"Kenneth" <kdw### [at] gmailcom> wrote:
>
> This particular example-- multiplying two negative numbers and getting a
> positive answer-- has always given me pause, philosophically....
> ...when it comes to -2 X -3, it just
> doesn't seem 'intuitive' that it should produce a positive value. (Although,
> what *else* it should produce is certainly a mystery!)

Hey, I've come up with my own 'comfortably' intuitive way of understanding this
concept!

Take a negative number,  say -2

Now I want to negate that negative number:  -(-2)

There are two ways of 'seeing' or understanding -(-2)  The first is simple
'cancellation' of the signs (!). Because, since the leading negative sign is
just a symbol with no numerical quantity attached, the RESULT has to be *a*
value of 2, with some kind of unknown-for-now sign. But the result can't be the
original -2... if it was, then the leading minus-sign would have no purpose at
all(!) Not logical! So, therefore, the result needs to be positive... since it
can't be anything else, dammit! (well, it could *possibly* be zero, by a
quasi-physical rule...i.e., 'forcing' the -2 to go back toward zero on a number
line.... but I'll ignore *that* result...)

The other way of looking at it is as simple multiplication: the 'naked' negative
symbol  '-'  times -2. Even though this operation *in itself* is the
'non-intuitive' crux of the matter, the RESULT needs to be the same as with the
'cancellation' example above...  +2 ...  with no need to do any further
conceptualizing!

SO... Following from this 'equality of operations', it now seems obvious that
multiplying a negative with a negative equals a positive! Voila!

My little April Fool's joke, in December  :-P  Who says amateur philosophers
can't be brilliant?!

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Cousin Ricky <ric### [at] yahoocom> wrote:

>
> -2 x  3 = -6
> -2 x  2 = -4
> -2 x  1 = -2
> -2 x  0 =  0
> -2 x -1 =  2
> -2 x -2 =  4
> -2 x -3 =  6
>
> Seems intuitive enough.  I figured this out on my own as a kid, although
> I was suspicious of my own reasoning until it was confirmed in math class.

Hmm. But doesn't the intuitive nature of your construction depend on an implicit
*assumption* that the numerical results should simply go from negative to
positive (in the descending order of your example)? In other words: that the
result of -2 X -2 being positive should *be* positive simply because -2 X 2 was
negative? (or, that -2 X -2 should simply be 'different' from -2 X 2?) Or was it
the middle column of positive-to-negative values that gave you the clue?

I think my own (flawed!) intuition when *I* was a kid would have been that -2 X
-2 would have equaled -4 !  :-O  Thankfully, my smarter teachers prevailed. ;-)

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Worth watching, as this has some relationship to the topic.
.... and it's animated with POV-Ray.   :)

https://www.youtube.com/embed/6cpTEPT5i0A?list=PL3C690048E1531DC7

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