Tetration

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Tetration is the hyperoperation which comes after exponentiation.[1] Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle ^{x}{y}} means y exponentiated by itself, (x-1) times.[2][3] List of first 4 natural number hyperoperations:

  1. Addition
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle a + n = a\!\underbrace{''{}^{\cdots}{}'}_n}
    a succeeded n times.
  2. Multiplication
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle a \times n = \underbrace{a + a + \cdots + a}_n}
    a added to itself, n times.
  3. Exponentiation
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle a^n = \underbrace{a \times a \times \cdots \times a}_n}
    a multiplied by itself, n times.
  4. Tetration
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle {^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n}
    Note (operator associativity): Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle {^{n}a} = \underbrace{(a^{(a^{(\cdot^{\cdot^{(a)...)}}}}}_n}
    a exponentiated by itself, n-1 times.

The above example is read as "the nth tetration of a".

Example[change | change source]

For the example, addition is assumed.

  1. Failed to parse (Conversion error. Server ("https://simple.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {^{2}3}=}
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle {3^{3}} = }
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle {3 \times 3 \times 3} = }
    Failed to parse (Conversion error. Server ("https://simple.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {3\times (3+3+3)}=}
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle {3 \times {9}} = }
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle 27 }

References[change | change source]