Knuth's up-arrow notation

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Knuth's up-arrow notation is a way of expressing very big numbers.[1] It was made by Donald Knuth in 1976.[1] It is related to the hyperoperation sequence. The notation is used in Graham's number.

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

  1. Exponentiation
    [math]\displaystyle{ a \uparrow^{1} b = a^b = \underbrace{a \times a \times \cdots \times a}_{b \ times} }[/math]
    a multiplied by itself, b times.
  2. Tetration
    [math]\displaystyle{ a \uparrow^{2} b = a \uparrow \uparrow b = {^{b}a} = \underbrace{(a^{(a^{(\cdot^{\cdot^{(a)...)}}}}}_{b \ times} = \underbrace{(a \uparrow^1 (a \uparrow^1 (... \uparrow^1 a)...)}_{b \ times} }[/math]
    a exponentiated by itself, b times.
  3. Third level
    [math]\displaystyle{ a \uparrow^{3} b = a \uparrow \uparrow \uparrow b = \underbrace{a \uparrow \uparrow (a \uparrow \uparrow (a \uparrow \uparrow \ldots a) \ldots ) )}_{b \ times} }[/math]
  4. etc

This notation is used to describe the incredibly large Graham's Number

References

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