Addition Operator: Properties & Reciprocity

[Bob3141592]

At the very beginning of my Analysis book, the axioms for the real number system are given, beginning with the addition axioms. The sum of x and y is commutative, and associative. There exists a zero identity element, but it isn't claimed to be unique. And there exists a negative element, so the sum of x and it's negative is the zero identity element. Any operation which can satisfy these properties can be called addition, and the objects it operates on can potentially be called real numbers, right? The notion of integers and counting hasn't entered into anything yet.

Likewise multiplication has axioms of commutativity, associativity, an identity element different from the additive identity element, but not claimed to be unique. There's distributivity over addition. Then there's reciprocity, for all elements except the addition identity.

I hate exceptions! Is the exclusion absolutely necessary, or is it possible to have some kind of number system where the additive identity element can have a multiplicative reciprocal? On page 4 in Shilov's book, it says that the distributive property connects the operation of multiplication with addition, but isn't the distinctiveness of the two operator's identity elements and the reciprocal exclusion of zero a more forceful connection? Perhaps it's the same thing.

 

[ice109]

1 don't double post 2 i doubt it.

 

[CompuChip]

Uniqueness of the additive and multiplicative identities is easy to show (e.g. assume that 0 and 0' are both additive identities and derive 0 = 0'). In fact, what they are saying in the beginning of your book, is that the real numbers form a ring, and one property of rings is that they form a group under addition and the units of the group (which is in this case all numbers except 0) are a group under multiplication. Then the exception is not really an exception anymore, and one can show the uniqueness properties from general group theory. For more information, also see wikipedia. And indeed, the properties you mention do not uniquely define the real numbers: they just say that they are an example of a ring, not that they're the only example (otherwise, indeed, the whole concept would be a bit useless).
To rigorously define the reals, you would have to used Dedekind cuts, etc. For all practical purposes in analysis, the given axioms will do though.

 

[arildno]

OP:
You have forgotten one exceedingly important axiom, namely:
$$0\neq{1}$$
without this axiom, you are perfectly able to create a consistent number system with all the standard laws, along with the existence of the reciprocal of the multiplicative neutral element.

 

[Vid]

That can be derived from the others for any ring with more than two elements.

 

[tiny-tim]

Hi Bob!

Well, it couldn't be unique could it, because you could add any other number to it, and get another inverse:

If there is an element, ∞ say, such that ∞0 = 1,

then (∞ + a)0 = ∞0 + a0 = 1 + 0 = 1. ​

 

[HallsofIvy]

If e1 and e2 are both 'additive identities', then what is e1+ e2? Since e1 is an additive identity, e1+ a= a for all a so e1+ e2= e2. But since e2 is an additive identity, a+ e2= a for all so e1+ e2= e1. Therefore e1= e2.

You can do the same with multiplicative identities.

By the distributive law, we must have a(b+ 0)= ab+ a0. But, since 0 is the additive inverse, b+ 0= b so a(b+0)= ab. That is, ab+ a0= ab so a0= 0 for all a.

Now, suppose we have a field in which all members, including 0, the additive identity, have multiplicative inverses. Call the multiplicative inverse of 0, 0'. Then 0(0')= 1 by definition of multipicative inverse, but 0(0')= 0 since 0(a)= 0 for all a. That is, if 0 has a multiplicative inverse then 0= 1. But then a1= a= a0= 0. If 0 has a multipicative inverse then it is the only member of the field!

Some texts include the "axiom" that a field must contain more than 1 element so that 0 cannot have a multiplicative inverse. Even if you don't do that, the only case in which 0 has a multiplicative inverse is trivial.

 

[Bob3141592]

Thanks to all for your replies. At this point, I'm not struggling with the material per se, but I am trying to bring a fresh eye to it after all these years. I'm trying not to think about what I think I know numbers are, or what the operators are. If addition is a black box to me now, acting on objects also enclosed in black boxes, when their properties emerge more fully I hope to understand them better.

tiny-tim, your comment is interesting. I gather you're saying that if the additive identity element is not excluded from multiplicative inverses, then all numbers would be the inverse of all other numbers. That rather makes sense, especially as HallsOfIvy shows all of those numbers would have to be 0. The wiki link on rings is also very appreciated, as it touches on systems where some of the basic assumptions are not made. I'll have to dig around in there. More to explore is always good!

