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. Jim Propp
Better yet, can anyone write about the usefulness of such a mathematical object? -- Gene From: James Propp <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Wednesday, August 24, 2016 3:18 PM Subject: [math-fun] Bizarro arithmetic 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. Jim Propp
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".
From: James Propp <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Wednesday, August 24, 2016 3:18 PM Subject: [math-fun] Bizarro arithmetic
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.
Jim Propp
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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.
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> wrote:
On Aug 24, 2016, at 3:56 PM, Mike Stay <metaweta@gmail.com <javascript:;>> wrote:
On Wed, Aug 24, 2016 at 4:22 PM, Eugene Salamin via math-fun <math-fun@mailman.xmission.com <javascript:;>> 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 <javascript:;>>
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.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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What's a good symbol to use for bizarro multiplication, in contexts where I want to use both it and normal multiplication (for instance, showing that each of them can be defined in terms of the other)? The ideal symbol should be witty (without being offensive). Maybe I should just use whatever Martinez uses, since presumably he addresses this issue of two-way translation, but I don't have a copy of the book in hand yet. Jim Propp On Thursday, August 25, 2016, 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 <javascript:_e(%7B%7D,'cvml','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 <javascript:_e(%7B%7D,'cvml','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 <javascript:_e(%7B%7D,'cvml','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');> <javascript:_e(%7B%7D,'cvml','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 <javascript:_e(%7B%7D,'cvml','metaweta@gmail.com');>> wrote:
On Wed, Aug 24, 2016 at 4:22 PM, Eugene Salamin via math-fun <math-fun@mailman.xmission.com <javascript:_e(%7B%7D,'cvml','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 <javascript:_e(%7B%7D,'cvml','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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-- Mike Stay - metaweta@gmail.com <javascript:_e(%7B%7D,'cvml','metaweta@gmail.com');> http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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I'd use \cdot · for multiplication and \ boxdot ⊡ for the weird multiplication. On Thu, Aug 25, 2016 at 9:51 AM, James Propp <jamespropp@gmail.com> wrote:
What's a good symbol to use for bizarro multiplication, in contexts where I want to use both it and normal multiplication (for instance, showing that each of them can be defined in terms of the other)? The ideal symbol should be witty (without being offensive).
Maybe I should just use whatever Martinez uses, since presumably he addresses this issue of two-way translation, but I don't have a copy of the book in hand yet.
Jim Propp
On Thursday, August 25, 2016, 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 <javascript:_e(%7B%7D,'cvml','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 <javascript:_e(%7B%7D,'cvml','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 <javascript:_e(%7B%7D,'cvml','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');> <javascript:_e(%7B%7D,'cvml','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 <javascript:_e(%7B%7D,'cvml','metaweta@gmail.com');>> wrote: > > On Wed, Aug 24, 2016 at 4:22 PM, Eugene Salamin via math-fun > <math-fun@mailman.xmission.com <javascript:_e(%7B%7D,'cvml','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 <javascript:_e(%7B%7D,'cvml','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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-- Mike Stay - metaweta@gmail.com <javascript:_e(%7B%7D,'cvml','metaweta@gmail.com');> http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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.
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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There's a SIAM review of the book here. https://www.siam.org/pdf/news/1042.pdf On Wed, Aug 24, 2016 at 9:20 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.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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participants (6)
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Andy Latto -
Dan Asimov -
Eugene Salamin -
James Propp -
Mike Stay -
Paul Palmer