000 < E(10^10)#100,000 = E1#100,002 = 10^^100,002.
This is a number that came up in the xkcd forum "My number is, in fact, bigger!", an unofficial sequel to the "My Number is Bigger!" thread. The new competition was initiated by Vytron. This number was defined by "Earthling on Mars" as a part of a larger naive attempt to beat the number <10,10,googol> using iterated power towers.
This number is larger than 10^^1,000,000 and so is also larger than 10^^100,002. On the other hand, 1,000,000^^1,000,000 < (10^^2)^^1,000,000 < 10^^1,000,002. At the same time we have 10^^1,000,002 is less than (3^^3)^^1,000,002 < 3^^1,000,005 and therefore much less than 3^^7,625,597,484,987 = 3^^^3.
Jonathan Bowers' tritri is a relatively small pentational number. Expanding it reveals it's formidable size. We have...
3^^^3 = 3^^3^^3 = 3^^3^3^3 = 3^^3^27 = 3^^7,625,597,484,987
It's a power tower of 3's 7,625,597,484,987 terms high. This makes it unfathomably larger than 10^^100,002, but vanishingly small compare to 2^^(2^256).
This is a naive upper-bound on the Mega. It is roughly equal to a power tower of 10s E77 terms high. This makes it less than a googol-stack, but more than a grangolgong.
10^^(10100)
googol-stack
This is another number I found on Cantor's attic. n-stack is defined as 10^^n. In other words, n-stack is a power tower of 10s "n" terms high. Having a power tower of a googol tens certainly seems pretty impressive. Yet this is still small for a pentational number. It is by necessity much larger than a "Mega" since a mega must be less than a power tower of tens only 258 terms high. Yet this number must also be vanishingly small compared to a grangoldex, because a grangoldex is greater than a "stack" of tens a grangol terms high, where a grangol is the 100th member of the googol series!
Therefore between the googol-stack and the grangoldex must be a vast sea of numbers!
E100#1#2
googoldex
The googoldex is a number I coined to illustrate just how many kinds of numbers can be named between my numbers using Hyper-E Notation. Hyper-E Notation has the advantage of more easily defining numbers between numbers in other systems.
The -dex prefix simply takes some number of the form Ea#b and returns Ea#b#2. So we let googol = E100#1, and so googoldex becomes E100#1#2. What does this mean? Working it out we obtain:
E100#1#2 = E100#(E100#1#1) = E100#(E100) = E100#googol =
EEEEEEEEEEEEE ... ... EEEEEEEEEEEEEEEE100 w/googol Es
= 10^10^10^10^ ... ... ^10^10^10^10^100 w/googol 10s
In other words, a googoldex is a power tower of 10s a googol terms high topped off with a 100. It's the googolth member of the googol series.
It's just "a little" larger than a googol-stack. In fact it's greater than 10^^(googol+1) but less than 10^^(googol+2).
(10100)^^(10100)
megafugagoogol
A megafugagoogol is only slightly larger than a googoldex. To see why consider the folowing:
googol^^googol > E(10^100)#(10^100-1) = E100#(10^100) = googoldex.
10^((10100)^^(10100))
megafugagoogol-plexed
This is a non-standard way to interpret "megafugagoogolplex". One can image the "megafuga" being applied before the "plex". To distinguish the cases I coin this number as megafugagoogol-plexed. This number is only slightly larger than a googoldex and vastly smaller than a googolplexidex. This can be demonstrated easily using the Left Associate Polyates Lemma (LAPL):
10^((10^100)^^(10^100)) < 10^((10^10^10)^^(10^100))
= 10^((10^^3)^^(10^100)) < 10^10^^(3+10^100) = 10^^(4+10^100)
< 10^^(10^10^100) = E1#(googolplex) < E100#(googolplex) = E100#(E100#2) = E100#2#2 = googolplexidex
:: megafugagoogol-plexed << googolplexidex
10^^(2^1024)
Limit of break_eternity.js
This is the limit of the break_eternity.js number library. It consists of a power tower of tens whose height is represented with a double-precision floating point number, which has a limit of 2^1024 (in fact this is slightly larger than the largest normal number it can actual represent). This is approximately equal to 10^^(1.797x10^308). It should be noted that at this size the precision must represent an integer, so this value is well defined and doesn't require a real valued tetration to make sense. This entry is mostly notable (to me) for being the very largest Entry of Robert Munafo's famous "Numbers List". So at this point and onwards we are in unexplored territory.
E100#2#2
googolplexidex
This number is a power tower of 10s a googolplex terms high topped off with a 100.
E100#(1+10^10^100)
googolplexidexiplex
This number serves a a lower-bound on the megafugagoogolplex. It is also an example of a combinatorial googolism. There is a wealth of numbers that can be derived from various googological systems, an only a tiny fraction of them are ever explicitly stated. These tend to aggregate in certain vicinities, because of the vast differences of power of the different word components.
(E100#2#2)^(E100#2#2)
fzgoogolplexidex
This number is incredibly close to a megafugagoogolplex, yet it is still slightly smaller.
E(1+10^100)#(10^10^100)
This number is an upperbound on googolplexidex^^2.
E(100+10^100)#(10^10^100)
A very accurate lower-bound on a megafugagoogolplex.
(10^10^100)^^(10^10^100)
megafugagoogolplex
This number is derived from the work of Alistair Cockburn. It combines his megafuga- prefix with the number googolplex. n-plex = 10^n while megafuga-n = n^^n. Note that the definition here is actually ambiguous. Is a megafugagoogolplex equal to megafuga(googolplex) or plex(megafuga(googol). That is, we can read it grammatically as meaning either megafuga-"googolplex" or as "megafugagoogol"-plex. These result in different numbers (See megafugagoogol-plexed). The original intent of Cockburns work however is that the prefixes are being applied after the suffixes. Thus a megafugagoogolplex is intended to mean googolplex^^googolplex. This number has come up independently from a few sources , usually as an example of the largest kind of number the average person would think of to try to trump Graham's Number (psst ... it doesn't even come close. See Graham's Number far below). That being the case it's nice to have a name, any name, for this number.
This number is pretty insanely huge, although it has more to do with the height of the stack than the terms being a googolplex. It confers just enough benefit so that it goes slightly past a googolplexidex. A googolplexidex is a power tower with a googolplex+1 terms, where as a megafugagoogolplex has only a googolplex terms, but the megafugagoogolplex ends up being ever so slightly larger (from a googologist's perspective) mainly because of the leading exponent. It can be shown that megafugagoogolplex lies between E100#(googolplex+1) and E100#(googolplex+2), and is greater than googolplexidex^googolplexidex. For a worked out proof click here.
E(101+10^100)#(10^10^100)
This is an accurate upper-bound on the megafugagoogolplex.
E100#(2+10^10^100)
googolplexidexiduplex
This number serves as a "weak" upper-bound on a megafugagoogolplex. It's also an example of a combinatorial googolism.
E100#3#2
googolduplexidex
This number is a power tower of 10s a googolduplex terms high topped off with a 100. This number is still vastly smaller than a giggolplex.
E100#4#2
googoltriplexidex
E100#(googoltriplex) = E100#(10^10^10^10^100).
E100#5#2
googolquadriplexidex
E100#(googolquadriplex) = E100#(E100#5).
E100#6#2
googolquintiplexidex
E100#(googolquintiplex) = E100#(E100#6).
E100#7#2
googolsextiplexidex
E100#(googolsextiplex) = E100#(E100#7).
E2#9#2
This number looks like it's equal to E100#8#2, but in fact this is not the case. Expanding we have E2#9#2 = E2#(E2#9) = E2#(E100#8) < E100#(E100#8) = E100#8#2. Thus we prove E2#9#2 < E100#8#2 even though E2#9 = E100#8. The number E2#9#2 can be used as an approximation of googolseptiplexidex using the Hyper-E Number Format, in which the arguments must be less than or equal to 10.
E100#8#2
googolseptiplexidex
E100#(googolseptiplex) = E100#(E100#8). E100#(E100#8) = E(10^2)#(E100#8) < E(10^10)#(E100#8) = E1#(2+E100#8) < E1#(E101#8) < E1#(E(10^10)#8) = E1#(E1#10) = E1#(E1#(E1#1)) = E1#1#3 = 10^^^3 = 10^^10^^10 = triataxis. Thus a googolseptiplexidex is smaller than a triataxis, the limit of the Tetronomical Epoch ...
10^^(10^10^10^10^10^10^10^10^10^9)
E1#(E9#9)
This number makes the limit of break_eternity.js seem small by comparison. It's just a little bit smaller than a triataxis.
E9#9#2
... only ... wait a minute ... with Hyper-E Notation we can actually get even closer to a triataxis than even that! Observe: E9#9#2 = E9#(E9#9) < E10#(E9#9) = E1#(1+E9#9) < E1#(E10#9) = E1#(E1#10) = E1#(E1#(E1#1)) = E1#1#3 = 10^^^3 = 10^^10^^10 = triataxis.
10^^10^^10
triataxis
This number is 10^^^3. It is also the 13th official competitor in the "My Number is Bigger" competition, and the 15th valid number. This number is the 3rd entry by Gmalivuk, the starter of the competition. It is after this number that Rodan "shuffles out".
It also happens to be a number I call triataxis. This means you have to apply the "taxarithmic" function (superlogarithm) 3 times to reduce it to 1. The reason I call it the "taxarithmic" function is because I think of it as classifying numbers into taxonomic categories (similar to Robert Munafo's Number classes concept). Let taxa(x) be the taxarithmic function defined by taxa(x) = taxa(log(x))+1 : x > 1 , taxa(x) = 0 : x <= 1. It follows from the definition that taxa(10^^n) = n. Thus we have that: taxa(taxa(taxa(triataxis))) = taxa(taxa(taxa(10^^10^^10))) = taxa(taxa(10^^10)) = taxa(10) = taxa(1)+1 = 0+1 = 1.
E100#9#2
googoloctiplexidex
10^10^...^10^10^100 w/googoloctiplex 10s.
E100#10#2
googolnoniplexidex
10^10^...^10^10^100 w/googolnoniplex 10s.
E100#11#2 = E100#(E100#11)
googoldeciplexidex
We are still a very long way from a ...
[10^^10^^100,3^^^^3)
Entries: 113
This Epoch covers most of the numbers that would be considered of pentational size. It includes such notable numbers as gaggol, greagol, dekataxis, and megiston. If these numbers are expressed as power towers describing power towers, then the number of power towers stays within the astronomical range. Another way to look at it, is these numbers can be expressed as astronomical sized "tetra-towers" (left-leaning towers of repeated tetration).
10^^10^^100
giggolplex
The giggolplex is a number coined by Jonathan Bower's as an extension of the googol naming conventions. A giggolplex is a power tower of 10s a giggol terms high, where a giggol is itself a power tower of 10s 100 terms high. It can be notated as:
giggolplex = 10^10^10^ ... ^10^10^10 w/giggol 10s
E100#100#2
grangoldex
Here's a HUGE number, very similar in spirit to the once great googolplex. A grangoldex is a power tower of 10s a grangol terms high, with a 100 on top of all this! Another way to think about it is that a grangoldex is the grangolth member of the googol series. We can also notate it as:
grangoldex = 10^10^10^ ... ^10^10^10^100 w/grangol 10s
This number is just slighly larger than a giggolplex. This can be seen since giggol < grangol, it follows that a grangoldex has more 10s than a giggolplex.
E1#102#2 = 10^^10^^102
An elementary upperbound on the grangoldex. Proving this is fairly straightforward:
E100#100#2 = E100#(E100#100) < E(10^10)#(E100#100) = E1#(2+E100#100)
To continue we simply observe that a+Eb@ < E(b+1)@ provided a is smaller than or equal to Eb@, regardless of any other factors, because every expression of the form Eb@ must be a power of 10, and if a <= Eb@ we have a+Eb@ <= 2Eb@ < 10Eb@ <= E(b+1)@. The upshot of this is we can always get an upperbound just by increasing the top exponent by 1. So we have:
E1#(2+E100#100) < E1#(E101#100) < E1#(E(10^10)#100) = E1#(E1#102) = E1#102#2
Thus E100#100#2 < E1#102#2. It shouldn't be too hard to see that this process can be iterated an arbitary number of times, carrying changes upwards, implying that:
E100#100#n < E1#102#n
This will be important for later.
(1,000,000^^1,000,000)^^(1,000,000^^1,000,000)
mungo
This is a googolism coined by "Earthling on Mars" as part of a naive attempt to beat <10,10,googol> using power towers. He describes 1,000,000^^1,000,000 as a "mega" and mungo as mega^^mega. It turns out however not be any faster an iteration than pentation. From the Knuth-Arrow theorem (see my paper "A Theorem for Knuth-Arrows") it follows that...
