Tetration

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Welcome to the Home of Tetration

         This site is about Tetration. Tetration is an obscure mathematical operation that currently falls under the category of Pure Mathematics, because it has almost no application. Even so, many mathematicians have been interested in tetration, because of the historical significance of its relatives in the hyper-operation sequence, and because it is easy to make very large numbers with tetration.
         The hyper-operation sequence is the technical name for the list of mathematical operators you already know: addition (a + b) is the first, multiplication (a*b = ab) is the second, and exponentiation (a^b = a^b) is the third. The reason why these are in the same list, and in this order is because you can think of multiplication as repeated addition, and you can think of exponentiation as repeated multiplication. The fourth operation in this sequence is called tetration, because it is the fourth (tetra). Tetration then, would be repeated exponentiation. The problem is that there are two ways to do that, lets say for repeating exponentiation on 3 threes:

The bottom-up method	Evaluate 3^3^3 as 27^3 = 19683
The top-down method	Evaluate 3^3^3 as 3^27 = 7625597484987

         From Algebra, we know that (a^b)^c = a^(bc), so the bottom-up method is not really a new operator, but is simply expressible in terms of exponentiation. Munafo uses the notation x₍₄₎y for this "lower" form of repeated exponentiation, and gives the relationship:

The bottom-up method in general	x(4)y = x^x^(y−1)
The top-down method in general	x(4)y = yx

         When we talk about tetration, we are only talking about the top-down method, because this makes a new operation. Now that we know what tetration is, it is easy to see why such an obscure mathematical operation would be studied for hundreds of years, even without any substantial progress.
         Another problem with tetration is that the values at real y in ^yx are unknown. Many extensions of tetration have been devised, but none so far have satisfied all the requirements tetration should have. Hopefully, my paper will provide new insight into tetration and its inverse, the super-logarithm, by providing a definition of an extension that does satisfy those requirements, and a method that approximates that definition. For more information, read my paper, available from the menu above.