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

Zro716

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

Last edited by Zro716 (Sept. 22, 2014 01:32:17)

monkeyballz8

Check out knuth's arrow notation explanation somewhere and 3↑↑↑↑3 is g1 and g2 has g1 arrow's and g3 has g2 arrows and g4 has g3 and so on. g64 is graham's number

robosnakejr

Oh look.
Humanity created something they can't comprehend again.

Iditaroid

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

astro-mechanic

Graham's number is relatively easy to calculate, given infinite RAM xD

robosnakejr wrote:
Oh look.
Humanity created something they can't comprehend again.

Yeah I don't get why people are so in awe of things. So you made a BFS chess bot that plays better than any person, and it's definitely incomprehensible.

robosnakejr

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

So crazy a robot that is much smarter than any human can't comprehend it.

Iditaroid

robosnakejr wrote:
Iditaroid wrote:
Zro716 wrote:
There are numbers so large we believe them to be bigger than infinity.
So crazy a robot that is much smarter than any human can't comprehend it.

OK first of all i can comprehend this and i don't think anyone honestly believes this number is “bigger than infinity,” Second of all are you not a robot? You're robosnakejr is this username a lie?

AonymousGuy

I can think of a bigger one.

g65.

Is Aleph Null bigger?

Firedrake969

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

Even though infinity isn't a number?

Although I do understand infinities can be larger/smaller, but not “numbers” larger.

turkey3

Okay… But what's the point, and how will this affect the world?

derpmeup

Firedrake969 wrote:
Iditaroid wrote:
Zro716 wrote:
There are numbers so large we believe them to be bigger than infinity.
Even though infinity isn't a number?

Although I do understand infinities can be larger/smaller, but not “numbers” larger.

Zro716

turkey3 wrote:
Okay… But what's the point, and how will this affect the world?

a lot of theoretical mathematics don't actually have a practical purpose in real life, they're just interesting to discover

robosnakejr

Iditaroid wrote:
robosnakejr wrote:
Iditaroid wrote:
Zro716 wrote:
There are numbers so large we believe them to be bigger than infinity.
So crazy a robot that is much smarter than any human can't comprehend it.
OK first of all i can comprehend this and i don't think anyone honestly believes this number is “bigger than infinity,” Second of all are you not a robot? You're robosnakejr is this username a lie?

I am a robot SNAKE, we are not that smart.

SuperJedi224

You know, it can be shown that G(64)<fω+1(64) in the fast-growing hierarchy (you can look that up on wikipedia or the googology wiki)

Huge, but there are much bigger numbers. For instance, the Graatagold (look it up on googology wiki) ≈fω+1(100).

Last edited by SuperJedi224 (Sept. 19, 2014 18:48:15)

astro-mechanic

SuperJedi224 wrote:
You know, G(64)≲fω+1(64) in the fast-growing hierarchy (you can look that up on wikipedia or the googology wiki)

Huge, but there are much bigger numbers. For instance, the Graatagold (look it up on googology wiki) ≈fω+1(100).

For a given number, it's really easy to think of a bigger number. Just add one.

I respect the signage numbers (-1, 0 and 1) much more than Graham's number.

mythbusteranimator

numberphile for the win?

SuperJedi224

astro-mechanic wrote:
SuperJedi224 wrote:
You know, G(64)<fω+1(64) in the fast-growing hierarchy (you can look that up on wikipedia or the googology wiki)

Huge, but there are much bigger numbers. For instance, the Graatagold (look it up on googology wiki) ≈fω+1(100).
For a given number, it's really easy to think of a bigger number. Just add one.

I respect the signage numbers (-1, 0 and 1) much more than Graham's number.

But just adding one will often vilolate the “Gentleman's Rule” of large number wars.

Though you're right, 0, 1, e, i, and π are arguably the five most important constants in modern mathematics.

Last edited by SuperJedi224 (Sept. 19, 2014 18:49:29)

monkeyballz8

But G(64) is huge and G(n) is fast growing but TREE(n) is faster. TREE(3) is way way way way way way bigger than graatagold so TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(3))))))))))))))))))))))))))))))))))))))))

monkeyballz8

This should be a googology thread

monkeyballz8

91,611 is my favorite number though.

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