No, Graham’s number is not the biggest number in the world, but it is one of the largest numbers ever used in a mathematical proof.
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Graham’s number is a massive number that was named after Ronald Graham, a mathematician who used it in a proof related to Ramsey theory. Despite its size, it is not the biggest number in the world, but it is one of the largest numbers ever used in a mathematical proof.
According to Dr. Aaron Cicourel, a mathematician at the University of California, San Diego, “To express Graham’s number in standard form, like with numbers we use every daywould require more space than the known universe has available.”
Here are some interesting facts about Graham’s number:
- The number is so large that even trying to comprehend it is practically impossible for the human mind.
- Using standard mathematical notation, it would take an enormous number of digits just to write out the “power tower” that is part of the definition of Graham’s number.
- The number was first described by Graham in 1971 and was used in a proof in 1977.
- The number is so big that it is impossible to calculate its digits using normal methods.
- Scientific American called Graham’s number “the largest specific positive integer ever to have served a definite purpose in a mathematical proof.”
- Graham’s number is not the biggest number in the world. In fact, there are numbers that are even larger, such as TREE(3) from the field of combinatorics.
While Graham’s number is not the biggest number in the world, it remains a fascinating and mind-boggling concept in the field of mathematics.
| Symbol | Definition |
|---|---|
x ↑ y |
exponentiation (x to the power of y) |
Kn |
the complete graph with n vertices (each vertex is connected to every other vertex) |
g |
the Graham function, which gives the smallest number of colors needed for a graph with n vertices and no single-color complete sub-graph with n vertices |
g1 |
the smallest number of colors needed for a graph with 6 vertices and no single-color complete sub-graph with 6 vertices |
g2 |
the smallest number of colors needed for a graph with g1 vertices and no single-color complete sub-graph with g1 vertices |
g3 |
the smallest number of colors needed for a graph with g2 vertices and no single-color complete sub-graph with g2 vertices |
g64 |
the smallest number of colors needed for a graph with g63 vertices and no single-color complete sub-graph with g63 vertices |
Graham's number |
g64 ↑↑↑↑ g64 (using Kn as n in the definition of g) |
Answer in the video
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.
There are also other opinions
Graham’s Number is so huge that it cannot be written down – the universe is simply not big enough. In fact, even specifying this number defies what’s possible using common mathematical notation. Instead, special notation has had to be developed.
Graham’s number holds the Guinness World Record for the biggest specific integer used in a published mathematical proof.
The number gained a degree of popular attention when Martin Gardner described it in the "Mathematical Games" section of Scientific American in November 1977, writing that Graham had recently established, in an unpublished proof, "a bound so vast that it holds the record for the largest number ever used in a serious mathematical proof."
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 4.2217times 10^ {-105}text { m}^ {3} 4.2217× 10−105 m3.
The word ‘Big’ is entirely inadequate to even begin to describe the enormity of Graham’s number….!
People like to ask: “How many digits in Graham’s number?”
See: How many digits are there in Graham’s Number? Is it possible to know how many digits that number has? [ https://www.quora.com/How-many-digits-are-there-in-Grahams-Number-Is-it-possible-to-know-how-many-digits-that-number-has ]
That question also doesn’t even begin to get how BIG Graham’s number is, because the answer (how many digits?) is itself a number so incredibly vast as to have no meaning.
In fact, you could probably spend all eternity asking “How many digits in the number that states how many digits are in the number that states how many digits are in the number… etc… that states how many digits are in Graham’s number” before you got to a number that made any kind of sense!
Just consider the first term in Graham’s Number, g1.
The first term is [math]3\uparrow \uparrow \uparrow \uparrow3[/math].
(See: Knuth’…
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So it’s a power tower that has 3 3 3 occurrences of in it, or about 7.6 trillion 3s. And there you have it. We just broke our brain.