Graham’s Number

Graham’s number is a number invented by Ronald Graham. In order to explain what it is, the notation must be understood. It’s called up-arrow notation, denoted by the ↑ symbol. One up-arrow just denotes that the second number is an exponent. For example, 3↑3 is 3^3, or 27. Using two arrow creates the fourth thing in the sequence of addition, multiplication, and exponentiation. Some call this math operation tetration. 3↑↑3 is 3^(3^3), 3^27, or 7,625,597,484,987. Using a third arrow, you can probably predict what happens. 3↑↑↑3 is 3↑↑(3↑↑3), or 3↑↑7,625,597,484,987. This means that you have (3^(3^(3^(...(3^3)...)))), and there are 7,625,597,484,987 3’s. For perspective, 3↑↑4, or 3^7,625,597,484,987, contains 3,638,334,640,024 digits. I’m not kidding, that is the actual number of digits, compute it using the Big Online Calculator. And yet, despite how far blown out of proportion this thing has been, it’s still not large enough. We need a fourth arrow. Don’t even get me started on the size of 3↑↑↑↑3, or 3↑↑↑(3↑↑↑3). And that number is called G(1). G(2) is 3↑↑↑...↑↑↑3. There are G(1) arrows. G(3) is 3↑↑↑...↑↑↑3, with G(2) 3’s. You get it now? Graham’s number is defined as G(64). And despite its immense size, it actually has a purpose. Suppose you had higher-dimensional hypercubes, and you had two colors for edges, and you wanted to know how many dimensions it took before a square where all lines were the same color was forced. The upper bound on that answer is Graham’s number.

Graham’s number is a number which was once considered the largest of all time.

14👍 2👎

Graham's Number

1. The largest useful number in real mathematics.

2. upper-bound solution to Ramsey theory.

3. Makes googolplex look like a pussy.

...9404248265018193851562535

7963996189939679054966380

0322234872396701848518643

9059104575627262464195387

...yeah that's the first 100 digits.

Graham's Number is enough to make Chuck Norris blink.

66👍 13👎

Graham's Number

This number is so purely indescribably large, that no mind in a fzgargagoltriplex years could fathom purely how monstrously large this number is. It may as well be infinity!
Here’s references for purely how large the number is:

(For Reference, on the basic scale, from addition to beyond decation.)
Level 1 - Addition; example 10+10 = 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 = 20
Level 2 - Multiplication; example 10*10 (or 10x10) (or 10•10) = 10+10+10+10+10+10+10+10+10+10 = 100
Level 3 - Triation; example 10^10 = 10*10*10*10*10*10*10*10*10*10 = 10,000,000,000
Level 4 - Tetration; example 10^^10 = 10^10^10^10^10^10^10^10^10^10
Level 5 - Pentation; example 10^^^10 = 10^^10^^10^^10^^10^^10^^10^^10^^10^^10
Level 6 - Hexation; example 10^^^^10
- Now we use: 10{n}10, n represents the amount of ^'s in the number.
Level 7 - Heptation; example 10^^^^^10 (or 10{5}10)
Level 8 - Octation; example 10^^^^^^10 (or 10{6}10)
Level 9 - Ennation; example 10^^^^^^^10 (or 10{7}10)
Level 10 - Decation; example 10^^^^^^^^10 (or 10{8}10)
- Now we get to 10{10{n}10}10 numbers.
10{10{10}10}10
10{10{10{10}10}10}10
10{{1}}10 = 10{10{10{10…(10*){10}…(10*)10}10}10}10
With operator notation (which was just used) we can go further, but that’s all we need to know for graham’s number.

Without further ado, here’s the definition using that system.
G0 = 3{4}3
G1 = 3{3{4}3}3
G2 = 3{3{3{4}3}3}3
G3 = 3{3{3{3{4}3}3}3}3
…
G64 = 3{3{3{3{3…(65*){4}…(65*)3}3}3}3}3 or Graham's Number

Person 1: Hey, Person 2!
Person 2: What is it?
Person 1: Did you know that there’s a number called Graham's Number, and it’s a really large number and-
Person Graham's Number: Did someone say my name?