1. Question: How is Graham's number gotten from the Knuth up-arrow notationi?
Answer: As someone mentioned, here is what Graham's notation means:
(I'm using ^ to mean up-arrow. For exponentiation, ^ and up-arrow mean the same thing.)
g(1) is defined as 3^^^^3
g(n) is defined as 3^^^...^^^3 ...where the number of ^'s written is equal to g(n-1)
That g(n) function increases _very_ rapidly, as n increases.
Graham's number is g(64).
2. Question: What font-size is assumed when it's said that the observable universe isn't big enough to write Graham's number?
Answer: Writing Graham's number as a 1 followed by zeros, suppose you could write one of the zeros in every Planck volume in the observable universe. There aren't enough of them to write it.
If, instead of 1 followed by a string of zeros, you wrote "A trillion trillion trillion...trillion trillion trillion. ...with a long enough string of "trillion" to write Graham's number you'd still need 2/3 as many characters, and so it would still be impossible to write all that string of "trillion", by writing one character in of the Planck volumes in the observable universe,
How small is a Planck volume? It's a hundredth of a millionth of a trillionth of the volume of a proton.
In fact, not only could you not write Graham's number by writing a zero or or a letter of "trillion" in every Planck volume in the universe. It's said that you couldn't even write the number of digits in the number of digits,
...or the number of digits in the number of digits in the number of digits.
...and so on, repeating that number-of-digits thing as many times as the number of Planck volumes in the observable universe.
I'm just quoting something said at Wikipedia. I can't personally guarantee it.
But I can guarantee the first claim: You can't write the Graham number even if you could write a zero, or a letter in "trillion" in each Plank volume of the observable universe,
But that's achievable by much smaller numbers that Graham's number.
How many orders can 200 objects be arranged in? The number is greater than the number of Planck volumes in the universe,
Say you numbered and listed all those orders. How many orders could that list be arranged in? The number of orders for that would be too big to be written even if you could write a zero or a letter of the word "trillion" in every Planck volume in the observable universe
You know that scan-pattern called the QR code, that accesses an advertiser's website? At least one kind of QR code scan pattern uses a 35X35 square pattern of smaller squares
How many combinations of shaded and unshaded squares are there?
There are more than the number of Planck volumes in the observable universe.
In how many orders could the world's human population be arranged? About 10 to the power of 72 billion.
In how many orders could you arrange a number of objects equal to 10 to the 72 billionth power?
That number of orders is about the 400 millionth power of the number of Planck volumes in the observable universe.
Graham's number is incomparably larger than any of these numbers, But those order-numbers can be gotten in a familiar way, ordering a certain number of objects, And it doesn't take many objects (about 200) to have more orders than the number of Planck volumes in the observable universe.
I'd like to answer to questions:
1. Question: How is Graham's number gotten from the Knuth up-arrow notationi?
Answer: As someone mentioned, here is what Graham's notation means:
(I'm using ^ to mean up-arrow. For exponentiation, ^ and up-arrow mean the same thing.)
g(1) is defined as 3^^^^3
g(n) is defined as 3^^^...^^^3 ...where the number of ^'s written is equal to g(n-1)
That g(n) function increases _very_ rapidly, as n increases.
Graham's number is g(64).
2. Question: What font-size is assumed when it's said that the observable universe isn't big enough to write Graham's number?
Answer: Writing Graham's number as a 1 followed by zeros, suppose you could write one of the zeros in every Planck volume in the observable universe. There aren't enough of them to write it.
If, instead of 1 followed by a string of zeros, you wrote "A trillion trillion trillion...trillion trillion trillion. ...with a long enough string of "trillion" to write Graham's number you'd still need 2/3 as many characters, and so it would still be impossible to write all that string of "trillion", by writing one character in of the Planck volumes in the observable universe,
How small is a Planck volume? It's a hundredth of a millionth of a trillionth of the volume of a proton.
In fact, not only could you not write Graham's number by writing a zero or or a letter of "trillion" in every Planck volume in the universe. It's said that you couldn't even write the number of digits in the number of digits,
...or the number of digits in the number of digits in the number of digits.
...and so on, repeating that number-of-digits thing as many times as the number of Planck volumes in the observable universe.
I'm just quoting something said at Wikipedia. I can't personally guarantee it.
But I can guarantee the first claim: You can't write the Graham number even if you could write a zero, or a letter in "trillion" in each Plank volume of the observable universe,
But that's achievable by much smaller numbers that Graham's number.
How many orders can 200 objects be arranged in? The number is greater than the number of Planck volumes in the universe,
Say you numbered and listed all those orders. How many orders could that list be arranged in? The number of orders for that would be too big to be written even if you could write a zero or a letter of the word "trillion" in every Planck volume in the observable universe
You know that scan-pattern called the QR code, that accesses an advertiser's website? At least one kind of QR code scan pattern uses a 35X35 square pattern of smaller squares
How many combinations of shaded and unshaded squares are there?
There are more than the number of Planck volumes in the observable universe.
In how many orders could the world's human population be arranged? About 10 to the power of 72 billion.
In how many orders could you arrange a number of objects equal to 10 to the 72 billionth power?
That number of orders is about the 400 millionth power of the number of Planck volumes in the observable universe.
Graham's number is incomparably larger than any of these numbers, But those order-numbers can be gotten in a familiar way, ordering a certain number of objects, And it doesn't take many objects (about 200) to have more orders than the number of Planck volumes in the observable universe.