Holmesian Truth

How do you explain to a little kid why -1*-1=1? In other words, why does a negative times a negative equal a positive? I liked this explanation from Dr. Math:

So the real question is,

   (-1)(-1) = ?

and the answer is that the following convention has been adopted:

   (-1)(-1) = +1

This convention has been adopted for the simple reason that any other convention would cause something to break.

For example, if we adopted the convention that (-1)(-1) = -1, the distributive property of multiplication wouldn't work for negative numbers:

   (-1)(1 + -1) = (-1)(1) + (-1)(-1)           (-1)(0) = -1 + -1                0 = -2

As Sherlock Holmes observed, "When you have excluded the impossible, whatever remains, however improbable, must be the truth."

I wonder if Sir Arthur Conan Doyle was thinking of negative numbers when he wrote that line. 

I also liked how Dr. Math pointed out that visualizing was not the same thing as understanding. Wait...would analogies count as "visualizing"? But I think almost everything we learn is through drawing analogies- mapping from a known set to an unknown set.