The size of Graham's number

[thetexan]

Wow. I started out thinking I might be able to estimate the size if Graham's number but I have reached my limit of effort.

After repeated work I believe I have gone part way.

Realize that the number is G64. I won't try to explain. Suffice it to say that Knuth's notation makes the logarithmic scale seem inconceivably inadequate to use as a reference.

Anyway. I read that if you fill the observable universe with grains of sand and on each of those grains use a microscope to write ten billion zeroes you would have the representation of a google. A Googol is incomprehensibly infinitesimal compared to G1 much less G64.

By my crude estimation you would need a sphere so large in scale to the observable universe as to be equal in ratio as a proton is to the observable universe filled with grains of sand each with 10 billion zeroes written on them to approximate a number roughly 3!4 shy of G1. (Up arrows). Also roughly equal to the US debt in 2030 by the way!

My head hurts to even try to finish G1. G64 is impossible.

tex

 

[thetexan]

Im sorry I meant ...3!3 shy of G4...

 

[thetexan]

I screwed up again. Sorry. I meant 3!3 shy of G1 not G4.

 

[Mentallic]

A few things...

Please keep in mind that you can edit your posts, as opposed to replying over and over.

Anyway. I read that if you fill the observable universe with grains of sand and on each of those grains use a microscope to write ten billion zeroes you would have the representation of a google. A Googol is incomprehensibly infinitesimal compared to G1 much less G64.

You mean a googolplex. A googol is a 1 followed by 100 zeroes. A googolplex is a 1 followed by a googol zeroes.

By my crude estimation you would need a sphere so large in scale to the observable universe as to be equal in ratio as a proton is to the observable universe filled with grains of sand each with 10 billion zeroes written on them to approximate a number roughly 3!4 shy of G1.

No, your estimate is way off. You need to understand that up arrow notation is a lot more powerful than you think, and for many of those that first delve into the topic, they almost always vastly misunderstand and underestimate the sheer magnitude of up arrows. For starters, don't bother trying to represent their scale with "a grain of sand expanded to the universe, with all its grains of sand expanded into another universe, etc. etc. with every grain of sand having trillions of 0's on it". This operation of grains representing universes and grains in that universe representing another universe is simply multiplication. You need to up your game beyond exponentiation!

My head hurts to even try to finish G1. G64 is impossible.

Yes, yes it is.

You need a firm grasp of exponential towers and their power, so just google tetration to begin with, and once you feel as though you understand that, then you're ready to move on further.