I ordered Prof. Alberto A. Martinez' "Negative Math" book - b/c of its noncommutative secret!

 Negative Math: How Mathematical Rules Can Be Positively Bent

https://www.martinezwritings.com/m/Kinematics.html 

I was very impressed by Prof. Alberto Martinez book on the background of Einstein's relativity.

 In that book he mentions working in Dinkytown by University of Minnesota at the Italian restaurant Mama Rosa's. I think he was the dishwasher - I can't remember and there is no way to read or search the book - maybe through Amazon? Nope.

Anyway I had read that book through Interlibrary loan - it's very cool.

So I was just reading the googlebook preview about the Replacements - and Bob Stinson also worked at Mama Rosa's as a dishwasher - and he was forced to quit until he reformed himself and got his job back...

The amazon book description of "Negative math" makes no mention of noncommutativity! But Martinez' website does.

 In my book Negative Math (Princeton University Press, 2005), I analyzed the history of negative numbers and I formulated a coherent algebra in which minus times minus is minus. In this new algebra, multiplication of numbers with different signs is non-commutative, which introduces symmetries that are missing in traditional numerical algebra.

amazon review excerpts: 

 A very accessible (relatively speaking) way to understand, question and contextualize the most important idea in Math: why is -2 -2 = 4 and not -4 ?

And once you read the book, you realize you can go both ways (with pros and cons).

 fascinating!

 an arbitrary decision that makes some operations easier and some way more confusing (the need for i, other asymmetries, etc)..

  The author's project in this book is to motivate then introduce an alternative, non-commutative arithmetic which he feels may have use in some mathematical descriptions of the physical world. He does not, however, present this in a mathematically sophisticated way - for example, there is no mention of groups, rings, or fields, and there is no appendix containing a concise presentation of his ideas.

The book is written as a popularization and it remains at that level throughout. His attempt to motivate his alternative arithmetic involves trying to create cognitive dissonance in the reader: to convince the reader there is something unsatisfactory with the way we calculate with negative and imaginary numbers. He does this by obdurately insisting upon questionable physical interpretations of arithmetical equations, then complaining about their obscurity and pointing at the mathematics as the culprit behind the confusion.

 . It not only explains WHY a minus times a minus is a plus, but also reveals that (essentially) the same reason is responsible for imaginary numbers and also vector algebra and its seemingly archaic rules. It opens the doors to a very deep understanding of mathematics.

 Martinez then shows why mathematicians had troubles with negatives by demonstrating what happens if a negative times a negative _isn't_ positive. This _REALLY_ helped me because I learn best when I can see the "wrong" way of doing something and what kind of results it produces. It showed that you can have two systems that are different but equally valid. It also helped reveal why vector algebra is the way it is; it is essentially a "wrong" way of doing regular algebra but that, critically, works.