Graham's number (g64) and other extremely big numbers

MrPenguin35230

excelguru wrote:
gigamushroom wrote:
ChocolatePi wrote:
PrincessPanda_test_ wrote:
I know the biggest number:

g10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.
g10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001
Multiply g64 by g64 g64 times to obtain the largest number. I call it.. THE ULTIEXCESSIOGOL. I know the name sucks
That's just g64^g64. Still really big.

G1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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0000000000000000000000
say <[MrPenguin35230] = [(cool) ]> for (2) secs

excelguru

goldfish678 wrote:
excelguru wrote:
G(G(G(…(G(64))…)))
with G(G(G(…(G(64))…))) layers
with G(G(G(…(G(64))…))) layers
with G(G(G(…(G(64))…))) layers
…
with G(G(G(…(G(64))…))) layers
with G(G(G(…(G(64))…))) layers
with G(64) layers

with G(64) rows

BEAT THAT!!!!!!!!!!!!!!
How about Aleph Omega to the infinity?

aleph null is the same thing as infinity.

excelguru

Zro716 wrote:
There are numbers so large we believe them to be bigger than infinity. If such were true, Graham's number would take the #1 spot. It is the largest number ever used to solve an actual problem, and suffice to say there are no words to describe its size. While we can easily say infinity is simply that, an endless number, we cannot even comprehend finite numbers beyond what we can count.

So what exactly is Graham's number? g64, as it is identified, is the solution to the upper bound in this problem in Ramsey theory:
Let N* be the smallest dimension n of a hypercube such that if the lines joining all pairs of corners are two-colored for any n>=N*, a complete graph K4 of one color with coplanar vertices will be forced. Stated colloquially, this definition is equivalent to considering every possible committee from some number of people n and enumerating every pair of committees. Now assign each pair of committees to one of two groups, and find N* the smallest n that will guarantee that there are four committees in which all pairs fall in the same group and all the people belong to an even number of committees (Hoffman 1998, p. 54).
(Source)
What was that gibberish? I don't know, it's probably saying if there exists a solution between two unbelievably huge numbers. And Graham's number takes that place.

How is Graham's number calculated? Heheh, well, that's not easy to put into English.

Let's start by stacking 3 over seven trillion times, or 3^(3^(3^(3^(…….^3)…))), and call it g1. In perspective, 3^^3 = 3^3^3 = 3^27 = 7625597484987, which is already pretty big. Whatever incredibly huge number g1 becomes, g2 equals 3 stacked that many times. And we don't stop there: Every iteration of Gn is 3 stacked Gn-1 times. We keep doing this until we get g64.

By the time we calculate g1, we have a number with more digits than atoms in the known universe. Feeling small already? Now take 3 stacked g1 times, and you can hear every mathematician's mind explode.

So, uh, talk about how insignificant you are to a number so big math can't even… ok you get the point

Here is a correct definition:

a^<1>b = a^b
a^<b>1 = a
a^<b>c = a^<b-1>(a^<b>(c-1))
G(1) = 3^<4>3
G(x) = 3^<G(x-1)>3
graham's number = G(64)

Last edited by excelguru (April 28, 2015 21:10:36)

excelguru

Ackermann's number is the biggest number ever.

Last edited by excelguru (June 25, 2015 13:58:33)

excelguru

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0

excelguru

excelguru wrote:

1000000000... (see the previous post for all the zeroes) ...00000000000000000

That was 100,000 zeroes!!!!

Last edited by excelguru (May 14, 2015 22:04:34)

excelguru

Knuth up arrows

Knuth up-arrows are a really powerful notation, and pretty much only Conway chained arrows can beat them. Here we go:

a↑b = a^b
a↑↑b = a↑(a↑(a↑… with b a's
a↑↑↑b = a↑↑(a↑↑(a↑↑… with b a's
a↑↑↑↑b = a↑↑↑(a↑↑↑(a↑↑↑… with b a's
…

These numbers are huge! Not just huge in the way a million or even a googol is, but HUGE.

