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        riddles >> general problem-solving / chatting / whatever >> Redefining Negative Arithmetic
        
(Message started by: Sir Col on Jul 7^th, 2008, 9:52am)

Title: Redefining Negative Arithmetic
Post by Sir Col on Jul 7^th, 2008, 9:52am

A colleague of mine mentioned a conversation he had with one of his student's today who asked the question, "Why can't we define the product of two negatives to be negative?"

This is far from a naive question and was presented by a very astute learner who is currently studying complex numbers...

Suppose by definition that x*y = -|x*y| where x,y < 0.
For example, -4*-4 = -16. But then sqrt(-16) = -4 and there would be no need for imaginary numbers in this context.

Would this "definition" lead to a closed system with the real number arithmetic, or would the need for "imaginary" numbers appear elsewhere?

Of course, it would change a number of results that we have become used to, but would it still be a consistent and coherent system?

There are certainly going to be some problems...

A graph like y = x² would resemble the cubic graph, but is this really a problem or just a matter of shifting conventions? Similarly with lines having negative gradients; they would "bounce" at the y-axis.

Although square roots would have one solution, division with negatives would be problematic: if -4*-4 = -16 and 4*-4 = -16 then -16/-4 = +-4.

(4^-1)^-1 = (1/4)^-1 = 4, but using laws of indices, (4^-1)^-1 = 4^-1*-1 = 4^-1 = 1/4.

Any thoughts?

Title: Re: Redefining Negative Arithmetic
Post by towr on Jul 7^th, 2008, 11:24am

It seems a bit inconvenient.
-1*(2 + -3) = -1*2 + -1*-3 = -2 + -3 = -5; so we can scratch distribution.
-1 - (-1) = -1 + -1*(-1) = -1+-1 = -2; so subtraction is no longer adding the additive inverse.
etc.

Title: Re: Redefining Negative Arithmetic
Post by Sir Col on Jul 7^th, 2008, 12:18pm

The distributive law could be rescued by revised definitions:
+ times + = +
+ times - = -
- times + = +
- times - = -

E.g.-1(2-3) = -1*-1 = -1
-1(2-3) = -1*2 + -1*-3 = 2 - 3 = -1

2(3-4) = 2*-1 = -2
2(3-4) = 2*3 + 2*-4 = 6 - 8 = -2

And although associativity still holds, can commutativity be sacrificed?
4*-4 <> -4*4

Also division becomes horribly ambiguous:
4*4 = 16 => 16/4 = 4 (1)
4*-4 = -16 => -16/-4 = 4 or -16/4 = -4
-4*4 = 16 => 16/4 = -4 (2) or 16/-4 = 4
-4*-4 = -16 => -16/-4 = -4

So 16/4 (1) does not necessarily equal 16/4 (2).

However, problems are encountered with division in modular arithmetic.

Title: Re: Redefining Negative Arithmetic
Post by Grimbal on Jul 8^th, 2008, 12:42am

on 07/07/08 at 12:18:25, Sir Col wrote:

The distributive law could be rescued by revised definitions:
+ times + = +
+ times - = -
- times + = +
- times - = -

Then I guess we have to do without an identity element for multiplication. :-/

Title: Re: Redefining Negative Arithmetic
Post by Eigenray on Jul 8^th, 2008, 5:05am

on 07/07/08 at 12:18:25, Sir Col wrote:

The distributive law could be rescued by revised definitions:
+ times + = +
+ times - = -
- times + = +
- times - = -

But then you lose distributivity from the right:
(a - b)c http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/ne.gif ac + bc.
All in all, it seems like a lot to give up just to stop people from asking why "a negative times a negative equals a positive."

Title: Re: Redefining Negative Arithmetic
Post by Grimbal on Jul 8^th, 2008, 5:16am

Yet, GF(2) has -a·-b = -|a·b| and it still has some use ;).