(1,000,000^^1,000,000)^^(1,000,000^^1,000,000) < 1,000,000^^(1,000,000+1,000,000^^1,000,000)
< (10^^2)^^(1,000,000+1,000,000^^1,000,000) < 10^^(1,000,002+1,000,000^^1,000,000)
< 10^^(1,000,002+10^^1,000,002) << 10^^10^^1,000,003
This means this number is still massively smaller than 10^^10^^10^^100 or giggolduplex. On the other hand, grangoldex = E100#100#2 = E100#(E100#100) < E1#(2+E1#102) < E1#(E1#103) = 10^^10^^103, and this is much smaller than (1,000,000^^1,000,000)^^(1,000,000^^1,000,000), so we know a mungo is larger than grangoldex. Thus we can say...
grangoldex << mungo << giggolduplex
E100#1#3 = E100#(10^100)#2
googoldudex
E100#1#3 = E100#googoldex. We've only just begun our climb to a grangoldudex ...
E40#2#3
Lower-bound on Triton
This is a lower-bound on triton that can be proved and expressed compactly in Hyper-E Notation. It's a value I've commonly used as an approximation, in the same way I've used E619#256 to approximate a mega. The proof of this isn't too difficult. We note that 3[5] = 3[4][4][4] and that 3[4] = 3^3^85 > 10^10^40. From here we note that N[3] > 10^N provided N > 10. So we know that (10^10^40)[3] > 10^10^10^40. This means that (10^10^40)[4] > E40#(2+10^10^40) > E40#(10^10^40) = E40#2#2. It follows from this that (E40#2#2)[4] > E40#2#3. This value is also greater than E100#1#3, since E100#1#3 = E100#(10^100)#2 while E40#2#3 = E40#(10^10^40)#2.
3[5]
triton
~ E40#2#3
Triton is the name I coined for "3 in a circle" from Steinhaus's circle function. Using Stepney's Bracket Notation we can expand it thus:
3[5] = 3[4][4][4] = 3[3][3][3][4][4] = (3^3)[3][3][4][4] = 27[3][3][4][4] = (27^27)[3][4][4] = (3^81)[3][4][4] = (3^3^4)[3][4][4] = ((3^3^4)^(3^3^4))[4][4] = (3^3^85)[4][4]
3^3^85 ~ 10^10^40. As a consequence a triton can be approximated fairly closely in ExE as E40#2#3. This means it falls between googoldudex and googolplexidudex. This is a far far far larger number than mega, or "2 in a circle". While Steinhaus named 2[5] "mega" and 10[5] "megiston" he did not provide any other names for numbers. Moser later defined a much much bigger number now simply called Moser.
E41#2#3
Upper-bound on triton
Although a bit trickier it can also be proven that this is an upperbound on a triton.
E100#2#3 = E100#(10^10^100)#2
googolplexidudex
It's not too hard to see that this would be larger than a triton. To do this we will use the upper-bound. 3[5] < E41#2#3 < E100#2#3.
E100#3#3 = E100#(E100#3)#2
googolduplexidudex
Post-fixes should be evaluated left-to-right, so we apply the two plexes first then the two dexes. Using this googol-x-plex-y-dex format we can create numbers of the form E100#(x+1)#(y+1). This has the unique property of filling out entries between what are normally singular recursive steps, as in for example, that gap between a giggolplex (E1#100#2) and a giggolduplex (E1#100#3), with Bowers' naming conventions.
E100#4#3 = E100#(E100#4)#2
googoltriplexidudex
Next ExE googolism in order.
E100#5#3 = E100#(E100#5)#2
googolquadriplexidudex
Googolquadriplex "dexed" twice. That means, start with a googolquadriplex, have that many 10s topped off with 100, then have that many 10s topped off with 100. Incomprehensible. Still, tame for pentational standards ...
E100#6#3 = E100#(E100#6)#2
googolquintiplexidudex
Googolquintiplex (E100#6) "dexed" twice.
E100#7#3 = E100#(E100#7)#2
googolsextiplexidudex
Googolsextiplex (E100#7) "dexed" twice. It's a long and mostly empty space between a giggolplex and a giggolduplex. Most of that space is taken up by these intermediate ExE googolism's.
E100#8#3 = E100#(E100#8)#2
googolseptiplexidudex
Googolseptiplex (E100#8) "dexed" twice.
E100#9#3 = E100#(E100#9)#2
googoloctiplexidudex
Googoloctiplex (E100#9) "dexed" twice.
10^^10^^10^^10
tetrataxis
The tetrataxis is ten pentated to the fourth, 10^^^4. It is therefore one of the basic Knuth arrow outputs. It can be proven with very little difficultly that E100#10#3, which can be named googolnoniplexidudex, is actually "slightly" larger, though it's difficult to say exactly how (numbers at this scale already have no easy way to compare them in ordinary terms). First we can write tetrataxis in Hyper-E Notation as E1#1#4. From there we can show that this is equal to E1#10#3. To do so, we simply note that if @ = 'E1#' then E1#1#4 may be expanded to @@@(@1). @1 = E1#1 = E1 = 10. So we have @@@10. This is E1#(E1#(E1#10)) which can be written compactly as E1#10#3. From here we can use the Hyper-E lemma that if we increase any argument in a Hyper-E expression, the resulting expression is larger. Thus we have E1#10#3 < E100#10#3. Thus we conclude: tetrataxis = 10^^^4 = E1#10#3 < E100#10#3 = googolnoniplexidudex. Therefore tetrataxis is smaller than googolnoniplexidudex. With a little bit more effort it can be shown that E1#10#3 > E100#9#3, the previous entry.
E100#10#3 = E100#(E100#10)#2
googolnoniplexidudex
Still a long long way from a giggolduplex actually. A grangoldudex comes "shortly" after that.
E100#11#3 = E100#(E100#11)#2
googoldeciplexidudex
Still a long long way from a giggolduplex. Remember that second argument needs to reach 100 before we get to a giggolduplex range. Let's just skip ahead to ...
10^^10^^10^^100
giggolduplex
A Bowerism found in the "Giggol Group". Not to be confused with applying the ordinary "plex" to a giggol twice, which is only E1#102, which is much much much much smaller. Instead Bowers uses "plex" to mean any additional step of recursion at the current recursive level. So for giggol, the "plex" here is equivalent to my "dex".
E100#100#3
grangoldudex
grangoldudex = E100#100#3 = E100#(E100#100#2)
= E100#grangoldex = EEE...EEE100 w/grangoldex Es.
The grangoldudex is smaller than 10^^^5. This can be seen as follows:
E100#100#3 = E100#(E100#(E100#100)) < E100#(E100#(E10,000,000,000#100)) = E100#(E100#(E1#102))
< E100#(E1#(2+E1#102)) < E1#(2+E1#(2+E1#102)) < E1#(E1#(3+E1#102)) < E1#(E1#(E1#103)) = E1#103#3
< E1#(10^^10)#3 = E1#1#5 = 10^^^5
E1#102#3
An upperbound on grangoldudex, and part of a special upperbounding sequence. Again we can always move up while "rounding things up" to get upperbounds. So for grangoldudex we have:
E100#(E100#(E100#100)) < E1#(E101#(E100#100)) < E1#(E1#(E101#100))
< E1#(E1#(E1#102)) = E1#102#3.
This pattern continues.
((1,000,000^^^2)^^^2)^^^2
humungo
This googolism was coined by "Earthling on Mars" as part of the "My number is, in fact, bigger" thread.
Bounding it from below we have ...
humungo > ((10^^1,000,000)^^^2)^^^2
> (10^^10^^1,000,000)^^^2 > 10^^10^^10^^1,000,000 = E1#1,000,000#2
Bounding it from above we have ...
humungo < 1,000,000^^^6 < (3^^^2)^^^6 < 3^^^8
Since humungo is less than 3^^^8 we know it's less than 3^^^10.
E100#1#4 = E100#(10^100)#3
googoltridex
A googol (E100) "dexed" thrice!
E100#2#4 = E100#(E100#2)#3
googolplexitridex
A googolplex (E100#2) "dexed" thrice ...
E100#3#4 = E100#(E100#3)#3
googolduplexitridex
A googolduplex (E100#3) "dexed" thrice ... oh wait here comes something to break the monotony ...
5^^^5 = {5,5,5} = ⟨5,5,3⟩
Tripent Jr.
~ E2184#3#4
Bowers' original value for a tripent, which is "five pentated to the fifth". In the old array notation this was written as {5,5,5}, hence "tri"(3)+"pent"(5), that is, three fives. In the new array notation this is equivalent to <5,5,3>. We can actually bound this quite accurately in Hyper-E notation as about E2184#3#4, the exact value being only "slightly" larger. This allows us to easily see this is above E100#3#4 yet below E100#4#4, the latter being approximately E(10^100)#3#4. Hyper-E thus allows us to pretty precisely measure different notations against each other.
E100#4#4 = E100#(E100#4)#3
googoltriplexitridex
Welp ... back to the grind. Googoltriplex (E100#4) "dexed" thrice ...
E100#5#4 = E100#(E100#5)#3
googolquadriplexitridex
Googolquadriplex (E100#5) "dexed" thrice ... still a long way to grangoltridex (E100#100#4).
E100#6#4 = E100#(E100#6)#3
googolquintiplexitridex
Googolquintiplex (E100#6) "dexed" thrice ...
E100#7#4 = E100#(E100#7)#3
googolsextiplexitridex
Googolsextiplex (E100#7) "dexed" thrice ...
E100#8#4 = E100#(E100#8)#3
googolseptiplexitridex
Googolseptiplex (E100#8) "dexed" thrice ...
E100#9#4 = E100#(E100#9)#3
googoloctiplexitridex
Googoloctiplex (E100#9) "dexed" thrice ...
E100#10#4 = E100#(E100#10)#3
googolnoniplexitridex
Googolnoniplex (E100#10) "dexed" thrice ...
E100#11#4 = E100#(E100#11)#3
googoldeciplexitridex
Googoldeciplex (E100#11) "dexed" thrice. Skipping ahead we have ...
E1#100#4
= 10^^10^^10^^10^^100
giggoltriplex
We are a long way from googoldeciplexitridex, but its only a short hop skip and a jump to ...
E100#100#4
grangoltridex
A grangoltridex = E100#grangoldudex. That is, a power tower of 10s grangoldudex terms high topped off with 100. What would Milton and Kasner say to that!
E1#102#4
= 10^^10^^10^^10^^102
An upperbound on a grangoltridex. It is equal to 10^^10^^10^^10^^102. This is trivially smaller than ...
E100#1#5 = E100#(10^100)#4
googolquadridex
A googol "dexed" four times ... yup ... we're just getting started on the next climb ...
E100#2#5 = E100#(10^10^100)#4
googolplexiquadridex
A googolplex "dexed" four times. At this point it's barely worth thinking about, but if you must know this would be the "fifth tower" beginning with 10^10^100, and each successive having as many 10s as the previous tower topped off with 100. This on the same level of Bowers' "a to the bth tower". The difference here is that with Hyper-E we can actually specify the starting value more precisely, thus we get much more precision ... not that we really "need" it. As you can see there are very few other numbers in this range.
E100#3#5
googolduplexiquadridex
A googolduplex "dexed" four times in a row ...
E100#4#5
googoltriplexiquadridex
A googoltriplex "dexed" four times in a row ...
E100#5#5
googolquadriplexiquadridex
A googolquadriplex "dexed" four times in a row ...
E100#6#5
googolquintiplexiquadridex
A googolquintiplex "dexed" four times in a row ...
E100#7#5
googolsextiplexiquadridex
A googolsextiplex "dexed" four times in a row ...
E100#8#5
googolseptiplexiquadridex
A googolseptiplex "dexed" four times in a row ...
E100#9#5
googoloctiplexiquadridex
A googoloctiplex "dexed" four times in a row ...
E100#10#5
googolnoniplexiquadridex
A googolnoniplex "dexed" four times in a row ...
E100#11#5
googoldeciplexiquadridex
A googoldeciplex "dexed" four times in a row ... let's skip on ahead to ...
E1#100#5
giggolquadriplex
There is some variation in this postfix. Bowers' tends to write it as "quadraplex", as in for example, googolquadraplex (see googolquadruplex, a long time ago at this point).
E100#100#5
grangolquadridex
grangolquadridex = E100#grangoltridex.
E1#102#5
An upperbound on a grangolquadridex.
E100#1#6
googolquintidex
A googol "dexed" five times ...
E100#2#6
googolplexiquintidex
A googolplex "dexed" five times ...
E100#3#6
googolduplexiquintidex
A googolduplex "dexed" five times ...
E100#4#6
googoltriplexiquintidex
A googoltriplex "dexed" five times ...
E100#5#6
googolquadriplexiquintidex
A googolquadriplex "dexed" five times ...
E100#6#6
googolquintiplexiquintidex
A googolquintiplex "dexed" fives times ...
E100#7#6
googolsextiplexiquintidex
A googolsextiplex "dexed" fives times ...
E100#8#6
googolseptiplexiquintidex
A googolseptiplex "dexed" fives times ...
E100#9#6
googoloctiplexiquintidex
A googoloctiplex "dexed" fives times ...
E100#10#6
googolnoniplexiquintidex
A googolnoniplex "dexed" fives times ...
E100#11#6
googoldeciplexiquintidex
A googoldeciplex "dexed" fives times ...