For example, take these numbers:

3↑↑2 = 3↑3 = 27
3↑↑3 = 3↑27 = 7,625,597,484,987
3↑↑4 = 3↑7,625,597,484,987 = … a number with 3.86 trillion digits
3↑↑5 has a 3.68-trillion-digit number of digits!!

3↑↑↑2 = 3↑↑3 = 7,625,597,484,987
3↑↑↑3 = 3↑(3↑(3↑(3↑… with 7,625,597,484,987 3's!!!!

Imagine what a monstrosity 3↑↑↑↑3 is!

Visualization of 3↑↑↑3:

3^3^3^…^3^3
7,625,597,484,987

WOW!

Here is a diagram of 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
3^3^3^…^3^3 |
3^3^3 |
3 |

An interesting thing is that up-arrows can actually be modified so that they're less powerful. Just change the parentheses to go to the left side. An interesting thing about this variation is that a↑↑b = a^a^(b-1). This isn't too difficult to see, using the fact that (a^b)^c = a^(b*c):

((…((a^a)^a)…)^a)^a (b a's) = a^(a*a*…*a*a) (b a's) = a^a^(b-1) (keep in mind that “b a's” counts the a on the outside too)

Last edited by excelguru (June 18, 2015 00:27:44)

excelguru

Conway chained arrows

Conway chained arrows are a notation for the biggest numbers ever. It is denoted by a finite string of positive integers, such as 3→4→5→6→7. It is very powerful.

Definition
Let p and q represent numbers, and X represent “all other numbers”.

The empty chain “” evaluates to 1. The singe-item chain p evaluates to p.
p→q = p^q
X→1→a = X→1 = X
X→p→q = X→(X→(p-1)→q)→(q-1)

These numbers are HUGE! They go beyond what I call the “exploding arrow” range, and into a whole new range! Take, for instance 3→3→3→2

3→3→3→2 = 3→3→(3→3→2→2)→1 = 3→3→(3→3→2→2) = 3→3→(3→3→(3→3→1→2)→1) = 3→3→(3→3→(3→3)→1) = 3→3→(3→3→(3→3)) = 3→3→(3→3→27) = ???

There is an equality with chained arrows:

a→b→c = a↑↑↑…↑↑↑b with c arrows

This means 3→3→3→2 is equal to:

3↑↑↑…↑↑↑3
3↑↑↑…↑↑↑3
27

3→3→3→3 = 3→3→(3→3→2→3)→2 = 3→3→(3→3→27→2)→2

It turns out this is equal to:

3↑↑↑…↑↑↑3 | 3↑↑↑…↑↑↑3 |
3↑↑↑…↑↑↑3 | 3↑↑↑…↑↑↑3 |
3↑↑↑…↑↑↑3 | 3↑↑↑…↑↑↑3 |
… | … | 27
3↑↑↑…↑↑↑3 | 3↑↑↑…↑↑↑3 |
3↑↑↑…↑↑↑3 | 3↑↑↑…↑↑↑3 |
27 | 27 |

Right at the end of the “exploding arrow” range - we skipped from the beginning of the exploding arrow range to the edge of it just by increasing one number by 1 - and multi-chains extend into even more ridiculous ranges…

excelguru

GRAHAM'S NUMBER

Graham's number is a huge number that is the very number that triggered this thread, although now it has become a discussion on big numbers in general.

Now, what exactly is Graham's number?

This is difficult to explain, but I'll try.

Now, let's have G(0) be 3↑↑↑↑3, that huge number from the Knuth arrow article.

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^…^3^3 |
3^3^3 |
3 |

Now, let's have G(1) have G(0) arrows. This means:

3↑↑↑….↑↑↑3
3↑↑↑↑3

Is this Graham's number? Nope.

Lets have G(2) have G(1) arrows, G(3) have G(2) arrows, G(4) have G(3) arrows - is G(4) Graham's number? NOT EVEN CLOSE.

