In addition to losing distrubutivity, you also lose multiplicative inverses, so the non-zero elements no longer form a group. People might intuitively not mind lack of inverses, but might mind the lack of cancellation that goes with it: -2x =- -2y does not imply x=y, so linear equations like -2x = 6, or -2x = -6, can have 0 or 2 solutions instead of 1. But for a non-mathematician, the properties of the structure as a whole are much less useful than it's ability to be used to calculate answers to real problems. From that point of view, a system where multiplication and addition yield almost the same results they do now, except that 234 * 456 is 106724 instead of 106704 is almost as good; it almost always gives the right answer. It's not easy to come up with a simple real-world problem that, in it's most natural mathematical formulation, involves multiplying two negative numbers. But I think the real question is what the benefits of the "negative times negative equals negative" rule are. Without some benefit, it seems likje the 234 * 456 = 106724 system, only worse, because it gives the wrong answer more often, and things like distributivity and cancellation fail more often. Andy On Thu, Aug 25, 2016 at 11:38 AM, James Propp <jamespropp@gmail.com> wrote:
That's right, distributivity is lost (or rather severely circumscribed) in the bizarro world. But I'm trying to write something for the sort of person whose reaction to "You lose distributivity" might be "So what?" or "Huh?"
Losing distributivity more concretely entails things like losing the ability to square 999 in your head (or at least having it be harder to do). This may not impress the "Huh?" crowd either, but I think it comes closer.
Jim
On Thu, Aug 25, 2016 at 10:18 AM, Mike Stay <metaweta@gmail.com> wrote:
The structure you describe violates distributivity: (-1)(5 + -3) = -2 (-1)(5) + (-1)(-3) = -8 If you mod out by distributivity, you end up with the characteristic 2 field GF(2).
A different approach would be to take the semiring R^{>= 0} = ([0, infty), +, 0, *, 1), then adjoin an idempotent element n (for 'negative') that commutes with everything. The multiplication rule becomes (a + nb) * (c + nd) = ac + n(ad + bc + bd). Here, modding out by a + na = 0 would give GF(2).
On Wed, Aug 24, 2016 at 8:20 PM, James Propp <jamespropp@gmail.com> wrote:
I just came across the book
Negative Math: How Mathematical Rules Can Be Positively Bent <http://www.goodreads.com/book/show/547416.Negative_Math#bookDetails> by Alberto Martinez. Or rather, I came across a description of it. Does anyone have a copy? I gather that it does more or less what I propose, although it appears to take a less judgmental view of the defects of "bizarro arithmetic" than I would.
Jim Propp
On Wednesday, August 24, 2016, James Propp <jamespropp@gmail.com> wrote:
What I mean is the set |R equipped with two operations + and *, with + defined in the usual way and with * defined in ALMOST the usual way but with the twist that for all a,b > 0, -a (aka 0-a) times -b equals -(ab).
Jim Propp
On Wednesday, August 24, 2016, Dan Asimov <dasimov@earthlink.net <javascript:_e(%7B%7D,'cvml','dasimov@earthlink.net');>> wrote:
On Aug 24, 2016, at 3:56 PM, Mike Stay <metaweta@gmail.com> wrote:
On Wed, Aug 24, 2016 at 4:22 PM, Eugene Salamin via math-fun <math-fun@mailman.xmission.com> wrote: > Better yet, can anyone write about the usefulness of such a mathematical object?
Characteristic 2 fields satisfy -1 * -1 = -1 and have lots of applications, but I don't know how they're "bad".
Many papers on algebra require fields to have "characteristic not equal to 2" and never look back.
These results holding only for p != 2 have got me very curious about what happens for p=2.
So: What are some basic results in algebra where the cases characteristic 2 and unequal to 2 come out different?
Dan
> On Aug 24, 2016 3:18 PM, James Propp <jamespropp@gmail.com> > > Has anyone written in an accessible vein about all the bad things that > happen when you decree that minus times minus equals minus instead of plus? > > I might do this in my September blog post but I'm hoping someone else has > beaten me to it.
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