[stuartj]

Originally posted by LubeMaster77
In one instance, you must agree that a negative number is just a positive number multiplied by -1, then we can always write the product of two negative numbers this way:
(-a)(-b) = (-1)(a)(-1)(b) = (-1)(-1)ab

Or
-2 * -3 = (-1)(2)(-1)(3)

= (-1)(-1)(2)(3)

= (-1)(-1) * 6
the distributive property of multiplication wouldn't work for negative numbers, for instance:
(-1)(1 + -1) = (-1)(1) + (-1)(-1)

(-1)(0) = -1 + -1

0 = -2
Since everything except +1 can be excluded as impossible, it follows that, however improbable it seems, (-1)(-1) = +1.
Let a and b be any two real numbers. Consider the number x defined by
x = ab + (-a)(b) + (-a)(-b).

We can write
x = ab + (-a)[ (b) + (-b) ] (factor out -a)
= ab + (-a)(0)
= ab + 0
= ab.
Also,
x = [ a + (-a) ]b + (-a)(-b) (factor out b)
= 0 * b + (-a)(-b)
= 0 + (-a)(-b)
= (-a)(-b).
So we have
x = ab
and
x = (-a)(-b)

Hence, by the transitivity of equality, we have
ab = (-a)(-b).


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