Graham's number is G(64)! Here is a diagram:

3↑↑↑….↑↑↑3 |
3↑↑↑….↑↑↑3 |
3↑↑↑….↑↑↑3 |
… | 64
3↑↑↑….↑↑↑3 |
3↑↑↑….↑↑↑3 |
3↑↑↑↑3 |

In terms of my ranges, this falls in the “exploding arrow” range.

Although it is obvious this is less than something like 3→3→3→3, it is still HUGE!

Other things that beat Graham's number:

G(65)
G(66)
G(100)
G(1000)
G(1000000)
G(10^100)
G(10^10^100)
G(3↑↑↑↑3)
G(G(2))
G(G(64))
G(G(G(64)))
G(G(G(…(G(64))…))) with G(64) layers
G(G(G(…(G(64))…))) with G(G(64)) layers

G(G(G(…(G(64))…)))
with G(G(G(…(G(64))…))) layers
with G(64) layers

G(G(G(…(G(64))…)))
with G(G(G(…(G(64))…))) layers
with G(G(G(…(G(64))…))) layers
with G(G(G(…(G(64))…))) layers
…
with G(G(G(…(G(64))…))) layers
with G(G(G(…(G(64))…))) layers
with G(64) layers

with G(64) rows

and so on.

And you still have time to invent a G2 function

Last edited by excelguru (June 18, 2015 18:36:49)

excelguru

Excelguru's big numbers I

Excelguru's extra-strong chained arrows

Excelguru's extra-strong chained arrows are an extension of Conway's chained arrows. The base rules go like this:

a→{b} = a→a→…→a→a with a a's
a→{b→c} = a→{(a→{(…{(a→{(a→{b})})}…)})} with c a's

Then to go further you look inside the curly braces and treat them like normal chains.

So we can have the following:

3→{3}
3→{3→3}
3→{3→{3}}
3→{3→{3→3}}
3→{3→{3→{3}}}
…

Note that things like 3→3→{3} and 3→{3}→3 are invalid. Also, a→{b→c→d→…} does not equal a→{(b→c→d→…)}.

The big googol

The big googol is a number defined with my extra-strong chained arrows.

10→100 = googol
10→10→100 = googoldubla
10→10→10→100 = googoltribla
10→10→10→10→100 = googolquabla
Then googolquibla, googolsebla, googolsepibla, googoloctabla, googolnonabla, and googoldecibla. Let goog(n) = 10→10→…→10→100 with n 10's

Goog(10) = googoldecibla
Goog(20) = googolicobla
Goog(30) = googoltriacongla
Goog(40) = googoltetracongla
Goog(50) = googolpentacongla
Goog(60) = googolhexacongla
Goog(70) = googolseptacongla
Goog(80) = googoloctacongla
Goog(90) = googolnonacongla
Goog(100) = googolcengibla

10→{100} = strongol
10→{10→100} = strongoogol
10→{10→10→100} = stronga-googoldubla
10→{10→10→10→100} = stronga-googoltribla
…
10→{10→{100}} = stronga-strongol
10→{10→{10→100}} = stronga-strongoogol
10→{10→{10→10→100}} = dublastronga-googoldubla
10→{10→{10→{100}}} = dublastronga-strongol = triblastrongol
10→{10→{10→{10→{100}}}} = triblastronga-strongol = quablastrongol

And finally:

big googol = 10→{10→{…{10→{10→{10}}}…}} with 100 10's - just under cengiblastrongol.

Ravioli7899

How about g65!

Ravioli7899

or g66!

Ravioli7899

g2000xg2000 g2000 times!

DUNDUNDUN

excelguru

Ravioli7899 wrote:
g2000xg2000 g2000 times! DUNDUNDUN

If you really want to beat graham's number, use chained arrows, like:

G→G→…→G→G with G G's where G is graham's number

DUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUNDUN

excelguru

Steinhaus-Moser notation I

Steinhaus-Moser notation is a notation for making big numbers.

