Tetration

Tetration is the hyperoperation which comes after exponentiation. $^{x}{y}$ means y exponentiated by itself, (x-1) times. List of first 4 natural number hyperoperations:

1. Addition

   $a+n=a\!\underbrace {''{}^{\cdots }{}'} _{n}$

   a succeeded n times.

2. Multiplication

   $a\times n=\underbrace {a+a+\cdots +a} _{n}$

   a added to itself, n times.

3. Exponentiation

   $a^{n}=\underbrace {a\times a\times \cdots \times a} _{n}$

   a multiplied by itself, n times.

4. Tetration

   ${^{n}a}=\underbrace {a^{a^{\cdot ^{\cdot ^{a}}}}} _{n}$

   Note (operator associativity): ${^{n}a}=\underbrace {(a^{(a^{(\cdot ^{\cdot ^{(a)...)}}}}} _{n}$

   a exponentiated by itself, n-1 times.

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

Example

For the example, addition is assumed.

1. ${^{2}3}=$

   ${3^{3}}=$

   ${3\times 3\times 3}=$

   ${3\times (3+3+3)}=$

   ${3\times {9}}=$

   ${3+3+3+3+3+3+3+3+3=9+9+9}=$

   $27$

References

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