any math uses for signed zero (like: x^0 = 1, x^-0=something else?)

[Indistinguishable]

In mathematics, you can make up whatever rules you want and study their consequences, and to the extent the rules you make up are analogous (even if not exactly the same as, and in some ways quite different from) to other rule-systems, you may want to describe them using similar terminology/notation. Thus, you are certainly free to imagine concepts such as distinct +0 and -0 and some rules for what you can say about them in a particular arithmetic framework.

Exactly what properties and uses such a framework would have depends on the particular framework you devise; of course, there are a number of other, quite well-studied frameworks in which terms like “zero”, “negation”, “positive”, etc., are used in such a way that there is no such thing, but no matter: the whole point is that we have an abundance of different abstract systems to study, as fits our interests and desires at any particular moment.

The designers of IEEE floating point, for example, had reason to want a separate +0 and -0 in the rule system defining IEEE floating point. This has not been a rule system which mathematicians have by and large found very interesting (it’s not particularly natural, clean, ubiquitous, etc.), but there’s nothing fundamentally amathematical about it.

It’s also fairly natural, and mathematically interesting, to study arithmetics which include such three distinct values as straight-up 0, positive 0, and negative 0 (the latter two representing something like infinitesimal deviations from straight-up 0); such arithmetics provide a convenient language for describing, say, the one-sided limiting behavior of functions, as noted above. (This goes hand in hand with also having reciprocal entities of positive infinity and negative infinity, as in the affinely extended numbers.)

How else might you use a distinct positive zero and negative zero? Well, it’s up to you: you describe what your imagination wants to do with such a concept, and then, poof, it exists. The only question is how your rules relate to other rule systems/what they fruitfully model. But they need no more founding legitimacy than your deciding to study them. And choosing to study a rule system doesn’t amount to some kind of rejection of other rule systems; you don’t have to “choose” between different accounts of the interplay between zero and sign. There’s no question of exclusion. Study everything!

[Topologist]

@ModernPrimate

…the generalizations aren’t a part of conventional math notation which is what OP is talking about.

We all immediately agree that, in the field of real numbers, which is probably what you’re referring to as “conventional math notation,” there is only one 0. If that were all the OP meant, it would be an uninteresting question. But, if by “conventional math” you mean what mathematicians do routinely, then we should ask, what interesting things can you say about algebraic structures that have multiple 0’s?

I haven’t given it enough thought to give much of an answer, but let me throw out an example that just occurred to me: the set of all matrices with, say, real entries. There are lots of 0 matrices. On the other hand, the operations are only partially defined. [Best to view the matrices as the morphisms in a category, but that’s already clear to anyone who understands that remark…]

[Asympotically_fat]

@ModernPrimate

Even if so, the generalizations aren’t a part of conventional math notation which is what OP is talking about. You can say 0^+ and 0^- all you want in limits, that is the orthodox way of treating it. However -0 and +0 aren’t recognized as different in conventional mathematics, even if by shorthand it’s done in certain problems. It would be like allowing 0/0, which can make sense in a lot of circumstances (once I even argued it should be allowed in certain situations), but it’s not formally regarded as possible. This isn’t a logical problem, it’s one of notation… it’s depending on how they’re defined.

The generalizations are part of conventional maths.How more conventional do you want than the study of objects such as groups, fields, rings, etc? There’s nothing unconventional about a ring.

However as I say even in these generalizations -0 = 0.

The best I can think of is that a left zero in a magma could have a left inverse that was not equal to itself (or equally a right zero in a magma could have a right inverse that was not equal to itself).

[psychonaut]

The Wikipedia article on signed zero makes the following claim, which it supports with one reference from a book on numerical analysis (probably available only in print) and one reference to some university lecture notes on calculus:

It is claimed that the inclusion of signed zero in IEEE 754 makes it much easier to achieve numerical accuracy in some critical problems, in particular when computing with complex elementary functions.

I haven’t read either reference, and numerical analysis isn’t my speciality, so I probably wouldn’t understand them very well even if I did… Here’s the relevant part of the lecture notes, which as you can see doesn’t really discuss the matter in detail but instead the reader to the aforementioned book:

Elementary Transcendental Functions

Non-algebraic analytic functions are called “Transcendental”. The Elementary Transcendental functions arise out of algebraic operations upon exp(z) and ln(z) . The latter’s multiplicity of values, each differing from others by integer multiples of 2πı , is suppressed by a notation that assigns one Principal Value to “ ln(z) ” in some contexts and, if done conscientiously, uses another notation like “ Ln(z) ” for the multi-valued version. One text, Complex Variables and Applications 6th. ed. (1996) by Brown & Churchill (McGraw-Hill), does just the opposite.

These formulas’ discontinuities, their slits, are located in the most commonly expected places. Also in accord with consensus are the values taken on the slits; acquiescence to the convention sign(0) := +1 is tantamount to attaching each slit to its side reached by going counter-clockwise around its one finite end. But this Counter-Clockwise Continuity is too simple a rule to work for a function whose slit is a finite line segment. Consequently some of the usual definitions of arcsec(z) := arccos(1/z) , arccsc(z) := arcsin(1/z) , arccot(z) = arctan(1/z) , arcsech(z) := arccosh(1/z) , arccsch(z) := arcsinh(1/z) and arccoth(z) := arctanh(1/z) may change one day as arccot did; it used to be arccot(z) := π/2 – arctan(z) until about 1967, but now its slit is a finite line segment joining logarithmic branch-points at z = ±ı and poked at z = 0 . The usual definition of arcsech(z) violates counter-clockwise continuity around z = 0 .

These and many other annoying anomalies, like √(1/z) ≠ 1/√z and arg(z) ≠ –arg(z) just when z < 0 , go away when a signed zero is introduced, though it brings a new anomaly many people find more annoying, namely that –4 + ı0 = –4 – ı0 but 2ı = √(–4 + ı0) ≠ √(–4 – ı0) = –2ı . This is treated, along with other perplexing examples and numerically stable algorithms for the formulas above, in my paper “Branch Cuts for Complex Elementary Functions, or Much Ado About Nothing’s Sign Bit”, pp. 165-211 in The State of the Art in Numerical Analysis (1987) ed. by A. Iserles & M.J.D. Powell for the Clarendon (Oxford Univ.) Press.