It works like this:

n in a triangle = n^n
n in a square = n in n triangles
n in a circle = n in n squares

And also two numbers were defined using it:

Mega = ②
Megiston = ⑩

The mega

Let's try to solve the mega.

Mega = ② = Square(Square(2)) = Square(Triangle(Triangle(2))) = Square(Triangle(2^2)) = Square(Triangle(4)) = Square(4^4) = Square(256) = 256 in 256 triangles

But x in a triangle is equal to x^x. This means the mega is f(f(…(f(f(256)))…)) with 256 f's, where f(x) = x^x!

So:

f(256) = (256^256)^(256^256)
f(f(256)) = ((256^256)^(256^256))^((256^256)^(256^256))
f(f(f(256))) = (((256^256)^(256^256))^((256^256)^(256^256)))^(((256^256)^(256^256))^((256^256)^(256^256)))

Does that mean this is equal to 256↑↑2^256? Nope. That's way smaller, because (a^b)^(c^d) > a^b^c^d, and it's been applied lots of times.

The megiston

The megiston is ⑩. Because its ten in a circle, it can't be much bigger than two in a circle, right. NO. 2 is just a small case of the circle, as it only generates two squares. The megiston has ten squares!!!

⑩ = Square(Square(Square(Square(Square(Square(Square(Square(Square(Square(10))))))))))

And even worse, the first square makes you solve ten triangles!!!!

And once you solved that, you have to go through that many triangles!!!!! AAAAGGGGHHHH!!!!!!!!!!!!!!!!!!!!!!!!! And that was only the second square…

Steinhaus-Moser notation II

We already went over Steinhaus' original notation, now we are going onto an extension by Leo Moser.

n in a triangle = n^n
n in a polygon with x sides (x>3) = n in n polygons with x-1 sides

There is no circle anymore, as it has been replaced with a pentagon.

There is a new number, called Moser's number. It is 2 in a polygon with mega sides! To solve it, you would have to go through lots of polygons, working with higher and higher numbers, solving increasingly complex expressions, and - you get it.

But we have bigger numbers:

10 in a polygon with megiston sides - I call this the Mosiston
2 in a polygon with Moser's number of sides - I call this the super-Moser
10 in a polygon with Mosiston sides - I call this the super-Mosiston

Super puts 2 or 10 in a polygon with x sides. Eventually we get to:

super-super-…-super-super-Moser with Moser's number of supers - I call this the hyper-Moser. To continue even further I think we have to invent super Steinhaus Moser notation!

Detriment2

Zro716 wrote:
There are numbers so large we believe them to be bigger than infinity.

This is wrong. There are no numbers bigger than infinity, that is the point of infinity… Absolutely nobody believes what you are claiming

excelguru

The Ackermann function

The Ackermann function, denoted A(x,y), is a 2-argument function that can generate large numbers. It is defined as:

/ y+1 if x=0
A(x,y) = | A(x-1,1) if x>0 and y=0
\ A(x-1,A(x,y-1)) if x>0 and y>0

Unlike Graham's number and Conway's chained arrows, this function was neither designed to make big numbers nor to have a direct application in professional mathematics, but to show that a total computable function does not have to be primitive recursive. What does that mean? I don't know myself.

Let's do an example:

A(3,3) = A(2,A(3,2)) = A(2,A(2,A(3,1))) = A(2,A(2,A(2,A(3,0)))) = A(2,A(2,A(2,A(2,1)))) = …

Let's try a smaller example:

