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They tend to be stable with time and never really go out of date | |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**