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Thursday, September 18, 2003

Negative x Negative = Positive

Sean asks for an explanation of why a negative times a negative is a positive.

Here we go...

-x * -y = (-1 * x) * (-1 * y) = (-1 * -1) * (x * y) = 1 * (x * y) = x * y

Obsessive Update: The above assumes -1 * -1 = 1. Why? The only thing I thought of, which really becomes a general rule is to create a contradiction. If -1 * -1 = -1 then...

-1 * (1 - 1) = (-1 * 1) + (-1 * -1)
-1 * (0) = (-1) + (-1)
0 = -2
...which is the contradiction we're hoping for.

Update: maybe Sean is looking for an applied explanation rather than a proof?

Hmm... that would be a good one. When do we do -x * -y in the real world?

Well, I do x * -y in the real world when I pay my mortgage (-y) for some number of months (x). So maybe I could ask how much I would have had had I not paid my mortgage (-y) the previous number of months (-x).

Clearly the result of these two formulas should be the opposite. If I paid -$N USD then if I didn't I would have saved $N USD.

1 comment:

Anonymous said...

LB123. [ Negative X Negative = Positive ], Proof
Proof of Liuhui Brahmagupta

1.
[ N = Number ]
[ Negative number = Negative = (-N) ]
[ Positive number = Positive = (+N) = (N) = N ]
[ Natural number = Natural ]

[ Dead Zero ( 0 ) ] = [ ( 0 ) ] = [ Nothing ]
[ Living Zero ( 0 ) ] = [ (+1) - (+1) ] = [ (+N) - (+N) ]
= [ (+1) + (-1) ] = [ (+N) + (-N) ]
= [ (-1) + (+1) ] = [ (-N) + (+N) ]
= [ (-1) - (-1) ] = [ (-N) - (-N) ]

[^^^] = [ (-N) = (+N) + (-2N) ]
= [ (-N) = { Living Zero ( 0 ) } + (-N) ]
= [ (-N) = { Living Zero ( 0 ) } - (+N) ]
= [ (-N) = { (+N) - (+N) } - (+N) ]
= [ (-N) = (+N) - (+N) - (+N) ]
= [ (-N) = (+N) - { (+N) + (+N) } ]
= [ (-N) = (+N) - { (+2N) } ]
= [ (-N) = (+N) - (+2N) ]
= [ (Negative) = (Subtraction of Positives) ]

Change Negative for (Subtraction of Positives).
[ (-N) = (+N) + (-N) + (-N) = ( 0 ) + (-N) = (+N) + (-2N) = (+N) - (+2N) ]
[ (-N) ] = [ (+N) - (+2N) ]
[ (-1) = (+1) + (-1) + (-1) = ( 0 ) + (-1) = (+1) + (-2) = (+1) - (+2) ]
[ (-1) ] = [ (+1) - (+2) ]

[ - (+N) = + (-N) ]
[ - (-N) = + (+N) ]
[ { - N - N } = { - ( N + N ) ]
[ + N - N ] = [ - N + N ]
[ ( A - B ) X ( A - B ) ] = [ ( A - B ) X A - ( A - B ) X B ]




2. Proof
[^^^] = [ Negative numer X Negative number = Positive number ]
= [ Negative X Negative = Positive ]
= [ (Negative) X (Negative) = (Positive) ]
= [ (-N) X (-N) = (+N) ]
= [ { (-N) } X { (-N) } = (+N) ]
= [ { (+N) + (-2N) } X { (+N) + (-2N) } = (+N) ]
= [ { (+N) - (+2N) } X { (+N) - (+2N) } = (+N) ]
= [ { (+N) - (+2N) } X (+N) - { (+N) - (+2N) } X (+2N) = (+N) ]
= [ { (+N^2) - (+2N^2) } - { (+2N^2) - (+4N^2) } = (+N) ]
= [ { (+N^2) - (+2N^2) } - { (+2N^2) + (-4N^2) } = (+N) ]
= [ (+N^2) - (+2N^2) - (+2N^2) - (-4N^2) = (+N) ]
= [ (+N^2) - (+2N^2) - (+2N^2) + (+4N^2) = (+N) ]
= [ (+N^2) + (+4N^2) - (+2N^2) - (+2N^2) = (+N) ]
= [ { (+N^2) + (+4N^2) } - { (+2N^2) + (+2N^2) } = (+N) ]
= [ { (+5N^2) } - { (+4N^2) } = (+N) ]
= [ (+5N^2) - (+4N^2) = (+N) ]
= [ (+N^2) = (+N) ]
= [ (+N) = (+N) ]
= [ (N) = (N) ]
= [ N = N ]


