Knuth's up-arrow notation

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Knuth's up-arrow notation is a way of saying 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
    a \uparrow^{1} b = a^b = \underbrace{a \times a \times \cdots \times a}_{b \ times}
    a multiplied by itself, b times.
  2. Tetration
    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}
    a exponentiated by itself, b times.
  3. Third level
    a \uparrow^{3} b = a \uparrow \uparrow \uparrow b = \underbrace{a \uparrow \uparrow (a \uparrow \uparrow (a \uparrow \uparrow \ldots a) \ldots ) )}_{b \ times}
  4. etc

References[change | edit source]

  1. 1.0 1.1 "Arrow Notation - from Wolfram MathWorld". Mathworld.wolfram.com. 2011-10-24. http://mathworld.wolfram.com/ArrowNotation.html. Retrieved 2011-10-30.
  2. "SS > factoids > big numbers". Users.cs.york.ac.uk. 1998-07-07. http://www-users.cs.york.ac.uk/susan/cyc/b/big.htm#up. Retrieved 2011-10-30.