It also isn't self-evident that x^(1/3) is the cube root, or that x^(-a) is 1/(x^a), but when you define the exponent operator in this way, it keeps things consistent (they don't break). That is the message I try to drive home. How to recognize convention and the way it is developed.
On Nov 27, 2012, at 10:12 AM, Peter Duveen <firstname.lastname@example.org> wrote:
> I have a new student, who was confused about negative numbers, both multiplication and division of the same. I wanted to demonstrate to him that a negative times a negative is a positive, but got flustered because I could not produce the demonstration. I promised to show him the demonstration at our next weekly meeting. > > Later, I sat down and derived the following demonstration: > > Proof that the multiplication of two negative numbers is a positive number: - 1 + 1 = 0 (Definition of -1); -1(-1 + 1) = 0 (0 times any number is 0); -1x-1 + -1x1 = 0 (distributive law of multiplication); -1x-1 + -1 = 0 (1 x any number is that number itself); -1x-1 = 1 (definition of -1; the same quantity added to equal quantities produce equal quantities). > > Main point is, it is not particularly self-evident that a negative number times a negative number yields a positive number. Some will undoubtedly argue that to demonstrate this property rather than to just state it will confuse the student. At the same time, I would argue that to not demonstrate it will confuse the student. > > I'm just throwing this out for discussion, but other properties that ought to be demonstrated are that 1/-a = -(1/a), another property that many may feel does not need to be demonstrated. On the contrary, I believe that not demonstrating it will lead to confusion.