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| They tend to be stable with time and never really go out of date |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!
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.
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© Ian R Scott 2007 - 2008
