Another way to think of this is that for each number there is a mapping from all real numbers to themselves defined by multiplying everything by it: f_c: R —> R via f_c(x) = cx . In some sense picking up the real line and putting it back on itself in the reverse direction is simpler than folding it in half. But also, the graph of f(x,y) = x o y is *very* strange (where o denotes multiplication where -- = -). —Dan
On Aug 25, 2016, at 8:38 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.