E1#100#6
10^^10^^10^^10^^10^^10^^100
giggolquintiplex
Bowers would call this number a "giggolquinplex".
E100#100#6
grangolquintidex
grangolquintidex = E100#grangolquadridex
E1#102#6
An upperbound on a grangolquintidex.
E100#1#7
googolsextidex
A googol "dexed" six times ...
E100#2#7
googolplexisextidex
A googolplex "dexed" six times ...
E100#3#7
googolduplexisextidex
A googolduplex "dexed" six times ...
E100#4#7
googoltriplexisextidex
A googoltriplex "dexed" six times ...
E100#5#7
googolquadriplexisextidex
A googolquadriplex "dexed" six times ...
E100#6#7
googolquintiplexisextidex
A googolquintiplex "dexed" six times ...
E100#7#7
googolsextiplexisextidex
A googolsextiplex "dexed" six times ...
E100#8#7
googolseptiplexisextidex
A googolseptiplex "dexed" six times ...
E100#9#7
googoloctiplexisextidex
A googoloctiplex "dexed" six times ...
E100#10#7
googolnoniplexisextidex
A googolnoniplex "dexed" six times ...
E100#11#7
googoldeciplexisextidex
A googoldeciplex "dexed" six times ...
E1#100#7
giggolsextiplex
Bowers doesn't have an official name for this number, but I've extended the latin much further than 5.
E100#100#7
grangolsextidex
grangolsextidex = E100#grangolquintidex = E100#(E100#100#6)
E1#102#7
A tight upperbound on a grangolsextidex.
E100#1#8
googolseptidex
A googol "dexed" seven times ...
E100#2#8
googolplexiseptidex
A googolplex "dexed" seven times ... at this point a googolplex is a tiny dot! Scratch that, even a grangoldex would be a tiny dot at this point!
E100#3#8
googolduplexiseptidex
A googolduplex "dexed" seven times ...
E100#4#8
googoltriplexiseptidex
A googoltriplex "dexed" seven times ...
E100#5#8
googolquadriplexiseptidex
A googolquadriplex "dexed" seven times ...
E100#6#8
googolquintiplexiseptidex
A googolquintiplex "dexed" seven times ...
E100#7#8
googolsextiplexiseptidex
A googolsextiplex "dexed" seven times ...
E100#8#8
googolseptiplexiseptidex
A googolseptiplex "dexed" seven times ...
E100#9#8
googoloctiplexiseptidex
A googoloctiplex "dexed" seven times ...
E100#10#8
googolnoniplexiseptidex
A googolnoniplex "dexed" seven times ...
E100#11#8
googoldeciplexiseptidex
A googoldeciplex "dexed" seven times ... or put another way ... a googol "plexed" ten times then "dexed" seven times. We've come along way from a googol, but we still are only getting started in googology ...
E1#100#8
giggolseptiplex
Bowers doesn't explicitly name this number, but it follows from the latin infixes.
E100#100#8
grangolseptidex
grangolseptidex is just below 3 pentated to the 10th. It's actually a pretty good lower bound.
E1#102#8
An upperbound on a grangolseptidex. To show this is less than 3^^^10 we note this is equal to 10^^10^^10^^10^^10^^10^^10^^10^^102. This is smaller than:
27^^27^^27^^27^^27^^27^^27^^27^^102
27 = 3^^2. Using the knuth arrow theorem this means its smaller than:
3^^(2+3^^(2+3^^(2+3^^(2+3^^(2+3^^(2+3^^(2+3^^104)))))))
By ascending each 2 to get a height of 1 greater we get this is all less than:
3^^3^^3^^3^^3^^3^^3^^3^^105
which is 3^1#105#8. 3^^3 = 7,625,597,484,987 >> 105. This means this last Hyper-E expression is less than 3^1#(3^^3)#8 = 3^1#1#10 = 3^^^10. Thus we have proven E1#102#8 is less than ...
3^^^10
Blatm's Pentational Number
The 14th competitor in the "My Number is Bigger" competition, and the 16th valid number. This number was entered by Blatm by reversing the order of the arguments of Gmalivuk's number. We are now well into pentational numbers. At this point elementary arithmetic expressions with exponents and factorials can no longer compete.
4^^^10
Xooll Shrug's
The 15th competitive entry in the "My Number is Bigger" competition, and the 17th valid entry. Xooll entered in response to Blatm's Pentational Number. Xooll typed *shrug* after it, as if to say, what's the big deal? But this response isn't terribly competitive and the real competition has only begun.
E1#1#10
10^^^10
dekataxis
This number is equal to 10^^10^^10^^10^^10^^10^^10^^10^^10^^10. To envision this number imagine Stage 1 as "10", Stage 2 as "10^10^10^10^10^10^10^10^10^10", Stage 3 as "10^10^ ... ^10^10" w/Stage 2 10s, ... and go all the way to Stage 10. This massive number is bigger than even the grangoldudex, but still smaller than the Megiston.
In Hyper-E this can be written as E1#1#10 which is also equivalent to E1#10#9.
(10^^^10)!
Ended's Salad Factorial
Technically this is the 16th competitive entry in the "My Number is Bigger" competition, and the 18th valid entry. It was entered by User Ended. Most of the strength of the number comes from pentation. At this point the numbers are so big that adding a factorial is so negligible that we can ignore it as a "salad factorial". To understand this, realize that at this scale N! ~ 10^N. Furthermore we have 10^^^10 = 10^^(10^^^9). So 10^^^10 is a power tower of 10s 10^^^9 terms high. 10^^^9 is an inconceivably vast number. From this we can gather that (10^^^10)! ~ 10^(10^^^10) = 10^^(1+10^^^9) ~ 10^^(10^^^9) = 10^^^10. So it has virtually no effect.
E10#9#9
This is a Hyper-E expression that is approximately (but not exactly) equivalent to E1#1#10. It's listed here to illustrate some principles of comparing Hyper-E Numbers. We can prove that this number is in fact larger than E1#1#10. To do so we first write E1#1#10 as E1#10#9. Next we expand both using the alternative recursive law:
E1#10#9 = E1#(E1#10)#8
E10#9#9 = E10#(E10#9)#8
We note here that E1#10 = E10#9 so this means the last two arguments of these numbers are identical, but the first argument is larger for E10#9#9 than E1#10#9, and so E10#9#9 is surprisingly larger. How much larger? In googological terms its extremely close and barely distinguishable. When expanded we have:
E1#(E1#10)#8 = E1#(E1#(E1#(E1#(E1#(E1#(E1#(E1#(E1#10))))))))
yet ...
E10#(E1#10)#8 = E10#(E10#(E10#(E10#(E10#(E10#(E10#(E10#(E1#10))))))))
Each of these 10s only has the effect of adding 1 to the next power towers height as opposed to using 1s. For example E10#(E1#10) = E1#(1+E1#10), which with a power tower this tall barely makes a difference. This then becomes the height of the next power tower which also gets topped off with one additional 10, and so on. The upshot is that these two numbers are incredibly close yet E10#9#9 is just a little bit larger. This still ends up becoming much more than a factorial. Why? Because that only increases the height of the last power tower by 1. In this case it would be equivalent to adding 1 at the first power tower, which will have a cascading effect, leading to a much larger end result. And yet even this is really nothing next to ...
10^^^11
E1#1#11 = E1#10#10
This number serves as a weak lower-bound for the Megiston.
E11#10#10
An even better lower-bound on a megiston using Hyper-E Notation. It is only "slightly" larger than 10^^^11 which is equivalent to E1#10#10, but it is actually closer to a Megiston while still being "slightly" smaller.
10[5]
Megiston
The Megiston is the lesser known of the two numbers Hugo Steinhaus defined with his circle notation. It is much much larger than a Mega, taking advantage of the full power of the circle operator, which is roughly on par with pentation. This number is much more difficult to bound than the Mega, due to various technical difficulties. It isn't too difficult however to show that it must lie somewhere between 10^^^11 and 10^^^12.
*Amusing side note: Bowers' incorrectly calls this number megaston on his infinity scrapers page.
E12#10#10
This is a tight upper-bound on a megiston we can express in Hyper-E Notation. This approximation is much much much closer to megiston than 10^^^12, which is equivalent to E1#10#11 = E1#(10^^10)#10. Written this way it's clear this is a much much tighter bound. Proving this bound is a little tricky but can be done using a "zipper" of +1s that can be zipped up to increase only the top exponent by 1.
10^^^12
E1#1#12 = E1#10#11
This number serves as a weak upper-bound for the megiston. It's actually a great deal weaker than 10^^^11. That is to say, this number is much much further away from a megiston compare to 10^^^11.
⟨10,100,3⟩
gaggol / hectataxis
= 10^^^100 = E1#1#100
This is Jonathan Bowers' gaggol, defined as 10^^^100, more commonly written in Array Notation as <10,100,3>. This makes it a very large pentational number and a very small hexational number. This number is larger than a megiston, but is "slightly" smaller than a greagol. In Hyper-E it can be expressed as E1#1#100. This number can also be called hectataxis in my naming scheme, indicating that you have to take the "taxarithm" 100 times to reduce this number to 1. That is, taxa^100(hectataxis) = 1. taxa(n) returns the "taxa" or "class" of the number, where taxa(1) = 0, and taxa(n) = taxa(log(n))+1. This yields the height of a power tower of tens that is passed as an argument. For example taxa(10^10^10) = 3, and in general taxa(10^^n) = n. This implies that taxa(10^^^n) = 10^^^(n-1) and taxa^100(10^^^100) = 10^^^0 = 1. At this point even the "taxa" of the numbers is far too large to comprehend. The taxa of hectataxis is 10^^^99. The order of magnitude of hectataxis can't even be described in our notations except to say it equals log(10^^^100) with is entirely circular as an explanation. What we can say however is that we have reached about the limit of usefulness for pentation. Soon we will need to start using hexation to go significantly further.
100^^^100
This number serves as a benchmark for the largest pentational number. Although this designation is arbitrary, a pentational number is usually understood as anything of the form a^^^b where a and b are relatively small arguments and the result is a number not already included in a smaller class of numbers. Hence if we let 100 be the limit of a "relatively small" argument, then 100^^^100 is the largest pentational number. Interestingly, this number is not that much larger than a gaggol or 10^^^100, relatively speaking. Next up ... the Ackermann Numbers ...
E3#100#99
This number serves as an upperbound on 100^^^100. Consider that 100^^n < E3#n. Therefore 100^^100 < E3#100, 100^^^3 = 100^^100^^100 < 100^^E3#100 < E3#(E3#100) = E3#100#2, 100^^^4 = 100^^100^^^3 < 100^^E3#100#2 < E3#(E3#100#2) = E3#100#3 ... and in general 100^^^n < E3#100#(n-1). Since E3#100#99 < E100#100#100 it follows that 100^^^100 < greagol. It can also be shown that 100^^^100 < E3#100#99 < 10^^^101. For a more detailed proof click here.
⟨10,101,3⟩ = 10^^^101 = E1#1#101
This serves as both an upper-bound on 100^^^100 and a lower bound on a greagol. The lower-bound is easier to demonstrate. Simply observe that 10^^^101 = E1#1#101 = E1#(E1#1#100) = E1#(E1#(E1#1#99)) = E1#(E1#1#99)#2 = E1#(E1#(E1#(E1#1#98))) = E1#(E1#1#98)#3 = ... = E1#(E1#1#1)#100 = E1#10#100 < E100#100#100 = greagol.
E100#100#100
greagol
A greagol, short for "great googol", is the 100th member of the grangol series. It is larger than and comparable to Jonathan Bowers' gaggol.
E1#102#100
This number is larger than a greagol, but less than 10^^^102. Hyper-E allows one to express more intermediate values than Knuth Up-arrow notation.
⟨10,102,3⟩ = 10^^^102 = E1#1#102
This is an upperbound on a greagol. In Hyper-E it can be written E1#1#102. This is one of the intervening steps in the proof that greagol << Folkman's Number.
Note that we can show that E1#102#100 < E1#1#102 easily as follows:
E1#102#100 < E1#(10^^10)#100 = E1#1#102.
16^^^102
This is another step in the proof greagol << Folkman's Number.
2^^^408
This is the final step in the proof greagol << Folkman's Number. By converting the base of the pentation to 2, it's made immediately apparent that this number must be less than Folkman's Number of 2^^^(2^901). Simply consider 2^^^408 < 2^^^512 = 2^^^(2^9) << 2^^^(2^901).
2^^^2^901
Folkman's Number
This moderately sized Ackermann class number was mentioned in the same article by Martin Gardner where he introduced the world to "Graham's Number" (See article here). Folkman was looking for a graph containing no K4s that forces a monochromatic K3 when it's two-colored. He devised an example of such a graph ... but it would contain 2^^^(2^901) points! This number is insanely large. Yet it's still smaller than G(1) of Graham's Number. Folkman's Number is somewhere between a greagol and G(1). Roughly speaking, the reason is because a greagol ~ 2^^^100 (actually larger) where as G(1) ~ 2^^^(3^^7,625,597,484,987) (actually larger). For a full proof click here.
