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.

 

[CellCoree]

I can understand your amazement and exhaustion in trying to comprehend the size of Graham's number. It is truly a mind-boggling concept that goes beyond our current understanding of numbers and their magnitude.

To give some context, Graham's number is a number that was first defined in a mathematical proof by Ronald Graham in 1977. It is so large that even writing it out would require more space than the observable universe could contain. It is estimated to be around 10^100 digits long, which is a number so large that it is difficult to even fathom.

To put it into perspective, a googol (10^100) is already a number that is beyond our everyday understanding. And as you mentioned, a googol is incomprehensibly infinitesimal compared to Graham's number. In fact, even if we were to use the entire observable universe to represent a googol, it would still be nowhere near the size of Graham's number.

To try and visualize the enormity of Graham's number, some have used analogies such as filling the entire observable universe with grains of sand and writing 10 billion zeros on each grain, or creating a sphere with a radius equal to the ratio of a proton to the observable universe, filled with grains of sand each with 10 billion zeros written on them. These analogies may help give a sense of scale, but they still fall short in truly grasping the magnitude of Graham's number.

In short, as you have discovered, trying to comprehend the size of Graham's number is a daunting task and can quickly become overwhelming. But as scientists, it is important to continue pushing the boundaries of our understanding and exploring concepts that may seem impossible to fully grasp. Who knows, perhaps one day we will have a better understanding of numbers and be able to tackle Graham's number with ease. Until then, let's marvel at its immensity and continue to be amazed by the wonders of mathematics.