Comments on Product of Negatives
Prunthaban said:
Nice proof. So now we should be able to talk in a very generic sense. The additive inverse of \( a \) is \( -a. \) From your proof it looks like in a field the multiplication of two values is equal to the multiplication of their additive inverses. That makes your proof independent of real numbers and assumes only that the system in question is a field (assuming your proof is not using anything other than the 9 field axioms).
Susam Pal said:
Prunthaban,
That is an interesting way to look at it. We can generalise it further. We neither need commutativity of multiplication in our proof nor do we need existence of multiplicative inverse (two of the field axioms), so our proof holds good for the elements of a ring as well.
Sunita Rajamani said:
Nice post. I really understood something this time! :-)