A(2,2) =
A(1,A(2,1)) =
A(1,A(1,A(2,0))) =
A(1,A(1,A(1,1))) =
A(1,A(1,A(0,A(1,0)))) =
A(1,A(1,A(0,A(0,1)))) =
A(1,A(1,A(0,2))) =
A(1,A(1,3)) =
A(1,A(0,A(1,2))) =
A(1,A(0,A(0,A(1,1)))) =
A(1,A(0,A(0,A(0,A(1,0))))) =
A(1,A(0,A(0,A(0,A(0,1))))) =
A(1,A(0,A(0,A(0,2)))) =
A(1,A(0,A(0,3))) =
A(1,A(0,4)) =
A(1,5) =
A(0,A(1,4)) =
A(0,A(0,A(1,3))) =
A(0,A(0,A(0,A(1,2)))) =
A(0,A(0,A(0,A(0,A(1,1))))) =
A(0,A(0,A(0,A(0,A(0,A(1,0)))))) =
A(0,A(0,A(0,A(0,A(0,A(0,1)))))) =
A(0,A(0,A(0,A(0,A(0,2))))) =
A(0,A(0,A(0,A(0,3)))) =
A(0,A(0,A(0,4))) =
A(0,A(0,5)) = A(0,6) = 7

Wait - it took all those steps just to collapse down to SEVEN???? There has to be a better way to do this…

It turns out that A(x,y) = 2↑<x-2>(y+3)-3. (I would use square brackets, but then it thinks it's some kind of formatting. Is there a way to deal with it?)

Last edited by excelguru (June 24, 2015 15:10:32)

excelguru

More of my stuff

Sorry if the definition of EESCAN was unclear. Here is an example to clear it up:

3→{3→{3}} = 3→{3→3→3} = 3→{3→(3→{3→2→3})→2} = 3→{3→(3→{3→(3→{3→1→3})→2})→2} =
3→{3→(3→{3→(3→{3})→2})→2} = 3→{3→(3→{3→(3→3→3)→2})→2} = …

Also here is an extension to up arrows:

a{↑}b = a↑↑…↑↑a with b arrows
a{↑}↑b = a{↑}a{↑}…{↑}a{↑}a with b a's
a{↑}↑↑b = a{↑}↑a{↑}↑…{↑}↑a{↑}↑a with b a's
a{↑}↑↑↑b = a{↑}↑↑a{↑}↑↑…{↑}↑↑a{↑}↑↑a with b a's
a{↑}{↑}b = a{↑}↑↑…↑↑a with b arrows after {↑}
a{↑}{↑}↑b = a{↑}{↑}a{↑}{↑}…{↑}{↑}a{↑}{↑}a with b a's
a{↑}{↑}{↑}b = a{↑}{↑}↑↑…↑↑a with b arrows after {↑}{↑}
a{↑↑}b = a{↑}{↑}…{↑}{↑}a with b {↑}'s
a{↑↑}↑b = a{↑↑}a{↑↑}…{↑↑}a{↑↑}a with b a's
a{↑↑}{↑}b = a{↑↑}↑↑…↑↑a with b arrows after {↑↑}
a{↑↑}{↑↑}b = a{↑↑}{↑}{↑}…{↑}{↑}a with b {↑}'s after {↑↑}
a{↑↑↑}b = a{↑↑}{↑↑}…{↑↑}{↑↑}a with b {↑↑}'s
…

Then there's {{↑}}, {{↑}↑}, {{↑}{↑}}, {{↑↑}}, {{↑↑↑}}, {{{↑}}}, {{{{↑}}}}, and so on. How they work should be obvious.

Last edited by excelguru (June 25, 2015 13:54:17)

Zro716

Detriment2 wrote:
Zro716 wrote:
There are numbers so large we believe them to be bigger than infinity.
This is wrong. There are no numbers bigger than infinity, that is the point of infinity… Absolutely nobody believes what you are claiming

It was a joke, okay. Of course I knew that, but my point is that when we try to visualize such numbers as strings, infinity produces “∞” and really big finite numbers become enormous strings of digits that our limited brains struggle to comprehend, and as a result having more difficulty imagining a finite number than an infinite one, we presume the latter to be “visually” smaller than the former.

excelguru

A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(100) =
Ackermann's number

A(n) = A(n,n)

It takes the human mind to comprehend a number as small as Ackermann's number.

- these smileys are not invading the forums o_o

Last edited by excelguru (June 25, 2015 00:42:51)

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