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Is graham’s number bigger than planck’s?

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Yes, Graham’s number is significantly larger than Planck’s constant.

And now, in greater depth

Yes, Graham’s number is significantly larger than Planck’s constant. In fact, it is so large that it is practically incomprehensible.

Graham’s number was first introduced by mathematician Ronald Graham in 1971. It is a theoretical upper bound for a problem in the mathematical field of Ramsey theory, and is often cited as the largest number ever used in a proof. Graham’s number is so large that even describing it is difficult. It is a product of a long chain of exponents and multiplications, and the final result is a number with so many digits that it is impossible to write down in full.

On the other hand, Planck’s constant is a fundamental constant of nature that plays a crucial role in quantum mechanics. It is a physical constant that relates the energy of a photon to its frequency, and is used in a variety of calculations involving atomic and subatomic particles.

To get a better sense of just how much larger Graham’s number is than Planck’s constant, we can look at some approximate values:

Constant Approximate Value
Graham’s number 3↑↑↑↑3 (or approximately 10^10^10^10)
Planck’s constant 6.626 x 10^-34 joule-seconds

As for a quote on the topic, here is what mathematician Ron Graham himself had to say about his famous number: “I was just thinking about some geometric Ramsey theory, and this number came up. And then, when I tried to explain it to someone, it took me half an hour to get the point across. So then I realized that this was something special.”

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In summary, while both Graham’s number and Planck’s constant are important in their respective fields, they are on vastly different scales of magnitude. Graham’s number is a theoretical construct that is so large it is virtually incomprehensible, while Planck’s constant is a fundamental constant of nature that plays a crucial role in quantum mechanics.

See a video about the subject

Mathematicians Tony Padilla and Matt Parker discuss arrow notation, used to represent very large numbers and particularly in combinatorics problems. They discuss the concept of Graham’s number and its development as the maximum possible number of people needed to be in committees with certain conditions on connections, using arrow notation to show how the number increases and its scale, which lies between 6 and Graham’s number. Despite being smaller than infinity, which is currently used in mathematical proofs, Graham’s number is shockingly large, with only its last 500 digits known and its first digit unknown. The video ends with an interesting anecdote about Graham, who was a mathematician and circus performer.

Many additional responses to your query

Even n, the mere number of towers in this formula for g1, is far greater than the number of Planck volumes (roughly 10 185 of them) into which one can imagine subdividing the observable universe.

A short answer: approximately, it would take Graham’s number of observable universes.

A bit longer answer: please see my comment to another (somewhat similar) question: https://www.quora.com/Is-a-googolplex-raised-to-the-googolplexed-power-a-googolplex-times-larger-than-any-of-these-numbers-Graham%E2%80%99s-number-TREE-3-SSCG-3/answer/Konstantin-Beloturkin/comment/44511681. You are making the mistake which many other people do who do not fully realize the power of hyper-operators. What you are trying to do is to reduce the value of Graham’s number by applying to it:

1. logarithmation by base 10 (considering the number of decimal digits in it rather than the number itself);
2. division by some arbitrary small numbers (like the number of Planck volumes in the observable Universe which is roughly [math]8.46 ∙ 10^{184}[/math]) – and yes, I am not mistaken – it is a googologically very tiny number.
These operations (division and even logarithmation) are so unbelievably weak as compared to…

More interesting questions on the topic

What is bigger than Graham's number?
Answer will be: Other specific integers (such as TREE(3)) known to be far larger than Graham’s number have since appeared in many serious mathematical proofs, for example in connection with Harvey Friedman’s various finite forms of Kruskal’s theorem.
Similar
What is Graham's number equal to?
The answer is: What Is the Graham Number? The Graham number (or Benjamin Graham’s number) measures a stock’s fundamental value by taking into account the company’s earnings per share (EPS) and book value per share (BVPS). The Graham number is the upper bound of the price range that a defensive investor should pay for the stock.
What is the largest theoretical number?
In reply to that: A googolplex, is, in fact, equal to 10googol and can only be written in exponential format. A googol, which is equal to 10100, may also be written as 1010^2; the Planck cube number containable within the Universe can also be written as 1010^2,27, however a googolplex is 1010^100!
Is there a number bigger than Googolplexian?
Answer will be: What’s bigger than a googolplex? Even though a googolplex is immense, Graham’s number and Skewes’ number are much larger. Named after mathematicians Ronald Graham and Stanley Skewes, both numbers are so large that they can’t be represented in the observable universe.
Is Graham's number bigger than Planck's?
As an answer to this: 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.
How big is Graham's number?
Graham’s number is much larger than any other number you can imagine. It is so large that the observable universe is far too small to contain an ordinary digital representation of Graham’s number, assuming that each digit occupies one Planck volume which equals to about 4.2217\times 10^ {-105}\text { m}^ {3} 4.2217× 10−105 m3.
Is Graham's number bigger than a googolplex?
Graham’s number is also bigger than a googolplex, which Milton initially defined as a 1, followed by writing zeroes until you get tired, but is now commonly accepted to be 10 googol =10 (10100). A googleplex is significantly larger than the 48th Mersenne prime.
What is a Grahams number growth rate?
The reply will be: Forget about TREE (3), your number isn’t even bigger than G (65) which is Grahams number followed by G (64) number of arrows in between them. A grahams number growth rate is not more than fw+1 while there are not enough finite ordinals to represent growth rate of TREE (3).

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