The discussion revolves around the properties of multiplication in rings, specifically examining the commutativity of multiplication and the implications for the equation (-x) * y = x * (-y). Participants explore the proof that -1 * -1 = 1, utilizing the distributive property and properties of additive inverses.
Participants generally agree on the steps to prove that -1 * -1 = 1, but there is contention regarding the characterization of this conclusion as being by definition.
Some assumptions regarding the properties of rings and the definitions of additive and multiplicative identities are implicit in the discussion. The proof relies on the distributive property and the nature of additive inverses, which may not be universally accepted without further context.
Cool,lurflurf said:Use the distributive property with
(-1)(1+(-1))=0
Not by definition.1MileCrash said:Cool,
(-1)(1) + (-1)(-1) = 0
-1 + (-1)(-1) = 0
(-1)(-1) = 1 by definition
Mark44 said:Not by definition.
1 + (-1) = 0 since 1 and -1 are additive inverses of each other
-1(1 + (-1)) = -1(0) = 0, since 0 times anything is 0.
-1(1) + (-1)(-1) = 0
Since -1(1) and (-1)(-1) add to zero, they are additive inverses.
We know that -1(1) = -1, since 1 is the multiplicative identity,
so -1(-1) must equal 1.