[3^^^^3,E100##100)
Entries: 63
This epoch covers the remainder of numbers generated by Hyper-E Notation beyond those expressible with astronomically sized tetra-towers. This epoch contains my numbers: gigangol, gorgegol, gulgol, gaspgol, ginorgol, garantuul, and googondol, as well as Bowers' numbers: geegol, gigol, goggol, and gagol, as well as both versions tridecal. Jonathan Bowers' infinity scrapers also begin in this Epoch.
G(1)
3^^^^3
This is 3 hexated to the 3rd. Evaluating it we have:
3^^^^3 = 3^^^3^^^3 = 3^^^3^^3^^3 = 3^^^3^^3^3^3 =
3^^^3^^3^27 = 3^^^3^^7,625,597,484,987 =
3^^^3^3^3^3^ ... ^3^3^3^3 w/7,625,597,484,987 3s after 3^^^ =
3^^3^^3^^3^^3^^ ... ^^3^^3^^3^^3^^3
w/3^3^3^ ... ^3^3^3 3s
w/7,625,597,484,987 3s
To imagine this, let stage 1 = 3. Let stage 2 = 3^3^3 or 7,625,597,484,987, let stage 3 = 3^3^ ... ^3^3 w/7,625,597,484,987 3s, and in general each new stage is a power tower of 3s with the previous stage number of terms. 3 hexated to the 3rd is Stage 3^3^3^ ...^3^3^3 w/7,625,597,484,987 3s. This number is also G(1), the first member of graham's sequence (See G(64)).
((...((1,000,000^^^2)^^^2)...)^^^2)^^^2
w/((1,000,000^^^2)^^^2)^^^2-1 "^^^2"s
Earthling on Mars Number
This is the final form of Earthling on Mars's Number. This was his attempt to come up with a number larger than <10,10,googol> using power towers. This number however can be demonstrated to be in the hexational range, much much smaller than <10,10,googol>. Firstly we can observe that...
Earthling on Mars Number < 1,000,000^^^(2*1,000,000^^^6) < 10^^^(2+2*1,000,000^^^6)
< 10^^^(3*1,000,000^^^6) < 10^^^(3*10^^^8) < 10^^^(10*10^^^8) < 10^^^(10^^10^^^8)
= 10^^^10^^^9 < 10^^^10^^^10 = 10^^^^3 < 10^^^^10 = <10,10,4>
So Earthling on Mars Number is less than even <10,10,4>. In fact it's less than 10^^^10^^^10 making it smaller than a gaggolplex. On the other hand we have...
Earthling on Mars Number > 10^^^10^^10^^10^^10 = 10^^^10^^^4 > 3^^^3^^^3 = 3^^^^3.
So the Earthling on Mars Number is bigger than 3^^^^3 or G(1).
10^^^10^^^10
This number can be written as 10^^^^3 using Knuth Arrows, <10,3,4> using Array notation, and E1#1#1#3 or E1#1#10#2 using Hyper-E Notation. We are finally getting up to 4-argument E# expressions.
10^^^10^^^100
gaggolplex
This number is massively larger than G(1), yet at this stage it starts becoming more obscure why. The reason is because 3^^^3 (tritri) is vastly smaller than 10^^^100 (gaggol). Therefore 3^^^(3^^^3) << 10^^^10^^^100. This number can be written as E1#1#100#2 using Hyper-E.
E100#100#100#2
greagolthrex
This is the first number to introduce the -threx operator. "three" + "plex" = "threx". For a number of the form n = Ea#b#c, threx-n = Ea#b#(Ea#b#c).
Therefore a greagolthrex = E100#greagol = E100#(E100#100#100) = E100#100#100#2.
E1#1#102#2
An important upperbound on a greagolthrex. Recall that E100#100#n < E1#102#n. From here we prove E100#100#100#2 < E1#1#102#2.
E100#100#100#2 = E100#(E100#100#100) < E1#(2+E100#100#100)
< E1#(E101#100#100) < E1#(E1#102#100) < E1#(E1#1#102) = E1#1#102#2.
10^^^10^^^10^^^100
gaggolduplex
Another Bowerism on the Infinity scrapers' page. In E# this can be written as E1#1#100#3 making it definitely smaller than E100#100#100#3 (though googologically in the same neighborhood basically). It is still vastly larger than a greagolthrex. Just observe:
E100#100#100#2 = E100#100#greagol < E100#(E100#1#1)#greagol = E100#1#(1+greagol)
E100#(E100#1#greagol) < E(10^10)#(E100#1#greagol) = E1#(2+E100#1#greagol)
= E1#(2+E100#(E100#(E100#(E100#(E100#(... E100#(E100#1)...))))))))
E1#(E1#(E1#(E1#(E1#(E1#(E1#(E1#(E1#( ... E1#(E101#1) ... ))))))))))
E101#1 < E(10^10)#1 = E1#3
so we get an upperbound of:
E1#3#(1+greagol)
which is less than E1#(E1#1#1)#(1+greagol) = E1#1#(2+greagol) = E1#1#(2+E100#100#100)
we can again ascend the 2 up through the various power towers eventually reaching the innermost level...
E101#100 which is still less than E(10^10)#100 = E1#102 so we get...
E1#1#(E1#102#100) and since 102 < 10^^10 = 10^^^2 = E1#1#2 we have:
E1#1#(E1#(E1#1#2)#100) = E1#1#(E1#1#102)
which is E1#1#102#2 which is less than E1#1#100#3 since this is equal to E1#1#gaggol#2.
E100#100#100#3
greagolduthrex
greagol "threxed" twice. What the heck does this mean?! Well it means starting out with 100 power towers, then having that many power towers, then having THAT many power towers. We can't really wrap our heads around this. And yet ... we keep going ...
E1#102#100#3
An upperbound on a greagolduthrex.
E1#1#102#3
10^^^10^^^10^^^102
An even larger upperbound on a greagolduthrex.
E1#1#1#5
10^^^^5 = <10,5,4>
An even weaker upperbound on a greagolduthrex. This one is a simple upperbound in up-arrow notation, which brings us to ...
⟨10,100,4⟩
geegol
= 10^^^^100 = E1#1#1#100
A Bowerism based an extension of the googol. This is the forth in the sequence: googol, giggol, gaggol, geegol. Unlike some of Bowers' other numbers, this one did not shift between Bowers' older array notation and newer array notation. Bowers simply decreased the third argument accordingly. On his original site it was denoted as {10,100,6} (10 hexated to the 100th). On the new site it's <10,100,4>. I'll be using the angle brackets to denote Bowers new array notation, and curly braces to denote the old notation. In Hyper-E geegol can be expressed exactly as E1#1#1#100. This makes it "slightly" smaller than ...
⟨10,101,4⟩ = 10^^^^101 = E1#1#1#101 = E1#1#10#100
This one might require a little explanation. Note that using the alternative expansion:
E1#1#1#101 = E1#1#(E1#1#1)#100 = E1#1#(E1)E100 = E1#1#10#100. This is analogous to the way E1#101 is larger than a giggol and smaller than a grangol, and E1#1#101 is larger than a gaggol yet smaller than a greagol. The proofs in these cases are the same: E1#101 = E10#100 < E100#100 , E1#1#101 = E1#10#100 < E100#100#100.
E100#100#100#100
gigangol
A gigangol, short for "gigantic googol", is the 100th member of the greagol series.
⟨10,102,4⟩ = 10^^^^102 = E1#1#1#102
A modest upperbound on a gigangol. This follows from E100#100#100#n < E1#1#102#n. So E100#100#100#100 < E1#1#102#100 < E1#1#(10^^^10)#100 = E1#1#1#102.
⟨10,⟨10,100,4⟩,4⟩
E1#1#1#100#2
geegolplex
The plex version of the geegol. It's worth noting that after the gaggolduplex, Bowers' never has more than one plex version for any number. At this point he races through the next couple of generations of googol with two items each.
E100#100#100#100#2
gigangoltetrex
The "tetrex" version of a gigangol. tetrex is the gigangol, as threx is to greagol, and dex is to grangol. These special operators allow the next level of recursion.
E100#100#100#100#3
gigangoldutetrex
I like to have at least two additional version after introducing a new number name. This brings us to the next level ...
{7,7,7} = ⟨7,7,5⟩
Trisept Jr.
The original "trisept" or {7,7,7}. It's now written as <7,7,5> in Bowers' new array notation. This value can be approximately fairly accurately in Hyper-E as E695,974#6#6#6#6. This places it above gigangoldutetrex but still below a gygol.
⟨10,100,5⟩
E1#1#1#1#100
gigol
Originally called a gygol, now 'gigol', this is the fifth level of the sequence: googol, giggol, gaggol, geegol, gigol. In the original array notation this was written as {10,100,7}. In the new notation it equals exactly <10,100,5>. At this point with 5 up-arrows this is inconvenient to write in up-arrow notation, so Bowers array notation becomes more useful.
E100#100#100#100#100
gorgegol
The gorgegol, short for "the gorged googol", is the 100th member of the gigangol sequence.
⟨10,⟨10,100,5⟩,5⟩
E1#1#1#1#100#2
gigolplex
The plex version of a gigol. Interestingly, although Bowers frequently uses this type of recursion, it doesn't come very naturally to array notation. He typically uses a name in place of another array expression describing gigolplex as {10,gigol,7} or <10,gigol,5>. However I prefer to avoid the potential ambiguity given that some names have different meanings depending on which version of array notation we are talking about. In any case Hyper-E allows us to easily apply additional levels of recursion by simply appending another argument at the end.
E100#100#100#100#100#2
gorgegolpentex
gorgegol+pentex.
E100#100#100#100#100#3
gorgegoldupentex
gorgegol+pentex+pentex.
⟨10,100,6⟩
goggol
Bowers sixth googol variant. A goggol is comparable to and smaller than a gulgol ...
E100#100#100#100#100#100
gulgol
The gulgol, short for the *gulp* googol, is the 100th member of the gorgegol series.
⟨10,⟨10,100,6⟩,6⟩
goggolplex
The obligatory "plex" version of a goggol.
E100#100#100#100#100#100#2
gulgolhex
hexa + plex = "hex". gugol may be appended with an arbitrary number of "hexes".
E100#100#100#100#100#100#3
gulgolduhex
gugol + hex + hex = gulgol(2)hex = gulgolduhex. On to the next group/regiment ...
⟨10,100,7⟩
gagol
Not to be confused with a "gaggol". The seventh and last of the official Bowers googol variants. A gagol is comparable to a gaspgol ...
E100#100#100#100#100#100#100
E100##7
gaspgol
The gaspgol, short for "gasp googol", is the 100th member of the gulgol series.
⟨10,⟨10,100,7⟩,7⟩
gagolplex
The plex version of a gagol. This concludes Bowers most basic googolisms ...
E100#100#100#100#100#100#100#2
gaspgolheptex
"hepta" + "plex" = "heptex". We can append as many heptexes to gaspgol as needed/wanted.
E100#100#100#100#100#100#100#3
gaspgolduheptex
gaspgol + heptex + heptex = gaspgol(2)heptex = gaspgolduheptex.
E1#1#1#1#1#1#1#5
⟨10,5,8⟩
An upperbound on a gaspgolduheptex. Using a "zipper"-trick we can get rid of all the 100s and show it's less than this value ... which leads us to ...
{10,10,10} = ⟨10,10,8⟩
Tridecal Jr.
Bowers' original value for the "tridecal". In the original array notation this would be {10,10,10}, or three-tens, hence the name "tri"(3) + "decal"(10). However when Bowers changed his notation this old value shifted to <10,10,8>. The new value is <10,10,10>. Tridecal Jr. is just a little smaller than a ginorgol, and would also be smaller than a hypothetical 8th Bowers' googol variant ...
⟨10,100,8⟩
E1#1#1#1#1#1#1#100
This would be the value for a hypothetical 8th Bowers' googol variant. It would be comparable and slightly smaller than a ginorgol ...
E100#100#100#100#100#100#100#100
ginorgol
The ginorgol, short for "ginormous googol", is the 100th member of the gaspgol series.
E100#100#100#100#100#100#100#100#2
ginorgoloctex
"octa" + "plex" = "octex". We may append octexes to ginorgols, forming the ginorgol sequence ...
E100#100#100#100#100#100#100#100#3
ginorgolduoctex
ginorgol-octex-octex.
⟨10,100,9⟩
A hypothetical 9th Bowers' googol variant to match with ...
E100##9
gargantuul
The ninth of my elementary googol extensions. It's the 100th member of the ginorgol sequence. A gargantuul comes from "gargantuan" + "googol". Approximately equal to 10<9>102. We are getting close to Bowers' new tridecal ...
E100#100#100#100#100#100#100#100#100#2
gargantuulennex
"enna" + "plex" = "ennex". Repeated applications of ennex to gargantuul produces the gargantuul sequence. Roughly equal to 10<9>10<9>102 which is less than 10<9>10<9>10<9>10 = 10<10>4 = <10,4,10>. Thus this number is just shy of a tridecal ...
⟨10,10,10⟩
Tridecal
This number can also be written as <10,10,10> using linear array notation. This googolism was coined by Jonathan Bowers. This Number is also Bowers' benchmark for an "Infinity Scraper" ... so ... I guess we are only just getting started ... STILL! ...
