Comments on: Climbing the ladder of hyper operators: tetration

By: sennahoj

sennahoj — Wed, 11 Mar 2015 12:05:56 +0000

begin with adition of natural numbers N and the inverse leads you to the negative numbers so you have to define Z after multiplication the universal inverse forces the set of Q and than with the inverse of exponentiation you end up with C

ok, you see where its going?

is there an bigger set of numbers you need to solve all inverse tetration?

of cause T(a,b) with a and b are complex numbers.

By: George Daccache

George Daccache — Sat, 10 Jan 2015 06:33:19 +0000

Whoops, I may have been misunderstood – I’m not at the level of a mathematics researcher (yet?). By ‘research’ I meant an unpublished derivation of facts I just sat down and, well, derived. I never submitted a paper to a mathematical journal about tetration (again, for now – would they even accept the topic?) I guess that wouldn’t count as a reference, but it’s how I found out about the properties, so I just included it for completeness.

By: George Daccache

George Daccache — Sat, 10 Jan 2015 06:24:35 +0000

Fixed.

By: vzn

vzn — Fri, 09 Jan 2015 19:10:08 +0000

nice, great to see a nice meaty section on applications, although of course optional esp in mathematics. the ackermann function has a lot of connections in computer science. re "My own brief research on the properties of second-order tetration." why dont you link to that or ref something?

By: George Daccache

George Daccache — Thu, 08 Jan 2015 14:31:20 +0000

The whole definition of each operation as an iteration of the previous one is really the motivation – it was the question intuition led us to follow which led to the original concept of tetration. Of course, multiplication isn’t really repeated addition (“what’s 0.1 x 0.1?”), and this problem shows up when trying to define tetration from its predecessors, which is part of the reason why a rigorous definition of tetration for all real numbers hasn’t been made yet. So no, there isn’t any definition for tetration for all real arguments, and finding such a definition is actually the biggest problem faced in figuring out tetration (ironic, right?). Perhaps we could define it similarly to the generalization of multiplication and addition to the reals, but I’m not too sure since that would require a definition of tetration for rational numbers, which I’m afraid also isn’t agreed on.

By: George Daccache

George Daccache — Thu, 08 Jan 2015 13:49:55 +0000

You're right; I can't edit the post though, so I'll see if anyone else can fix it. I'm glad you liked the post, it really is an interesting topic.

By: mixedmath

mixedmath — Tue, 06 Jan 2015 18:27:18 +0000

I'm amused that this defines multiplication as repeated addition. It makes me wonder how annoying it would be to define tetration truly rigorously, for real arguments. Do you happen to know?

By: Clayton

Clayton — Tue, 06 Jan 2015 18:16:09 +0000

At the end of the definition, you accidentally compute $3^3=9$, when it should be $3^3=27$. Aside from that small miscalculation, very nice post indeed.

By: George Daccache

George Daccache — Tue, 06 Jan 2015 13:47:39 +0000

Thank you! It certainly seems as though we can generalize it as such (just by directly substituting), which provides us with a rather nice formula indeed, as well the corollary that $${}^0 x = 1$$

By: George Daccache

George Daccache — Tue, 06 Jan 2015 13:41:06 +0000

I'm glad you liked it. :)