What comes after exponents?

We use multiplication for repeated addition, and in turn use exponents for repeated multiplication. What topic comes after this, for repeated exponentials? Is there something my teachers are hiding from me?

Answer

The best answer is perhaps tetration though you might want to investigate also Ackermann’s function, Knuth’s up-arrow notation, and Conway’s chained arrow notation all of which capture the idea of growth beyond exponential growth in different ways.

It isn’t really being hidden from you because the uses of these ideas seem to come up in computing and combinatorics (the Hales-Jewett theorem and similar can be proved with Ackermann type bounds, though sometimes better bounds are available by trickier methods). Here is a discussion of an apparently simple problem from the 2010 IMO which generates beyond exponential rates of growth. You might want to try it first before reading the detail.

On the whole, though, the ideas involved run into problems of notation (which I have always thought similar to the problem of naming all the ordinal numbers – the notation just runs out and you need something new).

Attribution
Source : Link , Question Author : Monte Carlo , Answer Author : Mark Bennet

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