On Dec 2, 2012, at 8:42 PM, Joe Niederberger <email@example.com> wrote:
>> People state the plus and minus sign when they mean the positive and negative sign. > > I'll simply note that Barry Manzur of Harvard (of the recently mentioned book "Imagining Numbers" -- thanks Dave Renfro) says "minus times minus" over and over. He's not hung up on these wordsmithing trivialities. > > What remains though, is that neither he nor anyone else here can convincingly make a case for the infamous rule to be either common sense or a mathematical certainty. > > Cheers, > Joe N >
Here are three cases for your enjoyment.
(-a)(-b) = (0 - a) * (0 - b) = 0 * (0 - b) - a * (0 - b) = 0 - a * (0 - b) = 0 - (a * 0 - a * b) = 0 - (0 - a * b) = 0 - 0 + ab = ab
Multiplication is repeated addition. If one term is a negative then it is repeated subtraction. If both terms are a negative then it is repeated subtraction twice, which is back to repeated addition. This can be extended to the rationals by simply noting that the LCD can be factored out and the rest is just integers like above. This can be extended to the irrationals by simply treating them as an infinite series of non repeating decimal digits.
A car travels at a speed of V for time T. The distance travelled by the car is VT. If V is negative (the car is in reverse) then D is negative. If V is positive (the car is driving forward) then D is positive. So far so good, except we have assumed all along that T is positive. What if D is negative (the car is in reverse) and T is negative (we want to know where the car was)? Then VT will be positive. A minus times a minus is a positive.