At 10:17 AM 4/2/96, Steve Cottrell wrote: >One of the standard theorems from Foundations of Algebra courses and >which follows in a few steps from the field axioms is > > (-a)(-b) = ab. The proof is not difficult and it's readily > >accessible in any algebra foundations text. The reason that the >product of two negatives is positive is a consequence of the field >axioms.
Careful now. This proves that the product of the opposites of two elements is equal to the product of those two elements. That is, -a is not necessarily a negative number! Which goes back to the point raised earlier on the overloading of the "-" sign.
Why is this such a big deal? Not necessarily because we need to systematically teach students the different meanings, BUT... if there are important mathematical issues that are being obscured, a muddled situation for students is likely to result. Even if students need not be explicitly instructed about all this, curriculum writers and teachers had best be VERY aware.
A very common example of this muddling occurs when dealing with functions like f(x) = -x ... more than a few students believe f(x) always has to be negative. Thus, calling -x to be "negative x" is not at all a good idea...
PS: To finish up Steve's argument, we need to bring in ordering. (i) If a>0 and b>0, then ab>0 ... BY DEFINITION (closure) (ii) a>0 iff -a<0 ... BY DEFINITON (from trichotomy) SO... If a<0 and b<0, then -a>0 and -b>0 ... by (ii) Which means (-a)(-b)>0 ... by (i) Now use the theorem Steve mentions above: ab = (-a)(-b) > 0 Thus, the product of two negatives is positive.
W. Gary Martin Curriculum Research and Development Group University of Hawaii