@iam.cool13: slaying x

kayla mclean
kayla mclean
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Sunday 15 January 2023 03:38:42 GMT
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thiccckynicccky
noob :
The pins lol
2023-01-15 09:07:17
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buddyhealey
Buddy healey :
Oh my 😜😜
2023-01-15 06:25:10
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thiccckynicccky
noob :
OMG
2023-01-15 09:07:12
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Graham’s number is one of the largest finite numbers ever used in a serious mathematical proof, introduced by mathematician Ronald Graham while working on a problem in an area of mathematics called Ramsey theory. It is not “infinite”—it is a specific, exact finite number—but it is so unimaginably large that the observable universe is nowhere near big enough to write it down in ordinary decimal notation. Even the number of atoms in the observable universe (about 10⁸⁰) is tiny compared with Graham’s number. To understand why it is so enormous, it helps to build up from smaller ideas. First, multiplication is repeated addition: 3 × 4 = 3 + 3 + 3 + 3 = 12. Exponentiation is repeated multiplication: 3⁴ = 3 × 3 × 3 × 3 = 81. Tetration is repeated exponentiation: ³↑↑3 = 3^(3^3) = 3²⁷ = 7,625,597,484,987. Then comes pentation, which repeats tetration, then hexation, and higher operations. Instead of inventing a new symbol for every operation, mathematician Donald Knuth created Knuth’s up-arrow notation. One arrow means exponentiation: 3 ↑ 3 = 3³ = 27. Two arrows mean tetration: 3 ↑↑ 3 = 3^(3³) = 3²⁷. Three arrows mean repeating tetration. Four arrows mean repeating the three-arrow operation, and every extra arrow creates an operation vastly more powerful than the last. Even something like 3 ↑↑↑ 3 is already incomprehensibly larger than numbers such as a googol (10¹⁰⁰) or even a googolplex (10^(10¹⁰⁰)). Graham’s number is built by repeatedly increasing the number of arrows themselves. First define G₁ = 3 ↑↑↑↑ 3 (four arrows). This number is already beyond any practical comprehension. Next define G₂ = 3 ↑^(G₁) 3, meaning the number of arrows between the 3s is not four anymore—it is G₁ arrows. Since G₁ is already unimaginably huge, G₂ uses that many arrows. Then G₃ uses G₂ arrows. This continues recursively: Gₙ = 3 ↑^(Gₙ₋₁) 3. Graham’s number is G₆₄. That means the process of replacing the number of arrows with the previous result is repeated 64 times. The overwhelming size of Graham’s number does not come from the 3s themselves but from the explosive growth in the number of arrows. By the time you reach G₂, the number of arrows is already so large that describing it is effectively impossible using ordinary mathematical language. By G₆₄, the result is far beyond anything that could ever be physically represented. You could never write all its digits because there are nowhere near enough particles, space, or time in the universe to do so. Even if every Planck-length-sized region of space stored trillions of digits and every moment since the Big Bang was spent writing more, you would not come remotely close. Despite its size, mathematicians can still determine some properties of Graham’s number. For example, its last decimal digit is known to be 7, and other ending digits can also be computed using modular arithmetic without calculating the entire number. Graham’s number was once listed in the Guinness World Records as the largest number ever used in a mathematical proof, although later proofs have involved even larger finite numbers such as those arising from the TREE sequence and Busy Beaver function, which grow incomparably faster. Graham’s number nevertheless remains one of the most famous examples of an unimaginably large finite number because it demonstrates how quickly mathematical operations can outgrow all physical intuition. #fyp #views #viral #kebab #rampage
Graham’s number is one of the largest finite numbers ever used in a serious mathematical proof, introduced by mathematician Ronald Graham while working on a problem in an area of mathematics called Ramsey theory. It is not “infinite”—it is a specific, exact finite number—but it is so unimaginably large that the observable universe is nowhere near big enough to write it down in ordinary decimal notation. Even the number of atoms in the observable universe (about 10⁸⁰) is tiny compared with Graham’s number. To understand why it is so enormous, it helps to build up from smaller ideas. First, multiplication is repeated addition: 3 × 4 = 3 + 3 + 3 + 3 = 12. Exponentiation is repeated multiplication: 3⁴ = 3 × 3 × 3 × 3 = 81. Tetration is repeated exponentiation: ³↑↑3 = 3^(3^3) = 3²⁷ = 7,625,597,484,987. Then comes pentation, which repeats tetration, then hexation, and higher operations. Instead of inventing a new symbol for every operation, mathematician Donald Knuth created Knuth’s up-arrow notation. One arrow means exponentiation: 3 ↑ 3 = 3³ = 27. Two arrows mean tetration: 3 ↑↑ 3 = 3^(3³) = 3²⁷. Three arrows mean repeating tetration. Four arrows mean repeating the three-arrow operation, and every extra arrow creates an operation vastly more powerful than the last. Even something like 3 ↑↑↑ 3 is already incomprehensibly larger than numbers such as a googol (10¹⁰⁰) or even a googolplex (10^(10¹⁰⁰)). Graham’s number is built by repeatedly increasing the number of arrows themselves. First define G₁ = 3 ↑↑↑↑ 3 (four arrows). This number is already beyond any practical comprehension. Next define G₂ = 3 ↑^(G₁) 3, meaning the number of arrows between the 3s is not four anymore—it is G₁ arrows. Since G₁ is already unimaginably huge, G₂ uses that many arrows. Then G₃ uses G₂ arrows. This continues recursively: Gₙ = 3 ↑^(Gₙ₋₁) 3. Graham’s number is G₆₄. That means the process of replacing the number of arrows with the previous result is repeated 64 times. The overwhelming size of Graham’s number does not come from the 3s themselves but from the explosive growth in the number of arrows. By the time you reach G₂, the number of arrows is already so large that describing it is effectively impossible using ordinary mathematical language. By G₆₄, the result is far beyond anything that could ever be physically represented. You could never write all its digits because there are nowhere near enough particles, space, or time in the universe to do so. Even if every Planck-length-sized region of space stored trillions of digits and every moment since the Big Bang was spent writing more, you would not come remotely close. Despite its size, mathematicians can still determine some properties of Graham’s number. For example, its last decimal digit is known to be 7, and other ending digits can also be computed using modular arithmetic without calculating the entire number. Graham’s number was once listed in the Guinness World Records as the largest number ever used in a mathematical proof, although later proofs have involved even larger finite numbers such as those arising from the TREE sequence and Busy Beaver function, which grow incomparably faster. Graham’s number nevertheless remains one of the most famous examples of an unimaginably large finite number because it demonstrates how quickly mathematical operations can outgrow all physical intuition. #fyp #views #viral #kebab #rampage

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