In a mathematical system like this, multiplication may not be associative, commutative, or distributive, and may lost other useful properties as well. This would make many basic mathematical proofs fail, or indeed would lead to them being false entirely. In my book that makes the current system much more useful.
The more I think about it, the more almost everything in mathematics in the past 500 years would fall apart. I fail to see ANY benefit for math overall.
Given such a system, I have to wonder what dx[sup]2[/sup]/dx would be. |x[sup]3[/sup]|?
… |2x|?
sigh
[QUOTE=groman]
Well yeah, all of proven mathematics is a logical chain that stems from several independant definitions. You can always change the definitions if you want and still have valid mathematics, just most of the time it’s not going to be all that interesting./QUOTE]
On the other hand, to break this property takes a lot more than just redifining that line. “negative times negative is positive” is not an axiom, but a theorem dependent upon a number of more basic statements. To change this you’d have to change those and pretty much nothing in the positive numbers would behave the way we think they should either.
[QUOTE=Mathochist]
Are you positive? You’re sounding pretty negative about all this. 
[QUOTE=DSYoungEsq]
Har. If you want I can show you how it all goes Pete Tong…
The algebra of R is best viewed visually as vectors beginning at the origin(0) and ending with the real number®. Hence, the real number “r” is really a vector starting at 0 and ending at r. If “vector” squicks you out, think “ray” from geometry. In this setting, negatives indicate direction relative to 0. Negative numbers lie to your left side of the real line, and the positive numbers lie to your right side…
<----------------(negative)----0--------(positive---------->
A single real number® is really just a vector in R, with a sign and a magnitude:
<<<------------------------------0---------->®--------------------->>> {r positive}
<<<-------------------®<------0------------------------------------>>> {r negative}
Then adding real numbers becomes a matter of displacement, s+r :
<<<------------------------------0---------->®-------->(s)------------->>> {s,r positive}
<<<------------------------------0----------------------->(s+r)---------->>>
<<<------------------------------0----------------------->®------------->>> {s,r positive}
<<<------------------------------0--------------->(s)--------- ----------->>> {s,r positive}
<<<------------------------------0------>(r-s)---------------------------->>>
Multiplication is scaling and possibly flipping:
<<<------------------------------0---------->®-------------------------->>> {r positive}
(-r) is r, flipped
-(-r) is (-r), flipped, which is the original r
<<<--------------(-r)<----------0---------->®-------------------------->>> {r positive}
(-k*r) is flipping and stretching r
<<<------------------------------0-------------------->(2r) ------------>>> {r positive}
<<<------------------------------0----->(.5r)--------------------------->>> {r positive}
<<<—(2r)<-------------------0-------------------->(2r) ------------>>> {r positive}
<<<---------------(-.5r)<-----0----->(.5r)--------------------------->>> {r positive}
So multiplying a negative number with another negative number is flipping and stretching a negative number:
s=-3*-2=6:
<<<-------------------------(-2)<-----0--------------------------------------->>>
<<<-------------------------(-2)<-----0----->(-1*-2 = 2------------------->>>
<<<-------------------------(-2)<-----0------------------->(3*2=6)-------->>>
You can also view it from a counting perspective … viewing the count as a stock count from the perspective of A and B:
+2 ~ A gets 2 thingies from B
-2 ~ A gives 2 thingies to B
22 = +4~ A gets 2 thingies from B, twice
2-2 = -4~ A gives 2 thingies to B, twice
-2*-2 = -(2*-2) = - (A gives 2 thingies to B, twice) = B gives 2 thingies to A, twice
The vector/ray approach works better in general, in particular when fractional numbers are in play.
That depends on the grammar. It’s a prescriptivist myth for English, by the way.
From the book that I mentioned, it seems like it took a few centuries. For some reason, negative numbers are not intuitive.
cereberus number line examples are helpful. It is fairly easy to represent addition of negative numbers on the number line, and even multiplication where at least one multiplicand is positive, but not multiplication of two negatives without flipping. Division of a postive by a postive, or a negative by a positive, can be easily represented, but not division of a positive by a negative or a negative by a negative. (Without flipping, that is.)
It seems that the contention of the author of the book mentioned in the OP is borne out, and maybe the reason is that we intuitively think in terms of number lines.
I’m a math teacher, and the rules of using negatives still confound many high schoolers.
On a test I just graded about Order of Operation, a student properly got to the point in a problem by stating “30 + (-2)” . Then for the last step, he got an answer of -32. sigh
I really want my students to do well. But their tests are full of little errors like that that throw off their whole answer.
It was more like a couple milleniums. For example, when Cardano (I think it was–I am too lazy to check it) worked out the solution to cubic equations, he had many cases:
ax^3 + bx^2 +cx = d, ax^3 + bx^2 = cx + d, and so on and each was dealt with independently, a tradition that goes back at least to the Babylonians, who preceded the Greeks by centuries. Negative numbers apparently didn’t “feel” right to them.
As for the OP, I recall one book (presumably in elementary school) explaining that you can think of a negative number as representing a debt and also as representing taking away and if someone takes away a debt of $1, the result is a gain of $1. While if someone takes away $1 or gives you a debt of a $1, you have lost $1. From a more mathematical point of view, it is to save the distributive law as more than one poster has explained.
Sheila Tobias in her book, Overcoming Math Anxiety goes on and on about the fact that x^{-2} is 1/x^2 as her bete noire of mathamatics. I don’t think she ever caught on to the fact that it is just there to save the law of exponents x^{a+b} = x^ax^b. When a mathematician comes to extend the domain of a function, he will want to preserve as many properties as possible. This is not parsimony, or not only parsimony; it is to preserve as many properties of the original structure as possible. For instance, you go from the real numbers to the complex numbers by adding a square root of -1 and all the arithmetic laws are satisfied. But throw in another square root of -1 (call it j and ij will be a third) and you discover that you cannot save commutativity (ij = -ji). Add yet another and you discover that you cannot save the associative law of multiplication. You cannot go beyond that and still have division (theorem of the late Frank Adams).
I think the question at hand is whether you can come up with a different system of numbers that behaves exactly the same for positive numbers but differently for negatives.
I had a chance to look at the book in question tonight. Since it’s aimed at a lay audience, it’s a bit difficult to get information out of, but it looks like the author is using {(0, 0)} [symbol]È[/symbol] (N[sup]+[/sup] [symbol]´[/symbol] 3) and interpreting that as a set that includes positive, negative and unsigned integers. The system fails to be a ring because not every element has a unique additive inverse.
It’s an interesting exercise in constructibility, but it’s not clear that it’s actually useful. I’d be interested in seeing a more formal treatment of it.
Okay, my browser chokes horribly on that. Any chance you can write it out in words?
It says something that reads to me like “The union of the set consisting just of the ordered pair (0,0) with a set consisting of (N with a little plus sign over it, multiplied by three.)”
Whatever that might mean. 
-FrL-
Oh, the first time we got “signs as part of numbers” and not just as “operators”, we were told straightaway that “you multiply the signs on one side and the actual number on the other”.
So it’s not that a negative times a negative makes a positive… it’s that “minus times minus gives plus”. Not any harder to learn than the multiplication tables themselves, and learned at the same age (4th grade).
(0,0) union natural numbers crossed with a three-element set. What browser are you using?
Safari on OS X. I know that it handles unicode just fine. Are you using Microsoft’s back-asswards non-standards-compliant extended character set?