Wikipedia:Reference desk/Archives/Mathematics/2022 November 12

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November 12

Graham's number g₁

After reading about Graham's number g₁, I learnt that a 3.6 trillion-digit number is still insufficient for it. So, limiting my non-math mind to understanding just g₁, I decided to take a number with centillion (short-scale) zeros (which, if I'm not mistaken, is 10 to centillionth power). What fraction or percentage of g₁ does that number constitute? Otherwise, is it possible to represent a fraction/percentage of g₁ using large number names? Thanks. 212.180.235.46 (talk) 12:18, 12 November 2022 (UTC)

The number is incomprehensibly larger than 10 to the centillionth power. Expressing the latter number as a fraction of g₁ would require more zeros after the decimal point than the number of quarks that can fit in 10 to the centillionth observable universes. (Insert sound effect of head exploding.)  --Lambiam 15:25, 12 November 2022 (UTC)

(edit conflict) Graham's number, and even the number of digits in Graham's number defy attempts to quantify it in ordinary notation. The percentage you're talking about would be equally hard to comprehend because it would be so small. The good news is that, other than being an example of a extremely large number used in a mathematical proof, it has no real use and the numbers you encounter in real life are more tractable in terms of notation. Humans really aren't evolved to comprehend large numbers. I can imagine the number of people in a stadium, say 100,000. Then the number of people on Earth is about a stadium full of people for each person in a stadium (10,000,000,000), but that's just a calculation, not something I can actually imagine. --RDBury (talk) 15:27, 12 November 2022 (UTC)

As to imagining Earth's population canned and ready for shipment, see "7.3 Billion People, One Building",  --Lambiam 15:56, 12 November 2022 (UTC)

Graham's number itself is kind of mostly of historical interest. The bound for that problem has since been greatly improved, and proofs of other problems have since involved even larger numbers. And given that the lower bound for the problem is only 13, there's always the chance that the actual answer to the problem that Graham's number came from is laughably small. :) Double sharp (talk) 00:22, 14 November 2022 (UTC)

Graham's number is g₆₄, not the g₁ the OP mentions. But $g_{1}=3\uparrow \uparrow \uparrow \uparrow 3$. -- SGBailey (talk) 13:10, 14 November 2022 (UTC)

Does it make much difference for us ;-) As a person opening a dam said after saying, this dam holds ten million gallons of water, and the engineer beside him said ten thousand million, Oh what is a thousand in so many millions ;-) By the way, the number of people in the world is projected to go over 8 billion sometime thie week. NadVolum (talk) 13:28, 14 November 2022 (UTC)

"Any large number is finite, and you can start thinking about it as 3." – John Conway. Double sharp (talk) 17:16, 14 November 2022 (UTC)

• I think the OP is asking "what proportion of g₆₄ is g₁?" Basically, the relationship between 1 and g₁ is the same as the relationship between g₁ and g₂; which is to say that if you start at 1, and build the set of operations to get to g₁, you then take that entire set of operations and apply them g₁ to get to g₂. Then, you take everything you did to get from 1 all the way to g₂ and apply that to g₂, and you get g₃. Repeat that process, each "g" step applying the full set of operations you took to get to g_n, and applying them to g_n to get to g_n+1, and do that ALL THE WAY to g₆₄. That's Graham's number. It should be noted that if I wanted to compare g₁ to g₂ as a fraction, I would need more atoms than exist in the known universe just to make the ink required to write the denominator of THAT fraction. These scales are stupid. --Jayron32 18:46, 14 November 2022 (UTC)