(10^^^^^^^^^^10)! * (10^^^^^^^^^10)!^^^^^^(10^^^^^^10)!
Twasbrillig's Up-arrow Salad
This is the 17th record setting entry in the "My Number is Bigger" competition. This number is a big jump from the 16th record setter. However it's a salad number. The factorials and multiplication add little to the number and it ends up being not much larger than a Tridecal. This is also the 19th number listed in the forum.
(11^^^^^^^^^^11)! * (11^^^^^^^^^11)!^^^^^^(11^^^^^^11)!
Blatm's Finesse
This is the 18th record setting entry in the "My Number is Bigger" competition, and the 20th official entry. This was Blatm's response to Twasbrillig's Up-arrow Salad. We can see that people recognize intuitively that salad numbers are sloppy because Blatm responded by saying "Not one for Finesse, are you?". Blatm simply took Twasbrillig's number and converted to base 11. This makes for a (slightly) larger value, though at this point this likely to be clobbered by whoever is willing to type out more up-arrows, such as a screen fill. As it turns out, this is the last number in the Primitive Recursive Epoch, and the next entry is much much larger!
⟨10,100,10⟩
This would almost certainly be larger than the last two salad numbers. This would be the hypothetical tenth Bowers' elementary googol extension ... which brings us to ...
E100##10
googondol
My tenth and final elementary googol extension.
E100#100#100#100#100#100#100#100#100#100#2
googondoldecex
"deca" + "plex" = "decex". This concludes the base operators: plex, dex, threx, tetrex, pentex, hex, heptex, octex, ennex, and decex.
g(1)
2^^^^^^^^^^^^3 = ⟨2,3,12⟩
This is the first value in the sequence used to construct Little Graham, the original Graham's Number used in RL Graham's 1971 paper "Ramsey Theory for n-parameter sets".
E100##20
googonkosol
After a googondol I did continue some further extensions, this time jumping by 10s. This name is derived from "googondol" + "icosi"(greek 20). It can be expanded (using Extended Hyper-E rules) to E100#100# ... #100 w/20 100s. This falls in a bit of a deadspot for the numbers, because we are transitioning into higher levels of recursion at this point.
3-->3-->2-->2
3^^^^^^^^^^^^^^^^^^^^^^^^^^^3
This is one of the smallest non-trivial cases of Conway's Chain Arrow Notation. Three argument chains are trivial as they are just an alternative to Knuth Arrow notation. 4 element chains are also trivial if any of it's elements are 1 (assuming the other arguments are no larger than say 10). Expanding this expression we get:
3-->3-->2-->2
3-->3-->(3-->3-->1-->2)-->1
3-->3-->(3-->3-->1)
3-->3-->(3^3)
3-->3-->27
That is, we get 3 followed by 27 Up-arrows followed by 3. This places this number somewhere between a tridecal (10-->10-->10) and a boogol (10-->10-->100). It also lies between G(1) and G(2). In Part II we will begin to go through Conway's Chain Arrow Notation in more depth.
E100##30
googontritol
googondol + "trito"(3) for the "30th googol".
E100##40
googonsartol
The 40th googol.
E100##50
googonpetol
The 50th googol.
E100##60
googonextol
The 60th googol.
E100##70
googonheptol
The 70th googol.
E100##80
googonogdol
The 80th googol.
E100##90
googonentol
The 90th googol.
{10,10,100} = ⟨10,10,98⟩
Boogol Jr.
The original boogol, a googolism coined by Jonathan Bowers. A boogol is 10 hecated to the 10th, or 10{100}10 using Bowers old array notation. However in 2007 Bowers changed the definition of his notation so that <b,p> = b^p instead of the former {b,p} = b+p. In the new notation therefore, {10,10,100} is actually <10,10,98>. That is, two tens between 98 up-arrows. At this scale however this barely seems significantly different than a boogol, and in fact the newer boogol will be the very next entry ...
{10,10,102} = ⟨10,10,100⟩ = 10⟨100⟩10
boogol
The post-2007 definition of a boogol, where it is 10^^^...^^^10 w/100 ^s, which honestly ... is pretty cool. This is only "slightly" larger than its original definition which involved being 10 to the 10 under the 100th operation. This boogol is technically 10 (102)-ated to the 10th, which doesn't sound as spiffy. The main reason why the new array notation is preferable, has nothing to do with the numbers being "larger". The "larger" here is so slight in googological terms its barely worth mentioning past this point. Rather it's because of consistency. With <b,p> = b^p we consistently have that <b,1...> = b regardless of whatever follows, since b^1 = b. Notice this is true of all the hyper-operators: b^^1 = b , b^^^1 = b , b^^^^1 = b , etc. With Bowers new array notation this becomes universal and brings great harmony to the whole.
In Hyper-E Notation this boogol can be written as:
E1#1#1# ... #1#1#1#10
w/99 "1#"s
This makes it just ever so slightly less than a gugold. Which brings us to ...
⟨10,100,100⟩ = 10⟨100⟩100
This would be the hypothetical 100th Bowers' googol extension. It could be written as:
E1#1#1# ... #1#1#1#100
w/99 "1#"s
This gets us blisteringly close to a gugold and yet ...
⟨10,101,100⟩ = 10⟨100⟩101
E1#1#...#1#1#10#100 w/98 "1#"s
This number is larger than a boogol but still smaller than a gugold! We can prove this by noting that E1#1#...#1#1#101 = E1#1#...#1#10#100 < E1#1#...#1#100#100 < E100#100#...#100#100#100. With that we have exhausted trientrical arrays ...
XIII. Tetrentrical Epoch
[E100##100,E100###100)
Entries: 143
In 1996 John Conway introduced the world to Chain Arrows in "The Book of Numbers". This notation picks up where Knuth's Up-Arrow notation leaves off, and ends at around an order-type of ω^2. This notation allows us to easily pass up Graham's Number, and yet is almost as simple as up-arrow notation. This is also the range of "tetrentrical arrays", which are 4 entry arrays in BEAF. In Extensible-E, these are ExE expressions with only two delimiters: # and ##, where there is at least one ## delimiter.
E100##100
gugold
This number is roughly equivalent to 10^^^...^^^100 w/100 ^s = 10<100>100 = <10,100,100>. In actuality it's larger than this. Since Jonathan Bower's boogol = 10<100>10 < 10<100>100 < E100##100, it follows that boogol < gugold.
A gugold is my smallest googolism to use Extended Hyper-E Notation. It can be written in ordinary Hyper-E as:
E100#100#100#100# ... ... ... ... #100#100#100#100#100 w/100 100s
⟨10,102,100⟩ = 10⟨100⟩102
This number will be ever so slightly larger than a gugold, and so acts as a strong upperbound in Array Notation. This number is still vastly smaller than ...
E100##1#2
great googol
The great googol is E100#100#100#100# ... #100#100#100 with a googol 100s. This can be written compactly as E100##1#2 by utilizing the power of the deutero-hyperion. This number is much larger than a gugold but much smaller than a Moser. It can be approximated as 10^^^...^^^100 in which there are a googol up-arrows.
2[2[5]]
Moser
This number is 2 inside a "mega-gon" (See mega). This number was popularly regarded as "the largest number", until Graham's Number came along. A Moser is less than G(2), but larger than G(1). It's approximately:
2<10^^257>3
G(2)
3<3<4>3>3
The Second stage of the modern Graham's Number.
g(2)
2<2<12>3>3
The second stage of the original Graham's Number.
10{10{100}10}10
boogolplex Jr.
The original value of a boogolplex. This is equivalent to 10<(10<98>10)-2>10 using the new array notation. This would be slightly smaller than the modern definition.
10⟨10⟨100⟩10⟩10
boogolplex
The plex version of a boogol. This value will be slightly smaller than a gugolda-suplex.
E100##100#2
gugolda-suplex
"super" + "plex" = "suplex". This is the first operator beyond the "first row" of operators: plex, dex, threx, tetrex, pentex, hex, heptex, octex, ennex, decex.
10⟨10⟨100⟩102⟩102
This odd looking number is rather exacting upperbound on a gugolda-suplex. It's very important for all further upperbounds of E# and xE# numbers in array notation, so it bares some explication. We may note that E100#100#...#100#100 < E1#1#...#1#102, regardless of how many 100s there are. We saw some earlier examples of this. It follows that E100##100 < E1#1#...#1#102 w/99 1s. It follows then that E100##(10<100>102) is larger than E100##gugold. We then replace all these 100s with 1 except for the last one which gets replaced with 102, and this can be written sucintly in array notation as 10<10<100>102>102. From here we note that we can turn this into a standard expanded expression by replacing all the numbers with 102. Thus we get 102<102<102>102>102. Which brings us to ...
⟨102,3,1,2⟩
From here we can quickly surmise that:
E100##100#n < <102,n+1,1,2>
From here we can create more upperbounds, literally accounting for all the additional boosts from all of those 100s.
G(3)
3<3<3^^^^3>3>3
The 3rd stage of the modern Graham's Number.
g(3)
2<2<2<12>3>3>3
The 3rd stage of the original Graham's Number.
E100##100#3
gugolda-dusuplex
gugolda-suplex-suplex. Put another way it's E100##gugolda-suplex.
⟨102,4,1,2⟩
A relatively strong upperbound on gugolda-dusuplex in array notation.
G(4)
3<3<3<3<4>3>3>3>3
The 4th Stage of the modern Graham's Number.
g(4)
2<2<2<2<12>3>3>3>3
The 4th Stage of the original Graham's Number.
g(7)
Little Graham
This is the original Graham's Number, which first appeared in RL Graham's 1971 paper "Ramsey Theory for n-parameter sets". This is currently the smallest known upper-bound for the problem in Ramsey theory for which "Graham's Number" as we know it today, was supposedly invented. It is significantly smaller than Graham's Number, but still much much bigger than a Moser. Little Graham can be defined most compactly as follows:
Little Graham = g(7)
where g(1) = 2^^^...^^^3 w/12 ^s
and g(n) = 2^^^...^^^3 w/g(n-1) ^s whenever n>1
Using Jonathan Bowers' hyper-operator notation we have:
Little Graham = 2<2<2<2<2<2<2<12>3>3>3>3>3>3>3 < <3,9,1,2> << <10,100,1,2>
G(8)
This is the smallest member of the Graham Series, G(n), greater than Little Graham. Consequently:
Little Graham = g(7) < G(8) << G(64) = Graham's Number
:: Little Graham << Graham's Number
G(64)
Graham's Number
Graham's Number, sometimes abbreviated as G, was first coined in 1977 when Ronald Graham used it as an upper bound to a problem in Ramsey theory. At the time it was the largest number anyone had seen arising from a serious mathematical proof. It quickly gained recognition in the Guinness Book of World Records for being the largest number in serious mathematics. But what is Graham's Number,...(READ MORE).
A(G64,G64)
xkcd Number
A well known joke by xkcd, shows mathematicians getting their minds blown when someone smashes Graham's Number and the Ackermann function together. I'm not sure how serious this was, but this joke didn't age very well. This is the sort of thing the average person thinks is "impressive" but a mathematician would probably just shrug and a googologist would face-palm. This is a little less than G(65). The A function is only equivalent to adding 1 to the G function. That's just how fast it grows. G(A(64,64)) would be much much larger, and is just as easy to come up with. Still, it's a well known number in googology lore, and it illustrates googology principles of why we don't smash different functions together, not because it "blows up" and makes "numbers too big for us to handle", that's silly. You WANT to make numbers as big as possible, but because it's actually really really weak and easy and unimpressive ... from a googology point of view. 'nuff said.
G(65)
G of sixty-five
A common retort to Graham's Number.
{10,100,1,2}
corporal jr.
This is the original Corporal (note the curly braces). This number can be written as:
10{10{ ... 10{10{10}10}10 ... }10}10
w/99 sets of { }
Insane. This number can be bounded in xE# quite easily. First we note that 10{10}10 < 10<10>10 = E1#1#1#1#1#1#1#1#1#10 < E1#1#1#1#1#1#1#1#1#100 < E100##10.
Next we have 10<10<10>10>10. It follows quickly this is less than E100##(E100##10) = E100##10#2. From here we can bound corporal ...
⟨10,100,1,2⟩
corporal
This is the modern corporal. It would be slightly larger than the "classic" version of corporal. Both versions would be smaller than ...
E100##10#99
A nifty upperbound for corporal. What's useful about this is it immediately follows that E100##100#100 is larger. Even crazier ... even this is still smaller than G(100) ...
G(100)
G of one hundred
Nice try. This is still technically smaller than a "graatagold", since a graatagold is the 100th member of the gugold series, and G(1) < gugold, G(2) < gugolda-suplex, G(3) < gugolda-dusuplex etc. Therefore: G(100) < graatagold. We can also show this is approximately equal to 3^3##4#100, since this is equal to
3^3##(3^3##4)#99 and 3^3##4 is much much larger than 10, it follows that this is larger than the upperbound just proposed for a corporal ... which brings us to ...
