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
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:
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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