
Re: Some important demonstrations on negative numbers
Posted:
Nov 27, 2012 11:42 AM


Your demo is valid if the postulates of real numbers are known by the student, since it uses the identity element, hence it is necessary for (1)(1) to be 1.
On Tue, Nov 27, 2012 at 8:12 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.
  Jim
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