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by thaumasiotes
2305 days ago
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I don't understand the point you're trying to make. You claimed that negative and positive are defined by absolute reference to zero. I claimed that they aren't. But you're presenting an example that assumes I'm right and you're wrong! > For example, the integers mod n is a ring, so (-a) * (-b) = a * b holds, but it doesn't make sense to call a number mod n positive or negative, since -a mod n effectively means n - a mod n. If negative numbers were defined by reference to a comparison to zero, then the expression (-a) * (-b) would be meaningless nonsense in Z mod 5 -- as you point out yourself, Z mod 5 is not ordered in that way. But it isn't nonsense, and you're not saying it is -- instead, you assume it's obviously valid when you observe that the equality (-a)(-b) = ab holds. |
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1) Positive and negative numbers (defined in terms of comparison to 0)
2) The negation of a number (i.e., the additive inverse)
They're related in that when both concepts are defined, a negative number is the negation of a positive number. However, the two concepts don't coincide. I'm sure you know this, but even over the reals '-x' is the negation of a number, but not necessarily a negative number.
(-a) * (-b) = a * b is an equation about #2, and it holds in any ring/field, even ones where #1 doesn't make sense, e.g. Z mod 5. If #1 makes sense, then this immediately implies that the product of two negative numbers is positive.
My original point was that the blog post is talking about real numbers, for which #1 and #2 are both defined. However, if it's generalized to arbitrary rings/fields, where only #2 is defined, then you can't really refer to the equation '(-a) * (-b) = a * b' as 'the product of two negative numbers is positive'.