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carloswilson | 6 years ago
-a is a standard way to represent additive inverse of an element in field.
The point about "unary negation operator" seems irrelevant.
In the real number field, additive inverse of a positive real number is indeed the negative of that number. The negative of that number is also obtained by the application of unary negation operator on the positive number.
The additive inverse of 3.14 is -3.14. Unary negation operator applied to 3.14 gives us -3.14. I don't see how conflating unary negation operator with negative numbers here is any issue here.
wsxcde|6 years ago
This is a fact that follows from the definition of +. But + needs to be defined before you can start making assumptions about what the additive inverse is. The set over which the field is defined (Z or R) already contains -3, -2 etc. and -3 * -2 or -3 + -2 needs to be defined when you're constructing the field. It then turns out that -3 is the additive inverse of 3. You can't use this when arguing about the definition of why applying * on negative 2 and negative 3 gives you the result positive 6. Because you need to define * over all members of the field before you construct a field in the first place.
carloswilson|6 years ago
What? The set of all integers, Z, is not a field! R is. But Z isn't. Z is a ring, a commutative ring. I doubt you understand what a field is!
> already contains -3, -2 etc.
Yes, and those elements are literally the additive inverses of their positive counterparts. If you disagree with this, then the numbers -3, -2, etc. literally have no meaning.
> It then turns out that -3 is the additive inverse of 3.
Are you making this all up with your original research or do you have any proper literature written by a professional mathematician to back it up?