[joehilton]

I was actually using the terms "real" and "imaginary" non-colloquially to denote real and complex numbers (5, -1, pi, e, -1^-2, etc.).

It's true that nonnegative numbers are just abstract ideas that we come up with to represent quantity, but that's just it: quantity is a real thing. Is there such a thing as negative quantity? That's my question.

I like the reference to the contention of what we currently denote as imaginary numbers and how they should be compared as just as real as other numbers. This is the same issue I'm questioning but from the other side. I'm questioning if we should count negative numbers as real as opposed to understanding them as a set that is at least imaginary, if not complex or irrational.

I also like the point about a counterclockwise rotation being just as real as a clockwise rotation. This is where the non/negative sign is used to indicate a vector. As long as you have a point of reference, this is always ok because it's simply referring to what type of amount of whatever quantity/magnitude/etc we're dealing with.

But I believe this is different than what we often do with negative numbers when working through many of our problems that involve unitless numbers/constants or take an abstraction and apply it as a physical theory - in these cases the negative numbers were treated as separate and real when in fact they should have been directions, unit movement, or other information about actual physical quantities. This is where I'm wondering if we should treat negative numbers as a different set altogether.

[twangist]

Fine point: in the case of "negative" (clockwise) rotations, the negative sign applies only to a scalar (an angle), not a vector. Surely even negative money is a real thing too: debt, or anyway debit. (Granted, debt can be less real than a pile of cash: if you're owed you might never be repaid, if you owe you might default. But that's a case of reality not matching theory well :)

In physical applications, relative to a coordinate system negative numbers have perfectly intuitive meanings. A negative spatial coordinate simply indicates a point in one of the quadrants or octants other than the all-positive-coords one. A negative acceleration means something is slowing down.

I see no good purpose served by special-casing "negative" integers, and many needless ensuing headaches.