Addition, Multiplication, and What’s Next, and What’s In Between

We all know about adding.

A+B
A+B = B+A
A+0 = A
(A+B)+C = A+(B+C)

A nice operation having pleasant symmetries and a do-nothing value.

We all learned early in elementary school that multiplying is repeated addition

A*B = A+A+…+A repeated for a total of B times

This too is a pleasant operation, much like addition, though with a different magic value that lets the other number through unchanged.

A*B = B*A
(A*B)*C = A*(B*C)
A*1 = A

Okay, cool, so the next thing any curious kid asks is: what is repeated multiplication?

Powers, we were told. Exponents. But yech, this is not pretty!

A^B ≠ B^A
(A^B)^C ≠ A^(B^C)
A¹=A but no ?^B=B

Fortunately, there is a better way to define a “next” operation after multiplication. With this, there is a way to define the next one after that, … I’ll skip the reasoning and just state it:

A op_0 B = A + B
A op_1 B = A * B = exp( log(A) + log(B) )
A op_2 B = A ✻ B = exp(exp( log(log(A)) + log(log(B)) )
A op_3 B = … should be obvious …