 

[Hurkyl]

If you're willing to abandon some of the other axioms, you could look at a wheel.


But the main point is that rings and fields are "good" objects of study, because their theory is much simpler than those of other similar structures, and they have wide applicability.

And for the purposes of analysis, the reals are especially useful as a fundamental object, which is why we study it.

 

[ice109]

If you're willing to abandon some of the other axioms, you could look at a wheel.


But the main point is that rings and fields are "good" objects of study, because their theory is much simpler than those of other similar structures, and they have wide applicability.

And for the purposes of analysis, the reals are especially useful as a fundamental object, which is why we study it.

know somewhere i can read about "wheels"

 

[D H]

know somewhere i can read about "wheels"

Simple. Click on the word "wheel" in Hurkyl's post. In general, when you some something http://www.merriam-webster.com/dictionary/highlighted" as opposed to underlined it means there is either a link associated with the highlighted text.

 

[ice109]

Simple. Click on the word "wheel" in Hurkyl's post. In general, when you some something http://www.merriam-webster.com/dictionary/highlighted" as opposed to underlined it means there is either a link associated with the highlighted text.

there's not much in that wikipost

 

[Hurkyl]

there's not much in that wikipost

Wikipedia has references. :tongue:

 

[ice109]

Wikipedia has references. :tongue:

one, and it requires subscription to a journal

 

[Hurkyl]

one, and it requires subscription to a journal

Did you notice the clause at the end of the reference?

(also available online here).​

You can also follow some links from the journal site to get to the same place.

 

[Bob3141592]

Hurkyl,

Thanks for the link. I printed out the wikipedia article, which was plenty interesting. I'm printing out the referenced paper, but it looks like it will be a hard read. I wonder how far into it I can get. I've got questions and notes all over the short wiki page. It's remarkable how the difference between the unary / operator and the normal reciprocal produces these "special zeroes". They seem to have the flavor of an uncertainty residual in physics, which resists being operated on by scale factors. My terms are probably not right, but I wonder if there could be any connection. Anyway, I'll try to get my questions written up soon.

 

[CompuChip]

Don't forget the http://www2.math.su.se/~jesper/research/wheels/ which also contains the full paper and a two-page summary.

 

[Hurkyl]

It's not too surprising...

The 1/0 thing comes from projective geometry; e.g. look up the "projective reals" and the "Riemann sphere".

And the 0/0 thing comes from the fact that given any mathematical structure and a collection of partial (possibly total) operations, you can always extend it by adding one new symbol to represent 'undef', and extend all of the operations via:
. If it involves 'undef', the result is 'undef'
. If the expression is normally undefined, in this new structure we give it the value 'undef'
. If the expression is normally defined, in the new structure we give it the same value
(Of course, this will change the form of some identities)

 

[Bob3141592]

I can't help but feel surprised by this property of wheels. I'm vaguely aware of the Riemann sphere, and how adding the point at infinity enabled it to do a lot of things, but I didn't think adding that point changed the fundamentals of the numbers it was based on. Perhaps it does and I never appreciated that fact. In my reading it seemed presented as a small alteration, and afterwards everything is pretty much business as normal. For example, trig and calculus apply unaltered, don't they? I'm sure that's a very superficial view, and I'll ask questions about that later.

But I doubt that for wheels, the way an inversion isn't quite a reciprocal, it can be business as usual. The property $$( x + 0y )z = xz + 0y$$ makes it clear $$0y$$ is no ordinary number, and the property $$xz + yz = ( x + y )z + 0z$$ makes me think that $$x$$, $$y$$, and $$z$$ aren't exactly normal either. Given that $$x - x = 0x^2$$ I'm sure calculus won't work the way I'm used to. I wonder how qualitatively different it would be, and if that depends on how much variance there is between an involution and an inversion. But there have to be such operations, don't there? No matter how transformed this system is, it must have some form of calculus, doesn't it? Is it known what that would look like, or is the field too new?

