# 1. Notes On Up and Down Properties

I was back-packing around Australia one year and had a bit of time on my hands but had thought to bring along a typewriter. In those days that was "high-tech" and "AT Computers" seemed like a scientific marvel stolen from the hands of E.T.

So I began to think about numbers because

 They were cheap and fitted inside my travel budget They tend to be stable with time and never really go out of date Their batteries never go flat

I soon realized that having only four binary operators (+, - , * , /) must be a bit limiting. After all would we only want 4 fingers (like the Simpsons). What if we only had 2?

The Roman empire was close to such a limitation. After all, how do you multiply

x = IV · MCX?

If a farmer has 7 fields of 20 sheep, surely 7 · 20 = 140 would be a nice trick to know

So I thought to extend the set of binary operators to include two more - Ûp and Ðown

(+, -, · , /, , )

Interestingly, I found that ( · , /, , )   had the same "group properties" as (+, -, · , / ). So I thought I'd prove these properties on my typewriter.

I even got them signed by a Justice of the Peace!

# 3. Summary

I have listed my proofs for the group properties , and  .

I have pursued these operators further further into the complex domain, a Up-Taylor series and later on to a new form of Calculus. I think I'll add these developments to my web chapters for Conjecturals later on.