# Define negative numbers to be greater than infinity

by Count Iblis
Tags: define, greater, infinity, negative, numbers
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 Quote by Count Iblis
So, if I empty my bank account, I am flat broke, but if I overdraw by \$1, I am the richest person in the universe?

Just what does it mean to be "greater than infinity"?
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 Quote by DecayProduct Just what does it mean to be "greater than infinity"?
Ask Rom R. Varshamov and Armen G. Bagdasaryan. They wrote the paper, apparently.

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## Define negative numbers to be greater than infinity

This sounds like the end result one would obtain by applying twos-compliment arithmetic to the set of {reals, infinity}.
 P: 48 Let a be any negative real number, and b be any positive real number. a < b Hence, how in the world is negative numbers greater than infinity?!
 P: 367 Because, in this paper, they define a new order relation (I'll write <<) such that 0 << 1 << 2 << 3 << ... << -3 << -2 << -1. In this way, for any negative a and positive b, we have b << a. If we were to add an element infinity to this, then we would have b << infinity << a for any negative a, positive b.
P: 48
 Quote by Moo Of Doom Because, in this paper, they define a new order relation (I'll write <<) such that 0 << 1 << 2 << 3 << ... << -3 << -2 << -1. In this way, for any negative a and positive b, we have b << a. If we were to add an element infinity to this, then we would have b << infinity << a for any negative a, positive b.
Hmm. I've never heard this kind of math before. Thanks for sharing with us, very interesting one.
 PF Patron HW Helper Sci Advisor Thanks P: 11,935 It sounds like complete nonsense. To begin with, the authors seems to confuse concepts like axioms and conditions.
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 Quote by arildno It sounds like complete nonsense. To begin with, the authors seems to confuse concepts like axioms and conditions.
What matters is if the results derived in the paper are useful. I mean, when Dirac wrote in his book: "principles of quantum mechanics" that the derivative of Log(x) should contain a term proportional to a so-called "delta function" that he had just invented out of thin air a few pages back, was complete nonsense too. The whole notion of a delta function in the way he explained it, was inconsistent in the first place.
 HW Helper Sci Advisor P: 3,682 They're giving up transitivity, which is a pretty big blow. What does their system gain?
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 Quote by CRGreathouse They're giving up transitivity, which is a pretty big blow. What does their system gain?

I only had a quick look yesterday. It seems to me that the gain is that you have a more efficient formalism for doing computations involving divergent series.
 Mentor P: 4,182 I'm stuck on definition 2.1. How is that supposed to work for 0?
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 Quote by Office_Shredder I'm stuck on definition 2.1. How is that supposed to work for 0?
Using footnote 2 and certain bad assumptions you can give it the intended order where 0 is less than any nonzero element. If their caviler attitude bothers you, let 2.1 apply only to nonzero numbers and adjoin 0 in such fashion.
 P: 607 I looked it over. There is some interesting material in there. The "new ordering" isn't the significant part. I think it works to their disadvantage to use existing notation like $$\sum_{n=a}^b f(n)$$ with their new meaning. Better would be using a different notation.
 P: 33 it doesnt make sense simply because it is a different mathematical system than the one weve become accustomed to, you cant compare its results with traditional mathematical problems because the value of infinity is more "numerous" than a negative. its abstract in a way that makes less realistic sense but more ordering efficiency. just as imaginary numbers are used in situations when real numbers cannot provide a solution.
 PF Patron Sci Advisor Emeritus P: 16,094 I do think you guys are being too hard on them. Constructing linear operators that extend the domain of summation is not that uncommon. I doubt the ordering on Z that they use is actually relevant -- it just for whatever reason happened to suggest a path.
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 Quote by ZacharyFino it doesnt make sense simply because it is a different mathematical system than the one weve become accustomed to, you cant compare its results with traditional mathematical problems because the value of infinity is more "numerous" than a negative. its abstract in a way that makes less realistic sense but more ordering efficiency. just as imaginary numbers are used in situations when real numbers cannot provide a solution.
You've managed to change my ideas on this paper from "majorly screwed up" to "some pretty cool stuff"

 a new method for ordering the integers, from which we get Z = [0, 1, 2, ...,−2,−1]
Where do the integers switch from positive to negative? In our accustomed number system, zero is basically the turning point, but for this system in my eyes it seems to be 1/0 which suggests there is no switch, but a grey fuzzy area of $$+\infty \rightarrow -\infty$$ ??
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