| >Since y = x/0, it follows that the product of y and 0 is equal to x, because division is the inverse of multiplication. Can you explain how this follows? I thought division was only the inverse of multiplication for all nonzero denominators, which would mean we can't use that definition for deduction in x/0. It might hinge on your next sentence: >By the field axioms, division does not exist if there is no multiplicative inverse with which to multiply. but I don't understand why that's necessarily true. I don't understand how the field axioms require division by x to require the existence of a multiplicative inverse of x when x is zero. Sorry to take a bunch of your Friday, but I'm very curious now. Explanation much appreciated. ------- edit: Come to think of it, couldn't I define x/y as cotton candy for all x,y in field F and still satisfy the field axioms? They just don't refer to division. Any connection between x/y and y's multiplicative inverse is just a nice convention. That convention states that x/y = x * mult_inv(y) when y != 0, but nothing else. That definition has nothing to do with the field axioms and changing it doesn't require that I change anything about multiplicative inverses. That means I don't touch the field axioms and my field is still a field. |