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# Hyper4

hyper4 is an notation that describes power towers and large numbers, in terms of an extension of standard operators.

## Derivation of the notation

It can be seen as an answer to the question "what's next in this sequence: summation (+), multiplication (×), exponentiation (^),…?" Noting that
• a+b = 1+(a+(b-1))
• a×b = a+(a×(b-1))
• a^b = a×(a^(b-1))
recursively define an infix triadic operator a(n+1)b = a(n)(a(n+1)(b-1)) with a(1)b = a+b
then define hypern(a,b)=a(n)b and hyper(a,n,b)=a(n)b

The hypern family and hyper are very closely related to Knuth's up-arrow notation.

The family has not been extended to real numbers for n>3, due to nonassociativity in the "obvious" ways of doing it.

Known aliases for hyper4 include tetration, superpower, superdegree, and powerlog; other notation, hyper4(a,b)=ba.

## Extensions to the notation

These operators can be generalised in another way: careful readers will ask "what about expansions to the opposite side?" Since
• a+b = (a+(b-1))+1
• a×b = (a×(b-1))+a
• a^b = (a^(b-1))×a
define a(n+1)b = (a(n+1)(b-1))(n)a with a(1)b = a+b

But this suffers a kind of collapse, failing to form the "power tower" traditionally expected of hyper4:
a(4)b = a^(a^(b-1))

How can a(n)b and a(n)b suddenly diverge for n>3? This because of a symmetry called associativity that's defined into + and × (see field) but which ^ lacks. It is more apt to say the two (n)s were decreed to be the same for n<4. (On the other hand, one can object that the field operations were defined to mimic what had been "observed in nature" and ask why "nature" suddenly objects to that symmetry…)

The other degrees do not collapse in this way, and so this family has some interest of its own as lower (perhaps lesser or inferior) hyper operators.