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ⓘ Knuths up-arrow notation is a way of expressing very big numbers. It was made by Donald Knuth in 1976. It is related to the hyperoperation sequence. The notatio ..




                                     

ⓘ Knuths up-arrow notation

Knuths up-arrow notation is a way of expressing very big numbers. It was made by Donald Knuth in 1976. It is related to the hyperoperation sequence. The notation is used in Grahams number.

One arrow represents exponentiation, 2 arrows represent tetration, 3 for pentation, etc.:

  • Exponentiation a ↑ 1 b = a b = a × a × ⋯ × a ⏟ b t i m e s {\displaystyle a\uparrow ^{1}b=a^{b}=\underbrace {a\times a\times \cdots \times a} _{b\ times}} a multiplied by itself, b times.
  • Tetration a ↑ 2 b = a ↑ ↑ b = b a = a ⋅ ⋅ a) ⏟ b t i m e s = a ↑ 1 a ↑ 1. ↑ 1 a) ⏟ b t i m e s {\displaystyle a\uparrow ^{2}b=a\\uparrow b={^{b}a}=\underbrace {a^{a^{\cdot ^{\cdot ^{a)}}}}} _{b\ times}=\underbrace {a\uparrow ^{1}a\uparrow ^{1}.\uparrow ^{1}a)} _{b\ times}} a exponentiated by itself, b times.
  • etc
  • Third level a ↑ 3 b = a ↑ ↑ ↑ b = a ↑ ↑ a ↑ ↑ a ↑ ↑ … a …) ⏟ b t i m e s {\displaystyle a\uparrow ^{3}b=a\\\uparrow b=\underbrace {a\\uparrow a\\uparrow a\\uparrow \ldots a\ldots)} _{b\ times}}

This notation is used to describe the incredibly large Grahams Number.

                                     
  • exponentiation, tetration, etc. They are often written using Knuth s up - arrow notation Natural number level hyperoperations can be defined recursively
  • big even to write in scientific notation In order to be able to write it down, we have to use Knuth s up - arrow notation We will write down a sequence