E100##100#100
graatagold
This is the smallest of my googolism's to exceed Graham's Number. It is comparable to Jonathan Bowers'Corporal = <10,100,1,2>, and no, it doesn't matter which version.
<102,101,1,2>
A relatively tight upper-bound on a graatagold.
g(10^10^102,1,1,2,100,10)
Joyce's googolplux
Joyce's mysterious googolplux. This number is defined using Joyce's 6-argument g-function and it's much larger than Graham's Number. It is approximately G(10^10^102).
G(3^^^^3)
Yudkowsky's Number
A slightly better retort to Graham's Number (See G(65)). I call this number Yudkowsky's Number. Elizier Yudkowsky mentioned it in his article "Staring at the singularity", to make the point that Graham's Number is probably not even the largest number at this particular complexity level.
Suffice it to say that this is rather underwhelming from a googologist's point of view. It's less than a single recursive step from Graham's Number.
Another form of this number is G(G(1)), which begs the question ... why stop there!
g(g(8,10^100,10^100),1,1,2,100,10)
Joyce's googolpluc
Here's a number to make a "singularities head spin", Joyce's googolpluc. It is approximately G(googol^^^^^^^googol).
g(g(7625597484986,10^100,10^100),1,1,2,100,10)
Joyce's googolplum
Joyce's googolplum is approximately G(googol^^^...^^^googol) w/7,625,597,484,985 ^s. Even so this number is still vastly smaller than the next entry ...
G(G(26))
A lower bound on Conway's tetratri in terms of Graham's function.
3-->3-->3-->3
Conway's tetratri
A number mentioned by Jonathan Bower's as the largest number in the professional literature, and referred to as Conway's three-three-three-three. This number first appeared in Conway's Book of Numbers (1996) as an example of a number larger than Graham's Number. It has since been surpassed by even larger numbers in the professional mathematical literature.
This is also the 19th record setting entry in the "My Number is Bigger" competition, and the 21st entry. It was entered by Gmalivuk after giving up on a modest response on the previous record holder and he just decided to jump to the introduction of Conway Chain-Arrows. From this point onwards, up-arrow notation will no longer be competitive! This is also the record holder by the end of the first page of the forum ... of which there are 35 pages!!!
G(G(27))
An upper bound on Conway's tetratri in terms of Graham's function.
G(G)
G of G
A slightly less naive response to Graham's Number (see G(3^^^^3)). This number can also be written G(G(64)). This number however is still less than G(G(100)), which is still less than a graatagolda-sudex!
{10,{10,100,1,2},1,2}
corporalplex
The og corporalplex ... still smaller than G(G(100)) which is still smaller than a graatagolda-sudex ...
G(G(100))
We're way past Yudkowsky's Number but we still are just getting started ...
E100##100#100#2
graatagolda-sudex
"super" + "dex" = "sudex". The continuation from here should be clear.
<102,3,2,2>
A tight upperbound on graatagolda-sudex. Using the previous result we can have a tigher upperbound of <102,<102,101,1,2>+1,1,2> < <102,<102,102,1,2>,1,2> = <102,3,2,2>.
⟨10,100,2,2⟩
mulporal
This entry is not a standard Bowerism. This would be the "array equivalent" of E100##100#100#100 (greegold). CookieFonster (Cedric Fausey), gave this one the name "mulporal" based on the operation ⟨⟨2⟩⟩, which Bowers' calls multi-expansion. A mulporal is 10 multi-expanded to 100. It's interesting that Bowers' skips numbers of this form because they are a fairly natural continuation. The next major Bowerism is a grand tridecal which is quite a way aways yet. Note that the angle-brackets indicate that this is the modern array notation where ⟨b,p⟩ = b^p.
E100##100#100#100
greegold
The next regiment after graatagold.
4-->4-->4-->4
Conway's tetratet
This is the largest explicitly defined finite number in Conway's Book of Numbers. It's also the 4th Conway Number.
⟨10,100,3,2⟩
powporal
Bowers' calls the ⟨⟨3⟩⟩ operator power-expansion. A powporal is 10 power-expanded to the 100th. The term was coined by Cedric Fausey as an expansion of Bowers' googolisms. The powporal is smaller than and comparable to a grinningold.
E100##100#100#100#100
grinningold
The next regiment after greegold.
⟨10,100,4,2⟩
terporal
Bowers' calls the ⟨⟨4⟩⟩ operator expando-tetration. A terporal is 10 expando-tetrated to the 100th. The term terporal was coined by Cedric Fausey. This number is smaller than and comparable to golaagold. A golaagold is E100##100##5. You may notice the shift by 1. This is because of the fact that BEAF skips the limit ordinals. Meaning, at <b,p,1,2> it has a grow rate of w+1, <b,p,2,2> has a growth rate of w+2, etc. Since Extensible-E does not do this there is an offset of 1.
E100##100##5
golaagold
The next regiment after grinningold.
⟨10,100,5,2⟩
pepporal
Bowers' calls the ⟨⟨5⟩⟩ operator expando-pentation. A terporal is 10 expando-pentated to the 100th. The term pepporal was coined by Cedric Fausey. This number is smaller than and comparable to gruelohgold.
E100##100##6
gruelohgold
The next regiment after golaagold.
⟨10,100,6,2⟩
hexporal
Bowers' calls the ⟨⟨6⟩⟩ operator expando-hexation. A terporal is 10 expando-hexated to the 100th. The term hexporal was coined by Cedric Fausey. This number is smaller than and comparable to gaspgold.
E100##100##7
gaspgold
The next regiment after gruelohgold. This number is approximately equal to <10,100,6,2>.
⟨10,100,7,2⟩
hepporal
Bowers' calls the ⟨⟨7⟩⟩ operator expando-heptation. A hepporal is 10 expando-heptated to the 100th. The term was coined by Cedric Fausey. A hepporal is smaller than and approximately equal to E100##100##8 in ExE.
E100##100##8
ginorgold
The next regiment after gaspgold. Ginorgold is greater than and approximately equal to ⟨10,100,7,2⟩ in BEAF.
{10,10,10,2}
grand tridecal
A Grand Tridecal smaller than E100##100##11, which definitely makes it smaller than a gugolthra. Note that from a corporal to a grand tridecal Bowers skips over all of the exploding operators, passing up a whole row of regiments in my system. This starts to happen more and more as we proceed. Next up is the third row of regiments, beginning with the gugolthra regiment.
⟨10,100,99,2⟩
This is an accurate lower-bound of gugolthra in Bowers' Linear Array Notation based on a theorem for converting Linear Array expressions to an upperbounding ExE expression.
E100##100##100
gugolthra
Gugolthra marks the beginning of a third row of regiments ...
⟨10,103,99,2⟩
This is an accurate upper-bound of gugolthra in Bowers' Linear Arrays based on a conjecture about such upperbounds. Namely that if an ExE expression corresponds with <10,p,k,...> then it is smaller than <10,p+3,k,...>.
⟨10,3,100,2⟩
This is an upperbound on the previous entry. Expanding it we obtain <10,3,100,2> = <10,<10,2,100,2>,99,2> = <10,<10,<10,1,100,2>,99,2>,99,2> = <10,<10,10,99,2>,99,2>. <10,10,99,2> is much much larger than 103. This value itself is much smaller than ...
⟨10,10,100,2⟩
biggol
The name biggol was actually first suggested by Chris Bird, but later adopted by Bowers in his naming scheme. The name is based on analogy with boogol is to googol as biggol is to giggol. Since googol = <10,100> , boogol = <10,10,100>, and giggol = <10,100,2>, the pattern suggests that biggol should be <10,10,100,2>. This provides some in-between values that Bowers' skipped over in his original naming scheme. boogol, biggol, begin a kind of "second row" of googol extensions in Bowers' system. In mine however we have already skipped through 2 whole rows of regiments and it will be a whole 'nother row before we even get to the next major milestone: baggol. A biggol is analogous to my gugolthra, but is in fact "slightly" larger, due to certain offsets. A gugolthra, as you saw is actually closer to <10,100,99,2>. This offset is actually due to the fact that Bowers arrays skip over an order-type. That is <10,n,1,2> as a function of n is not of order-type w, but rather of order type w+1. Thus <10,10,n,2> is a function of the same order-type as E100##100##n, but it lags behind by a difference of 1. This however, makes the functions of equivalent strength according to my usual definition of "strength". As big as this value is it pales in comparison to the next major BEAF expression ...
⟨10,100,1,3⟩
corplodal
Here is a cool number. It's like corporal, but it involves Bowers' "explosion" operation instead of expansion. A corplodal is 10 exploded to the 100th! It can also be written as 10<<<1>>>100. A corplodal is smaller than and comparable to a graatagolthra.
E100##100##100##2
graatagolthra
2nd regiment of the 3rd row ...
⟨10,100,2,3⟩
mulplodal
A mulplodal is 10 multi-exploded to the 100th. The term multi-exploded was coined by Jonathan Bowers. The term mulplodal was coined by Cedric Fausey. A mulplodal is smaller than and comparable to a greegolthra.
E100##100##100##3
greegolthra
This number is especially notable as being the last regiment just before Jonathan Bowers' tetratri. An explanation can be found in the following entries to tetratri ...
⟨102,101,2,3⟩
A very strong upper-bound on a greegolthra. From here we apply an extended version of the Knuth Arrow Theorem, The Generalized Knuth Arrow Theorem, to obtain an upperbound in base 3:
<102,101,2,3> = 102<2,3>101 < (3<2,3>2)<2,3>101 < 3<2,3>103 = <3,103,2,3>
From here we can observe:
3<2,3>103 < 3<2,3>3<2,3>3 = 3<3,3>3 = <3,3,3,3>
And thus, we have proven that a greegolthra is smaller than a tetratri.
{3,3,3,3}
tetratri
Bowers' tetratri. The etymology is simple "tetra"(4) + "tri"(3), meaning four 3s in array notation. This number is much much larger than Conway's tetratri, and is larger than a gugolthra.
⟨10,100,3,3⟩
powplodal
This number is trivially smaller than a tetratri since we've replaced the first two 3s with a 10 and 100 respectively. This however is a lower bound for a grinningolthra. CookieFonster coined the name powplodal for this number, based on the operator ⟨⟨⟨3⟩⟩⟩which is called powerexplosion. This is actually a type of number that Bowers' himself skipped over, even though it is arguably a more natural continuation of his googol extensions. A powplodal can be written as 10⟨⟨⟨3⟩⟩⟩100 which follows the general pattern 10@100 for 10 to the 100th under the @th operation.
E100##100##100##4
grinningolthra
The first regiment larger than Bowers' tetratri. This is proven by the fact that grinningolthra is larger than a powplodal which is trivially larger than a tetratri.
⟨10,100,4,3⟩
terplodal
A terplodal is 10 explodo-tetrated to the 100th. The term explodo-tetrated was coined by Jonathan Bowers. The term terplodal was coined by Cedric Fausey. A terplodal is smaller than and comparable to a golaagolthra.
E100##100##100##5
golaagolthra
The 5th regiment of the 3rd row. Approximately equal to <10,100,4,3>.
⟨10,100,5,3⟩
pepplodal
A pepplodal is 10 explodo-pentated to the 100th. The term explodo-pentated was coined by Jonathan Bowers. The term pepplodal was coined by Cedric Fausey. A pepplodal is smaller than and comparable to a gruelohgolthra.
E100##100##100##6
gruelohgolthra
The 6th regiment of the 3rd row. Approximately equal to <10,100,5,3>.
⟨10,100,6,3⟩
hexplodal
A hexplodal is 10 explodo-hexated to the 100th. The term explodo-hexated was coined by Jonathan Bowers. The term hexplodal was coined by Cedric Fausey. A hexplodal is smaller than and comparable to a gaspgolthra.
E100##100##100##7
gaspgolthra
The 7th regiment of the 3rd row. Approximately equal to <10,100,6,3>.
⟨10,100,7,3⟩
hepplodal
A hepplodal is 10 explodo-heptated to the 100th. The term explodo-heptated was coined by Jonathan Bowers. The term hepplodal was coined by Cedric Fausey. A hepplodal is smaller than and comparable to a ginorgolthra.
E100##100##100##8
ginorgolthra
The 8th regiment of the 3rd row. Approximately equal to <10,100,7,3>.
⟨10,100,99,3⟩
A strong lower-bound on a gugoltesla based on Linear Array Extensible-E upperbound theorem.
E100##100##100##100
gugoltesla
The 1st regiment of the 4th row. "tesla" is a corruption of "tessera", for 4.
⟨10,10,100,3⟩
baggol
Another Chris Bird inspired "Bowerism". Just as gugolthra < biggol, gugoltesla < baggol. This again has to do with the offsets where Bowers' skips the limits ordinals, w, w2, w3, etc. causing an off-by-one that causes my numbers at the corresponding order to lag behind by one. Next up we've got the "3th row" of Cedric Fausey's "Bowerisms" and the 4th row of my own regiments. This are getting really deeply nested already and yet we can describe all this with just 4 entries in array notation and with only two delimiters in ExE! By ordinary measures we've left little piddling stuff like Graham's Number way behind already, but in "googology" we still are only just warming up!
