Multiplying negatives = positive?

Here’s a bizarre concept that I just can’t believe didn’t cause early mathematicians moral consternation…

Multiplying two negatives produces a positive value.

Huh?

For instance, 10 trigger happy terrorists who have access to 10 nuclear bombs, is a good thing? 10 BAD x 10 BAD = 100 GOOD!

That doesn’t make any sense. Or how about another?

Increased global warming times increased human population = super duper outcome. I don’t think so.

Negative times negative should be REALLY BLOODY NEGATIVE!

-10 x -10 = -100. That seems logical.

But there’s this other scenario that is really the problem, a positive number times a negative number, or a negative number times a positive number (yes, the commutative property).

So what should happen when you multiply +10 x -10 ? 10 Mother Teresas times 10 jihadist fanatics? Well, I think that there should be something akin to addition there, 10 x -10 = 0. But let me show you another example.

5 natural disasters times 2 Nobel Peace prizes: -5 x 2? Well, the absolute product of this equation is obviously 10. But is ALL that ten negative? Or is there some positive in there somewhere? If you think about a number line, -10 to 10, centered on zero, where would the logical product of -5 x 2 fall? Negative eight maybe? Negative six?

How about this:

-5 * 2 = -6 = -(10 – (10 * 2/5))

If the larger number is positive the product will be positive.
If the larger number is negative the product will be negative.
To the absolute value of the product, subtract the product of the absolute value of the  product times the fraction of the small number over the larger number.

±x * ±y = ±(|±x * ±y| – (|±x * ±y| * x / y))
or
if x is negative and larger:
x * y = -(|x * y| – (|x * y|* y / x))
if x is negative and smaller:
x * y = +(|x * y| – (|x * y| * x / y))

Examples:

-10 * 1 = -9  = -(10 - (10 * 1/10))
-10 * 5 = -25 = -(50 - (50 * 5/10))
-10 * 9 = -9  = -(90 - (90 * 9/10))
-10 * 10 = 0  = (100 - (100 * 10/10))
 10 * -9 = 9  =  (90 - 90 * 9/10))
 10 * -5 = 25 =  (50 - (50 * 5/10))
 10 * -1 = 9  =  (10 - (10 * 1/10))

-7 * 1 = -6  = -(7  - (7 *  1/7))
-7 * 4 = -12 = -(28 - (28 * 4/7))
-7 * 6 = -6  = -(42 - (42 * 6/7))
-7 * 7 = 0   =  (49 - (49 * 7/7))
 7 * -5 = 10 =  (35 - (35 * 5/7))
 7 * -3 = 15 =  (21 - (21 * 3/7))
 7 * -1 = 6  =  (7  - (7 * 1/7))

 -37 * 12    = -300  = -(444 - (444 * 12 / 37))
  99 * -3    =  288  =  (297 - (297 * 3 / 99))
 273 * -111  = 17982 = (30303 - (30303 * 111/273))
  .5 * -.12  = .0456 = (.06 - (.06 * (.12 / .5)) 
-544 * .004321 = -2.350605328959

Positive times a positive is positive.
Negative times a negative is negative.
And the mixture of positive times a negative is somewhere in between.

That, to me, seems logical.

 

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