I have to try and read Carlstrom's paper, and with Wikipedia I can look up unfamiliar terms like 'semiring,' 'Zarinski subset,' 'monoid,' etc. That will be a big help. And I'd like to chart out a taxonomy of mathematical objects, to keep track of what makes a field different from a group or a ring or a pseudo-ring. Are there resources for this? I recently downloaded SageMath - does anyone know if that would be useful for my efforts?

Again, thanks for all your comments.

 

[Hurkyl]

I can't help but feel surprised by this property of wheels.

I guess it would be more accurate to say that it shouldn't be too surprising if you've ever done arithmetic using projective coordinates.

The projective 'line' over a field (which, for C, is the Riemann sphere. It's a 'line' because it has only one complex dimension) can be given by pairs of coordinates (a:b), where (0:0) is an illegal pair of coordinates, and the pairs (a:b) and (ca:cb) both denote the same point (for nonzero c).

If we choose (1:0) to be the infinite point, then other pairs of coordinates (a:b) correspond to the complex number a/b. (there are other correspondences, but this is the one most natural for our choice of coordinate chart)

+ and * are 'rational mappings' (i.e. given by algebraic formulas) on the finite points of the Riemann sphere, and so can be extended to all of the sphere by

(a : b) + (c : d) = (ad + bc : bd)
(a : b) * (c : d) = (ac : bd)

and you can see that these are only partial operators, because both (1:0)+(1:0) and (1:0)*(0:1) would result in the undefined coordinates (0:0).

If we use projective coordinates to define a number system, and we also allow (0:0) to be a number, then the resulting structure is precisely (isomorphic to) the wheel of fractions of complex numbers! (The involution is given by /(a:b) = (b:a))



I should point out that wheels probably won't be useful for your analysis course. Firstly, it isn't the projective reals that are interesting, but the extended reals, which adjoin two points at infinity ($\pm \infty$), and the primary technique of analysis is the art of approximation, so when faced with things like '0/0' or $(+\infty) + (-\infty)$, the usual need is to refine your approximation to avoid these singular values rather than try to treat them in an exact fashion.

but I didn't think adding that point changed the fundamentals of the numbers it was based on

And correctly so; extensions don't change a structure, they simply add to it. (Let W be the wheel of fractions for R) Everything you know about R is still applicable to the real numbers in W; x(y+z)=xy+xz is still a valid identity for real numbers, and 0x=x²+1 still doesn't have a real number solution. The 'change' is that these facts may fail when you consider arbitrary elements of W: e.g. x(y+z)=xy+xz can fail for some of the "new" elements of W, and 0x=x²+1 does have a solution in W. (the solution is 0/0)



Incidentally, as to the 'calculus' of these various structures...

The reals are diffeomorphic to the open interval (0,1)
The projective reals are diffeomorphic to the unit circle
The extended reals are diffeomorphic to the closed interval (0,1)
The complexes are diffeomorphic to (0,1) x (0,1)
The Riemann sphere is diffeomorphic to the unit sphere

I'm not sure what the 'right' topology is for a wheel. As the paper suggests, the name wheel came from drawing the projective reals as a circle, and then putting 0/0 as an additional point in the middle of the circle... but I think that if you want the operations to be continuous (a very important thing!), you have to have something more perverse, such as the topology where 0/0 is a member of every closed set.

And I'd like to chart out a taxonomy of mathematical objects, to keep track of what makes a field different from a group or a ring or a pseudo-ring. Are there resources for this?

Normally, you would learn about these structures in an abstrat algebra course. Depending on your level of interest, it might be best to just wait until then before paying much attention to groups and rings (or different kinds of fields). I don't really know a good resource for learning about these things on your own, aside from simply acquiring an abstract algebra book and working through it.