
Re: Some important demonstrations on negative numbers
Posted:
Nov 27, 2012 11:52 AM


It also isn't selfevident 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.
Bob Hansen
On Nov 27, 2012, at 10:12 AM, Peter Duveen <pduveen@yahoo.com> 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); 1x1 + 1x1 = 0 (distributive law of multiplication); 1x1 + 1 = 0 (1 x any number is that number itself); 1x1 = 1 (definition of 1; the same quantity added to equal quantities produce equal quantities). > > Main point is, it is not particularly selfevident 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.