⟨10,100,1,4⟩
cordetal
A cordetal is 10 detonated to the 100th. The term detonated was coined by Jonathan Bowers. The term cordetal was coined by Cedric Fausey. A cordetal is smaller than and comparable to a graatagoltesla.
E100##100##100##100##2
graatagoltesla
The 2nd regiment of the 4th row. Approximately equal to <10,100,1,4>.
⟨10,100,2,4⟩
muldetal
A muldetal is 10 multi-detonated to the 100th. The term multi-detonated was coined by Jonathan Bowers. The term muldetal was coined by Cedric Fausey. A muldetal is smaller than and comparable to a greegoltesla.
E100##100##100##100##3
greegoltesla
The 3rd regiment of the 4th row. Approximately equal to <10,100,2,4>.
⟨10,100,3,4⟩
powdetal
A powdetal is 10 power-detonated to the 100th. The term power-detonated was coined by Jonathan Bowers. The term powdetal was coined by Cedric Fausey. A powdetal is smaller than and comparable to a grinningoltesla.
E100##100##100##100##4
grinningoltesla
The 4th regiment of the 4th row. Approximately equal to <10,100,3,4>.
⟨10,100,4,4⟩
terdetal
A terdetal is 10 detonato-tetrated to the 100th. The term detonato-tetrated was coined by Jonathan Bowers. The term terdetal was coined by Cedric Fausey. A terdetal is smaller than and comparable to a golaagoltesla.
E100##100##100##100##5
golaagoltesla
The 5th regiment of the 4th row. Approximately equal to <10,100,4,4>.
⟨10,100,5,4⟩
pepdetal
A pepdetal is 10 detonato-pentated to the 100th. The term detonato-pentated was coined by Jonathan Bowers. The term pepdetal was coined by Cedric Fausey. A pepdetal is smaller than and comparable to a gruelohgoltesla.
E100##100##100##100##6
gruelohgoltesla
The 6th regiment of the 4th row. Approximately equal to <10,100,5,4>.
⟨10,100,6,4⟩
hexdetal
A hexdetal is 10 detonato-hexated to the 100th. The term detonato-hexated was coined by Jonathan Bowers. The term hexdetal was coined by Cedric Fausey. A hexdetal is smaller than and comparable to a gaspgoltesla.
E100##100##100##100##7
gaspgoltesla
The 7th regiment of the 4th row. Approximately equal to <10,100,6,4>
⟨10,100,7,4⟩
hepdetal
A hepdetal is 10 detonato-heptated to the 100th. The term detonato-heptated was coined by Jonathan Bowers. The term hepdetal was coined by Cedric Fausey. A hepdetal is smaller than and comparable to a ginorgoltesla.
E100##100##100##100##8
ginorgoltesla
The 8th regiment of the 4th row. Approximately equal to <10,100,7,4>.
⟨10,100,99,4⟩
The "associated linear array" of gugolpeta, from the "Associated Linear Array Theorem".
E100##100##100##100##100
gugolpeta
This number introduces the 5th root, "peta". This forms the "gugolpeta regiment", or sub-regiment as it were. We can continue as expected with graatagolpeta, greegolpeta, grinningolpeta, etc. A gugolpeta is larger than and approximately equal to <10,100,99,4>.
⟨10,10,100,4⟩
beegol
Beegol is larger than and comparable to a gugolpeta. The name is based on boogol + geegol, and is the 4th member of the second row of Bowers regiments.
⟨10,100,1,5⟩
corpental
A corpental is 10 pentonated to the 100th. The term corpental was coined by Cedric Fausey.
⟨10,100,99,5⟩
The "associated linear array" of gugolhexa, from the "Associated Linear Array Theorem".
E100##100##100##100##100##100
gugolhexa
This number introduces the 6th root, "hexa". This forms the "gugolhexa regiment", which contains graatagolhexa, greegolhexa, grinningolhexa, etc. At this point the name construction is routine. A gugolhexa is smaller than and comparable to <10,10,100,5> in array notation. <10,10,100,5> is a number Bowers' calls a bigol ... a big ol' number that's for sure :p
⟨10,10,100,5⟩
bigol
This number is larger than and comparable to a gugolhexa. This number is not pronounced "big"-gul. That's biggol (rhymes with giggle). Instead it's pronounced "BI"-gul ("BI" like in Bicycle). I asked Bowers' about pronunciation and he confirmed this directly.
⟨10,100,1,6⟩
corhexal
A corhexal is 10 hexonated to the 100th. The term corhexal was coined by Cedric Fausey.
E100##100##100##100##100##100##100
gugolhepta
Beginning of the gugolhepta regiment. A gugolhepta is comparable to Bowers' boggol = <10,10,100,6>.
⟨10,10,100,6⟩
boggol
It boggles the mind! From boogol + goggol.
⟨10,100,1,7⟩
corheptal
A corheptal is 10 heptonated to the 100th. The term corheptal was coined by Cedric Fausey.
E100##100##100##100##100##100##100##100
gugolocta
Beginning of the gugolocta regiment. A gugolocta is comparable to Bowers' bagol = <10,10,100,7>. Bowers jokes to never eat a bagol bagels. This implies that is pronounced /bae-gol/ not /bah-gol/.
⟨10,10,100,7⟩
bagol
Don't eat this many bagels! And yes, this number is pronounced "bay"-gull, not "bag"-gull, that's baggol. This is the last of the series boogol, biggol, baggol, beegol, bigol, boggol, and bagol. This completes the second row. The third row will begin with troogol, but we won't get there for a bit, as my other series go a little further than this.
⟨10,100,1,8⟩
coroctal
A coroctal is 10 octonated to the 100th. The term coroctal was coined by Cedric Fausey.
E100###9
gugolenna
Beginning of the gugolenna regiment. A gugolenna is comparable to Bowers' ... well actually Bowers' doesn't have a name for the number this is comparable to, however, it would be comparable to <10,10,100,8>. Actually it would be closer to <10,100,99,8>.
⟨10,10,100,8⟩
This would be the hypothetical 8th member of the second row of Bowers' googolisms.
⟨10,100,1,9⟩
corennal
A corennal is 10 ennonated to the 100th. The term corennal was coined by Cedric Fausey.
E100###10
gugoldeka
Beginning of the gugoldeka regiment. A gugoldeka is larger than and comparable to <10,10,100,9> in array notation, although its more accurate to compare it to <10,100,100,9>, which it is also larger than. We are now getting close to Bowers' general ... let's proceed more cautiously ...
⟨10,10,100,9⟩
This would be the hypothetical 9th member of the second row of Bowers' googolisms.
⟨10,100,1,10⟩
cordekal
A cordekal is 10 dekonated to the 100th. The term cordekal was coined by Cedric Fausey.
E100##100##100##100##100##100##100##100##100##100##2
graatagoldeka
At this point the deutero-hyperions are nearly bursting at the seams. In array notation this is larger than but comparable to <10,100,1,10>. A general is <10,10,10,10>. It might seem like that 100 might have something to say about that, but in reality in is vanishingly small. Even <10,3,10,10> would be vastly larger. So we must keep going ...
⟨10,100,2,10⟩
muldekal
A muldekal is 10 multi-dekonated to the 100th. The term muldekal can be extrapolated from Cedric Fausey's naming scheme.
E100##100##100##100##100##100##100##100##100##100##3
greegoldeka
Approximately equal to <10,100,2,10>.
⟨10,100,3,10⟩
powdekal
A powdekal is 10 power-dekonated to the 100th. The term powdekal can be extrapolated from Cedric Fausey's naming scheme.
E100##100##100##100##100##100##100##100##100##100##4
grinningoldeka
Approximately equal to <10,100,3,10>.
⟨10,100,4,10⟩
terdekal
A terdekal is 10 dekonato-tetrated to the 100th. The term terdekal can be extrapolated from Cedric Fausey's naming scheme.
E100##100##100##100##100##100##100##100##100##100##5
golaagoldeka
Approximately equal to <10,100,4,10>.
⟨10,100,5,10⟩
pepdekal
A pepdekal is 10 dekonato-pentated to the 100th. The term pepdekal can be extrapolated from Cedric Fausey's naming scheme.
E100##100##100##100##100##100##100##100##100##100##6
gruelohgoldeka
Approximately equal to <10,100,5,10>.
⟨10,100,6,10⟩
hexdekal
A hexdekal is 10 dekonato-hexated to the 100th. The term hexdekal can be extrapolated from Cedric Fausey's naming scheme.
E100##100##100##100##100##100##100##100##100##100##7
gaspgoldeka
Approximately equal to <10,100,6,10>.
⟨10,100,7,10⟩
hepdekal
A hepdekal is 10 dekonato-heptated to the 100th. The term hepdekal can be extrapolated from Cedric Fausey's naming scheme.
E100##100##100##100##100##100##100##100##100##100##8
ginorgoldeka
Approximately equal to <10,100,7,10>.
⟨10,100,8,10⟩
ocdekal
A ocdekal is 10 dekonato-octated to the 100th. The term ocdekal can be extrapolated from Cedric Fausey's naming scheme.
E100##100##100##100##100##100##100##100##100##100##9
gargantuuldeka
Approximately equal to <10,100,8,10>.
⟨10,100,9,10⟩
enndekal
A enndekal is 10 dekonato-ennated to the 100th. The term enndekal can be extrapolated from Cedric Fausey's naming scheme.
E100##100##100##100##100##100##100##100##100##100##10
googondoldeka
Approximately equal to <10,100,9,10>. This is notable for being the end of the (standard) regiments we can create. This can be thought as occupying position (10,10). This number however can be shown to still be smaller than <10,10,10,10>. Note that, <10,100,9,10> is smaller than <10,<10,10,9,10>,9,10> = <10,3,10,10> is smaller than <10,10,10,10>, which brings us at last to ...
{10,10,10,10}
general / tetradecal
Bowers' first introduces this number as general, following his military naming theme, but then also adds that it may also be called a tetradecal. At this point we have essentially passed up all the standard Saibianisms just before a throogol, but actually there is something even larger before we get there ...
⟨10,100,10,10⟩
dekdekal / gecoldeka
Technically larger than a general. The term gecoldeka I'm coining for this number. Using Cedric Fausey's naming scheme this can also be called dekdekal. This is still technically smaller than ...
E100##100##100##100##100##100##100##100##100##100##11
googoadoldeka
Although not explicitly mentioned, this name is pretty easily expressed following the established naming conventions. A googoadol was introduced as an 11th basic googol extension, and this is a natural continuation of that. This has the distinction of being the first "regiment" to clearly pass up a general, although it's basically in the same ballpark. With that we move onto ...
E100###11
gugolendeka
This is approximately equal to <10,100,99,10> in array notation. Because it's rather trivial to coin. All we need to continue is coming up with one additional root. Let's skip a bit though, to groups of 10 ...
⟨10,10,100,10⟩
This would be the hypothetical 10th member of the 2nd row of Bowers' googolisms. It does not have an official name. We won't see another Bowerism until we reach troogol. Let's wrap up this epoch ...
E100###20
gugolicosa
"icosa"(20) in greek. Approximately <10,100,99,19>.
E100###30
gugoltrianta
"trianta" meaning 30 (greek). Approximately <10,100,99,29>.
E100###40
gugolsaranta
"saranta" meaning 40 (greek). Approximately <10,100,99,39>.
E100###50
gugolpeninta
"peninta" meaning 50 (greek). Approximately <10,100,99,49>.
E100###60
gugolexinta
"exinta" meaning 60 (greek). Approximately <10,100,99,59>.
E100###70
gugolebdominta
"ebdominta" meaning 70 (greek). Approximately <10,100,99,69>.
E100###80
gugologdonta
"ogdonta" meaning 80 (greek). Approximately <10,100,99,79>.
E100###90
gugoleneninta
"eneninta" meaning 90 (greek). Approximately <10,100,99,89>. Which at last brings us to ...
⟨10,100,99,99⟩
A strong lower-bound on E100###100, which is a throogol. This brings us to ...
XIV. Linear Array Epoch
[E100###100,E100#^#100)
Entries: 92
In 2002 Jonathan Bowers introduced the internet to his original Array Notation. Like Conway's Chain Arrows, it's a polyadic function, meaning it can take on an arbitrary number of arguments, but unlike Chain Arrows, it's much much stronger. Arrays of just 4 entries are already able to keep pace with chains of any length, and arrays of 5 entries or more go beyond it. Array Notation uses a more sophistated recursion that incorporates all the entries, unlike chain notations like Chain arrows and Hyper-E that only focus on the last two entries. However, Extended Hyper-E is able to keep up with what are these days called "Linear Arrays", by using multiple delimiters instead of only one like in Chain Arrows and Hyper-E. Thus, this is also the epoch of Extended Hyper-E.
E100###100
throogol
Approximately equal to <10,100,99,99>, although slightly larger than this value. It however is much smaller than <10,10,10,100>, which is Jonathan Bowers troogol.
