** Using WeBWorK **

Thanks Davide,

Actually we've thought harder about how we want to use bizarro arithmetic in general, and it's led to some interesting realizations. I'm not sure if this is the right place to lay them out, but an overview would be that we basically want arithmetic operations that satisfy the field axioms but also interact with usual arithmetic in special ways. I can at least quickly describe one application:

For factoring problems, we are turning on bizarro + and -, but leaving * and ^ alone. Now (x+1)^2, (x+1)(1+x), (x+1)(x+1) are all equivalent, but x^2+2x+1 and x^2+x+x+1 are different from these (and to each other). Basically, regular * and bizarro + do not satisfy the distributive law, and that makes this approach to factoring (seem to) work (so far). However, we realized we still want regular * and bizarro + to obey the distributive law when the situation is "distributing a negation". Otherwise (x-1)(x-1) and (1-x)(1-x) are different.

(Side note: for factoring we still have to do a special attack for the response to (2x - 2)(x/2 - 1/2).)

Thinking more about it all, our latest proposal is that the bizarro operations be defined as:
where f is an odd one-to-one function on R. If f is a simple power function, there is too much interaction between the bizarro arithmetic and regular arithmetic. As you have noted, if f involves logarithms or exponentiation, there are machine arithmetic issues. So the simplest thing we've come up with is
and its inverse function that has an explicit formula found from applying the cubic formula. At all steps the numbers are kept within roughly their original value cubed.

Your observation about the numbers produced from your seed make me think we should go with f^-1(x) being x^3+x, rather than f(x). Then 16^8 will roughly translate to cube(cubeRoot(16)^cuberoot(8)), and it opens the door for slightly larger bases and exponents.

We're nearing the end of our basic algebra problem library development, and I look forward to tidying up our investigations like these and seeing if they have a place in the general distribution.