Define negative numbers to be greater than infinity

In summary, this conversation is about a paper written by Rom R. Varshamov and Armen G. Bagdasaryan, where they discuss a new order relation and number system with the inclusion of infinity. While the paper may seem nonsensical at first, it offers a more efficient formalism for computing with divergent series. There is some interesting material, but it may benefit from being written in a more acceptable form by a mathematician or mathematics student.
  • #1
Count Iblis
1,863
8
http://arxiv.org/abs/0907.1090" [Broken] :smile:
 
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  • #2
Count Iblis said:
http://arxiv.org/abs/0907.1090" [Broken] :smile:

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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  • #3
DecayProduct said:
Just what does it mean to be "greater than infinity"?

Ask Rom R. Varshamov and Armen G. Bagdasaryan. They wrote the paper, apparently.
 
  • #4
This sounds like the end result one would obtain by applying twos-compliment arithmetic to the set of {reals, infinity}.
 
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  • #5
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?!
 
  • #6
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.
 
  • #7
Moo Of Doom said:
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.
 
  • #8
It sounds like complete nonsense.

To begin with, the authors seems to confuse concepts like axioms and conditions.
 
  • #9
arildno said:
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. :biggrin:
 
  • #10
They're giving up transitivity, which is a pretty big blow. What does their system gain?
 
  • #11
CRGreathouse said:
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.
 
  • #12
I'm stuck on definition 2.1. How is that supposed to work for 0?
 
  • #13
Office_Shredder said:
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.
 
  • #14
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 [tex]\sum_{n=a}^b f(n)[/tex] with their new meaning. Better would be using a different notation.
 
  • #15
it doesn't make sense simply because it is a different mathematical system than the one we've become accustomed to, you can't 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.
 
  • #16
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.
 
  • #17
ZacharyFino said:
it doesn't make sense simply because it is a different mathematical system than the one we've become accustomed to, you can't 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" :smile:

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 [tex]+\infty \rightarrow -\infty[/tex] ??
 
  • #18
Count Iblis said:
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. :biggrin:
Sure enough, it just seemed extremely amateurish at first glance not the least the initial discussion concerning the "correctness" of the 18th century view, which they seemed to espouse.

If they had said that they had been INSPIRED by that view to construct a new number system, rather than pushing for its "correctness", I would have been less suspicious of it.

I haven't bothered to look much further into it, I'll admit.
 
  • #19
arildno said:
Sure enough, it just seemed extremely amateurish at first glance.

I think "amateurish" is apt. It seems clear neither author is a mathematician. But (unlike most papers with that characteristic) this seems to have some good content. Maybe what they need is a mathematician (or mathematics student) to take their material and write it in a more acceptable form. Maybe with some other notation... replace the new [tex]\Sigma_a^b \;f(n)[/tex] with [tex]{\mathbb{S}}_a^b \;f(n)[/tex] or [tex]{\oplus}_a^b \;f(n)[/tex] and something similar for the new limit
 
  • #20
Here are some telltale signs of crackpottery:

However, some properties of negative numbers had been remaining unclear for a long time, in particular, the order relation between positive and negative numbers.

I really don't see anything that is unclear about the usual ordering of the integers.

”nothing”

That is, they used the wrong quotation mark in LaTeX, should be ``nothing''

Definition 2.5 is nothing but a telescopic sum.

I stopped reading after that.
 
  • #21
Dragonfall said:
Here are some telltale signs of crackpottery:

I think inexperience ("amateurish" as arildno and g_edgar) + hubris suffice to explain those.

The paper seems fine to me. I don't know if it's interesting or not, though.
 
  • #22
That's true, I take it back.
 

1. What are negative numbers?

Negative numbers are numbers that are less than zero. They are represented by a minus sign (-) in front of the number.

2. How are negative numbers related to infinity?

Negative numbers are considered to be greater than infinity because they are infinitely far away from zero in the negative direction on the number line.

3. Why are negative numbers defined to be greater than infinity?

This definition allows for consistency in mathematical operations and helps to extend the number system to include values less than zero.

4. Can negative numbers ever be equal to infinity?

No, negative numbers cannot be equal to infinity. Infinity is a concept, not a specific number, and therefore cannot be equal to any other number, including negative numbers.

5. How do we use negative numbers in real life?

Negative numbers are used in real life to represent values that are less than zero, such as temperatures below freezing, debts, and elevations below sea level. They also play a role in mathematical concepts such as direction and distance.

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