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actually, this would imply that there is no infinity, just an illusion of a really big number. and your first example would equal A because:

say A=10

10 / (infinity - (infinity + 1))

remember order of operations...
10 / (infinity - infinity)
10 / 0 (cuz a value minus itself is always nothing)
10

so, A / (infinity - [infinity + 1]) = A
but you see, this is solved algebraically / mathematically, and does not use a branch specifically designed for infinite values. but its a good start, so try building from here, k?

Judging from the response in your S&T thread, I doubt the viability of this proposal.

I think there are a couple of things wrong with this thread tyrandan. First of all: 500g is too little for a full report, second of all: why don't you make this a general topic, you will get no responses for this. This is an interest based guild, people will respond if they are interested, and they do it for physics. No one is here to sell their knowledge. Ie, VorpalNeko is giving introductory lessons in special relativity for free. Now, you should do the same, and you should post this information for everyone.

I've edited your title, now please, dont' trail the S&T dirt over here, the 500g part was rather vulgar.

In regard to the 1/infinity example on the second post... It wouldn't be 0. if you were to graph 1/x you would see that it never equals 0. Though it would get very close.

Why? It makes no sense in the projective extension of the reals, since A - B is by definition A+(-B), wherease there is no negative infinity. In the affine extension, there is a negative infinite, but it cannot be added to infinity because that operation leads to contradictions. If you want to be able to subtract infinite values, then take a look at other systems, such as the surreals. However, what happens is that there will not be a unique positive "infinite" number--e.g., in the surreals, there is actually a proper class of numbers greater than every real number. So, for example, one may have
ω > x, for any real x
ω² > xω, for any real x
ω^ω > ω^x, for any real x,
... etc.


No! Division by zero being defined collapses the number system unless one gives too many of the standard properties of arithmetic. The only sane approach is for division by zero to be undefined (curiously, one can have zero divisors and still have a fairly meaningful system, i.e., nonzero a,b with ab = 0... this is obvious in modular arithmetic--take Z/nZ with nonprime n).


The function is not defined until the domain (and codomain, actually) is specified. If the domain is the reals, then of course you are correct--the function will never give 0. However, if the domain is (for example) either the projectively or affinely extended reals, then it will give a 0.


True, but it's really not the primary issue when we're dealing with extensions of the number system--if we're changing our arithmetic, there is no reason for 1/x not to give zero if we define things appropriately.

Personally, it surprises me quite a bit that when discussing things like operations with infinity, people automatically resort to calculus-like limits. There is really no reason to interpret expressions like "A/0" or "inf/inf" in terms of limits of functions. Such expressions certainly do not mention any limits of functions anywhere.

I'm willing to be that you have covered functions sufficiently to understand this. A function f:A→B maps each element from the domain A to exactly one element in the codomain B. If the sets are some types of numbers, this can be an algebraic relationship (e.g, 1/x), but the definition is much more general than that. For example, a function could map the students in your class to the last test grade that they received, or the set of all living people to their birthday. In those cases, there is nothing numeric about the domain (in both times, it is a set of people).

Take something like f(x) = sqrt(x). Is this a well-defined function? Well, not really, although it is frequently understood that the domain is the non-negative real numbers R*. Thus, we could say f:R*→R. But the statement "f(x) = sqrt(x)" contains no such information about the domain, nor does it force it. For example, we could take the domain to be all real numbers and the codomain to be the complex numbers, i.e., f:R→C. Or f:C→C, for that matter. The point is that the domain was not specified, so all those different domains give different functions.

Similarly, your "f(x) = 1/x" does not specify the domain. We could say the domain is all nonzero real numbers (or positive reals, or any other set of reals that does not include 0). That is the standard assumption, in which case it is indeed true that f never gives 0. But the objective of the OP is to make a another number system (however incoherently he is going about this task), and so there is no reason to assume that the domain of f refers to a set of reals--if it does, it's not relevant to the task at hand.

Given the arithmetic, it looks like the ordinals or the surreals would be the best bet.

See: http://mathworld.wolfram.com/OrdinalNumber.html
and http://mathworld.wolfram.com/SurrealNumber.html