3. Example
[^^^] = [ (-1) X (-1) ]
= [ {(-1)} X {(-1)} ]
= [ {(+1) + (-2)} X {(+1) + (-2)} ]
= [ {(+1) - (+2)} X {(+1) - (+2)} ]
= [ {(+1) - (+2)} X (+1) - {(+1) - (+2)} X (+2) ]
= [ {(+1) - (+2)} - {(+2) - (+4)} ]
= [ {(+1) - (+2)} - {(+2) + (-4)} ]
= [ (+1) - (+2) - (+2) - (-4) ]
= [ (+1) - (+2) - (+2) + (+4) ]
= [ (+1) + (+4) - (+2) - (+2) ]
= [ {(+1) + (+4)} - {(+2) + (+2)} ]
= [ {(+5)} - {(+4)} ]
= [ (+5) - (+4) ]
= [ (+1) ]
= [ (1) ]
= [ 1 ]


4. Example
[^^^] = [ (-2) X (-3) ]
= [ { (-2) } X { (-3) } ]
= [ { (+1) + (-3) } X { (+1) + (-4) } ]
= [ { (+1) - (+3) } X { (+1) - (+4) } ]
= [ { (+1) - (+3) } X (+1) - { (+1) - (+3) } X (+4) ]
= [ { (+1) - (+3) } - { (+4) - (+12) } ]
= [ { (+1) - (+3) } - { (+4) + (-12) } ]
= [ (+1) - (+3) - (+4) - (-12) ]
= [ (+1) - (+3) - (+4) + (+12) ]
= [ (+1) + (+12) - (+3) - (+4) ]
= [ { (+1) + (+12) } - { (+3) + (+4) } ]
= [ { (+13) } - { (+7) } ]
= [ (+13) - (+7) ]
= [ (+6) ]
= [ (6) ]
= [ 6 ]




5. [ (-2) X (-3) ], The meaning of economic action
[ (-2) X (-3) ] = [ One party of Offset, Let us Offset 3 cases in (Debt, Bill $2). ]
= [ One party of Offset, Let us Offset (Debt, Bill $6). ]

6.
[ (-2) X (-3) ] = [ (Debt, Bill $2) X (-3) ]
= [ (Debt, Bill $2), subtract to add 3 times. ]
= [ (Debt, Bill $2), subtract to multiply 3 times. ]
= [ (Debt, Bill $2), come down to add 3 times. ]
= [ (Debt, Bill $2), come down to multiply 3 times. ]
= [ 3 cases in (Debt, Bill $2), com down. ]
= [ 3 cases in (Debt, Bill $2), Let us offset. ]

7.
[^^^] = [ (-2) X (-3) ]
= [ - { (-2) + (-2) + (-2) } ]
= [ - { (-2) X (3) } ]
= [ - { (-6) } ]
= [ - (-6) ]
= [ + (+6) ]
= [ (+6) ]
= [ (6) ]
= [ 6 ]


8. Law
[ N X (-N) ] = [ - ( N X N ) ]
[ N X (+N) ] = [ + ( N X N ) ]

[ N / (-N) ] = [ - ( N / N ) ]
[ N / (+N) ] = [ + ( N / N ) ]


9.
[^^^] = [ (-a) X (-b) = (+a) X (+b) ]
= [ { (-a) } X { (-b) } = (+a) X (+b) ]
= [ { (+a) + (-2a) } X { (+b) + (-2b) } = (+a) X (+b) ]
= [ { (+a) - (+2a) } X { (+b) - (+2b) } = (+a) X (+b) ]
= [ { (+a) - (+2a) } X (+b) - { (+a) - (+2a) } X (+2b) = (+a) X (+b) ]
= [ { (+ab) - (+2ab) } - { (+2ab) - (+4ab) } = (+a) X (+b) ]
= [ { (+ab) - (+2ab) } - { (+2ab) + (-4ab) } = (+a) X (+b) ]
= [ (+ab) - (+2ab) - (+2ab) - (-4ab) = (+a) X (+b) ]
= [ (+ab) - (+2ab) - (+2ab) + (+4ab) = (+a) X (+b) ]
= [ (+ab) + (+4ab) - (+2ab) - (+2ab) = (+a) X (+b) ]
= [ { (+ab) + (+4ab) } - { (+2ab) + (+2ab) } = (+a) X (+b) ]
= [ { (+5ab) } - { (+4ab) } = (+a) X (+b) ]
= [ (+5ab) - (+4ab) = (+a) X (+b) ]
= [ (ab) = (+a) X (+b) ]
= [ (a) X (b) = (+a) X (+b) ]
= [ (+a) X (+b) = (+a) X (+b) ]
= [ (a) X (b) = (a) X (b) ]
= [ a X b = a X b ]
= [ ab = ab ]


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I'm usually writing from my favorite location on the planet, the pacific northwest of the u.s. I write for myself only and unless otherwise specified my posts here should not be taken as representing an official position of my employer. Contact me at my gee mail account, username patrickdlogan.