⟨10,103,99,99⟩
This is believed to be an upperbound on a throogol. From here it is trivial to show that <10,10,10,100> is larger. We can note that 10<99,99>103 < 10<99,99>10<99,99>10 = 10<100,99>3 << 10<10<10,99>10,99>10 = 10<1,100>3 = <10,3,1,100> < <10,10,1,100> < <10,2,2,100> < <10,10,10,100>.
⟨10,10,10,100⟩
troogol
One of Bowers' new googolisms introduced on his 2007 polytope.net website. This value is still vastly smaller than ...
{10,10,10,{10,10,10,10}}
generalplex / tetradecalplex
This number falls between a throogol and a throogola-thruplex. This is because a throogol is approximately {10,10,10,100} where as a throogola-thruplex is approximately {10,10,10,{10,10,10,100}}, making it larger than even the generalplex.
E100###100#2
throogola-thruplex
Equivalent to to E100###throogol or E100###(E100###100). Originally this is where the operators ended, with this number originally called a grand throogol, thus being the first introduction of the "grand". However I recently came up with using "thru" as a way to continue. "three" + "su(per)" = "thru". It should be obvious this can be continued with: thrudex, thruthrex, thrutetrex, ... , thrusuplex , thrusudex , etc. etc.
E100###100#100
thrangol
At this point we want to recycle names to avoid as much added complexity as possible. A thrangol is derived from "throogol" + "grangol". By replacing "g" or "gr" with "thr" we can create a whole slew of "throogol" variants of our existing googolisms to continue. Many of these have amusing names that are fun to say. Next up with threagol, then thrigangol, thorgegol, thrulgol, thraspgol, thringorgol, thrargantuul, throogondol, etc.
A thrangol is approximately equal to <10,100,1,1,2>.
cgG64(G64)
Mouffles Monster
This is the 20th record setting entry in the "My Number is Bigger" competition. It is also the first record setter on page 2. It was entered by Mouffles in response to Gmalivuk's chain-arrow entry. In order to describe it we need a new function. Let...
cg(1) = 1
cg(2) = 2-->2
cg(3) = 3-->3-->3
and in general
cg(n) = n-->n-->n--> ... ... -->n-->n-->n w/n n's
Mouffles Monster = cgGraham's Number(Graham's Number)
Note that Mouffles has simply created the "cg" function first described by Conway and Guy in the Book of Numbers. At this point chain-arrows are no longer competitive. Despite the terrific leap forward, in the larger recursive scheme of things this nesting of cg(n) only leads to an increase of an order-type or two from Chain-arrows which is already of order-type w^2. This means nesting cg(n) falls around w^2+2. I've estimated that it falls between a thrangol and a threagol. At this point all the popular large number notations are exhausted, so the contestants are forced to either start inventing notations or pull something more esoteric from the professional literature such as FGH.
E100###100#100#100
threagol
Approximately <10,100,2,1,2>.
E100###100#100#100#100
thrigangol
Approximately <10,100,3,1,2>.
E100###100#100#100#100#100
throrgegol
Approximately <10,100,4,1,2>.
E100###100#100#100#100#100#100
thrulgol
Approximately <10,100,5,1,2>.
E100###100#100#100#100#100#100#100
thraspgol
Approximately <10,100,6,1,2>.
E100###100#100#100#100#100#100#100#100
thrinorgol
Approximately <10,100,7,1,2>.
E100###100##100
thrugold
Approximately <10,100,99,1,2>.
E100###100##100##2
thraatagold
Approximately <10,100,1,2,2>.
E100###100##100##3
threegold
Approximately <10,100,2,2,2>.
E100###100##100##4
thrinningold
Approximately <10,100,3,2,2>.
E100###100##100##5
throlaagold
Approximately <10,100,4,2,2>.
E100###100##100##6
thruelohgold
Approximately <10,100,5,2,2>.
E100###100##100##7
thraspgold
Approximately <10,100,6,2,2>.
E100###100##100##8
thrinorgold
Approximately <10,100,7,2,2>.
E100###100##100##100
thrugolthra
Approximately <10,100,99,2,2>.
E100###100##100##100##100
thrugoltesla
Approximately <10,100,99,3,2>.
E100###100##100##100##100##100
thrugolpeta
Approximately <10,100,99,4,2>.
E100###100##100##100##100##100##100
thrugolhexa
Approximately <10,100,99,5,2>.
E100###100##100##100##100##100##100##100
thrugolhepta
Approximately <10,100,99,6,2>.
E100###100##100##100##100##100##100##100##100
thrugolocta
Approximately <10,100,99,7,2>.
E100###100###100
throotrigol
Approximately <10,100,99,99,2>.
⟨10,10,10,100,2⟩
triggol
A triggol (troogol + giggol) is larger than and comparable to a throotrigol. This number is smaller than a pentatri.
{3,3,3,3,3}
pentatri
Jonathan Bowers' pentatri. This was among Bowers original set of 121 original googolisms on his 2002 website. It falls somewhere between the large gulf of tristo-throogol and teristo-throogol. It is approximately equal to {10,100,3,3,3} which will be comparable to a saibianism.
E100###100###100###100
throotergol
Approximately <10,100,99,99,3>.
⟨10,10,10,100,3⟩
traggol
A traggol (troogol + gaggol) is larger than and comparable to a throotergol.
E100###100###100###100###100
throopetol
Approximately <10,100,99,99,4>.
⟨10,10,10,100,4⟩
treegol
A treegol (troogol + geegol) is larger than and comparable to a throopetol.
E100###100###100###100###100###100
throohexol
Approximately <10,100,99,99,5>.
⟨10,10,10,100,5⟩
trigol
A trigol (troogol + gigol), pronounced "TRI"-gull, is larger than and comparable to a throohexol.
E100###100###100###100###100###100###100
throoheptol
Approximately <10,100,99,99,6>.
⟨10,10,10,100,6⟩
troggol
A troggol (troogol + goggol) is larger than and comparable to a throoheptol.
E100###100###100###100###100###100###100###100
throogogdol
Approximately <10,100,99,99,7>.
⟨10,10,10,100,7⟩
tragol
A tragol (troogol + gagol), pronounced "TRAY"-gull, is larger than and comparable to a throoheptol.
{10,10,10,10,10}
pentadecal
Jonathan Bowers' Pentadecal. We have now passed up most of the Throogol Super-Regiment.
E100####100
tetroogol
Approximately <10,100,99,99,99>. On googology wiki this number is approximated to <100,100,100,100,99>.
⟨10,10,10,10,100⟩
quadroogol
A quadroogol is larger than and comparable to a tetroogol.
{10,10,10,10,{10,10,10,10,10}}
pentadecalplex
The plex version of a pentadecal. This falls between a teroogol and a teroogola-teruplex.
E100####100#2
tetroogola-teruplex
Approximately but larger than <10,100,99,99,<10,99,99,99,99>>, making it definitely larger than a pentadecalplex.
E100####100###100
tetri-throogol
teroogol + throogol = teri-throogol.
E100####100####100
tetrootrigol
Approximately <10,100,99,99,99,2>.
⟨10,10,10,10,100,2⟩
quadriggol
A quadriggol (quadroogol + giggol) is larger than and comparable to a tetrootrigol.
{3,3,3,3,3,3}
hexatri
One of Jonathan Bowers original 120 named numbers.
E100####100####100####100
tetrootergol
Approximately <10,100,99,99,99,3>.
⟨10,10,10,10,100,3⟩
quadraggol
A quadraggol (quadroogol + gaggol) is larger than and comparable to a tetrootergol.
E100####100####100####100####100
tetroopetol
Approximately <10,100,99,99,99,4>.
⟨10,10,10,10,100,4⟩
quadreegol
A quadreegol (quadroogol + geegol) is larger than and comparable to a tetroopetol.
E100####100####100####100####100####100
tetroohexol
Approximately <10,100,99,99,99,5>.
⟨10,10,10,10,100,5⟩
quadrigol
A quadrigol (quadroogol + gigol), pronounced "Quadry"-gull, is larger than and comparable to a tetroohexol.
E100####100####100####100####100####100####100
tetrooheptol
Approximately <10,100,99,99,99,6>.
⟨10,10,10,10,100,6⟩
quadroggol
A quadroggol (quadroogol + goggol) is larger than and comparable to a tetrooheptol.
E100####100####100####100####100####100####100####100
teroogogdol
Approximately <10,100,99,99,99,7>. This puts it "just below" ...
⟨10,10,10,10,100,7⟩
quadragol
A quadragol (quadroogol + gagol), pronounced "quadray"-gull, is larger than and comparable to a tetroogogdol.
{10,10,10,10,10,10}
hexadecal
The hexadecal, one of Bowers original 120 named numbers.
E100#####100
pentoogol
Approximately <10,100,99,99,99,99>.
⟨10,10,10,10,10,100⟩
quintoogol
A quintoogol is larger than and comparable to a pentoogol.
{10,10,10,10,10,{10,10,10,10,10,10}}
hexadecalplex
One of Bowers original 120 named numbers.
E100######100
hexoogol
Approximately <10,100,99,99,99,99,99>.
⟨10,10,10,10,10,10,100⟩
sextoogol
A sextoogol is larger than and comparable to a hexoogol.
E100#######100
heptoogol
Approximately <10,100,99,99,99,99,99,99>.
⟨10,10,10,10,10,10,10,100⟩
septoogol
A septoogol is larger than and comparable to a heptoogol.
E100########100
ogdoogol
Approximately <10,100,99,99,99,99,99,99,99>. On googology wiki this number is approximated as <100,100,100,100,100,100,100,100,99>.
⟨10,10,10,10,10,10,10,10,100⟩
octoogol
A octoogol is larger than and comparable to a ogdoogol.
{10,10(1)2}
{10,10,10,10,10,10,10,10,10,10}
iteral
Jonathan Bowers' iteral, one of his original 120 googolisms. This number passes up everything up to yottoogol.
⟨10,100,99,99,99,99,99,99,99,99⟩
This array is larger than an iteral, but is smaller than entoogol.
E100#^(9)100
entoogol
Approximately <10,100,99,99,99,99,99,99,99,99>. On googology wiki this number is approximated as <100,100,100,100,100,100,100,100,100,99>.
E100#^(10)100
dektoogol
Approximately <10,100,99,99,99,99,99,99,99,99,99>. On googology wiki this number is approximated as <100,100,100,100,100,100,100,100,100,100,99>.
⟨10,20(1)2⟩
This array is smaller than icosolus = E100#^(20)100.
E100#^(20)100
icosolus
Approximately <10,100,99,...,99> w/19 99s. This array as 21 entries. Therefore it's greater than <10,20(1)2>.
{3,27(1)2}
{3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
ultatri
An unusual googolism by Jonathan Bowers. This is the only one to use the "ulta" root, which seems to imply that this is {3,...,3} with {3,3,3} 3s, in Bowers' older array notation. In the older array notation this is equivalent to {3,3,2(1)2}. This implies that ultadecal would correspond to {10,10,2(1)2} which would be a pretty long array indeed.
⟨10,30(1)2⟩
This array is smaller than triantolus = E100#^(30)100, but larger than ultatri.
E100#^(30)100
triantolus
Approximately <10,100,99,...,99> w/29 99s. This array as 31 entries. Therefore it's greater than <10,30(1)2>.
⟨10,40(1)2⟩
This array is smaller than sarantolus = E100#^(40)100.
E100#^(40)100
sarantolus
Approximately <10,100,99,...,99> w/39 99s. This array as 41 entries. Therefore it's greater than <10,40(1)2>.
⟨10,50(1)2⟩
This array is smaller than penintolus = E100#^(50)100.
E100#^(50)100
penintolus
Approximately <10,100,99,...,99> w/49 99s. This array as 51 entries. Therefore it's greater than <10,50(1)2>.
⟨10,60(1)2⟩
This array is smaller than exintolus = E100#^(60)100.
E100#^(60)100
exintolus
Approximately <10,100,99,...,99> w/59 99s. This array as 61 entries. Therefore it's greater than <10,60(1)2>.
⟨10,70(1)2⟩
This array is smaller than ebdomintolus = E100#^(70)100.
E100#^(70)100
ebdomintolus
Approximately <10,100,99,...,99> w/69 99s. This array as 71 entries. Therefore it's greater than <10,70(1)2>.
⟨10,80(1)2⟩
This array is smaller than ogdontolus = E100#^(80)100.
E100#^(80)100
ogdontolus
Approximately <10,100,99,...,99> w/79 99s. This array as 81 entries. Therefore it's greater than <10,80(1)2>.
⟨10,90(1)2⟩
This array is smaller than enenintolus = E100#^(90)100.
E100#^(90)100
enenintolus
Approximately <10,100,99,...,99> w/89 99s. This array as 91 entries. Therefore it's greater than <10,90(1)2>.
⟨10,100(1)2⟩
goobol
This array is smaller than godgahlah = E100#^(100)100.
⟨10,100,99,99, ... ... ,99,99⟩ w/99 99s
This is a tight lower bound on godgahlah. It has 101 entries and is thus larger than a goobol yet smaller than a godgahlah. With that we move on to the next epoch ...
Continue on to Part III for numbers so large they can't even be expressed in linear array notation