# Infinity subtracted from infinity is undefined.

1. Jan 9, 2008

### Hippasos

Let n be any number.

infinity + n = infinity

n = infinity - infinity

n = undefined

!...?

2. Jan 9, 2008

### mgb_phys

If you're a mathematician - yes it's undefined
If you're a physicist then it can be zero or infinity depending on which makes your theory work!

3. Jan 9, 2008

### CRGreathouse

Yes, lots of operations in the extended reals are undefined.

4. Jan 9, 2008

5. Jan 9, 2008

### al-mahed

the second statement doesn't make any sense, am I right?

Could you explain?

6. Jan 9, 2008

### MaWM

Actually.. its alot more complicated. You need to know how quickly each of the infinities is diverging in order to compare them.

7. Jan 10, 2008

### HallsofIvy

Staff Emeritus
NEITHER statement makes any sense until you specify what what number system you are working in. "Infinity" itself is not defined in either the real number field nor the complex number field and so neither statement makes any sense in them. There are several different ways to extend the number systems to include "infinity" and operations involving "infinity". Those statements may or may not make sense depending on the system.

8. Jan 10, 2008

### nicksauce

Let n = be any number
cat + n = blue
n = cat - blue
n = undefined ?

Infinity is not a real number, so you cannot perform operations like infinity + n, just like you cannot perform operations like cat + n, or n = cat - blue.

9. Jan 11, 2008

### Gib Z

It was a joke, poking fun at the way physicists do mathematics.

10. Jan 11, 2008

### yasiru89

You do realise mgb phys that your little tricks of trade have extrapolated from the mathematical theory of limits instead of adding to it? In functional form an expression of the form $$\infty - \infty$$ may be reduced to the form $$\frac{\infty}{\infty}$$ and evaluated by a method such as l'Hopital's rule(as long as putting into the required form does not make an answer obvious) to obtain a finite, infinite or null answer.

The space being worked on may be altered to fit the theory and methods of renormalisation introduced to deal with these forms. Even the most sloppy work can possibly be made rigorous but sloppy is never elegant...

Last edited: Jan 11, 2008
11. Jan 11, 2008

### al-mahed

ohhh, all right!!!

12. Jan 12, 2008

### Gib Z

I have no idea what any of that meant...

13. Jan 12, 2008

### yasiru89

Say some expression is of the form $$\infty - \infty$$ and defined by a limit on the variable(say x approaches a) by $$f(x) - g(x)$$
Since this may be changed into,
$$g(x).[f(x)/g(x) - 1]$$

We can deal with this case using l'Hopital's rule. This works when the limit exists of course but some 'physical' theories require more, the most famous being Quantum Field theory I believe. Then we work with alternate definitions and methods like going through extra dimensions to drop off infinities along the way. In such ways we are left with sensational results like $$\prod_{p} p = 4\pi^{2}$$, (product over all primes denoted)
The way physicists do mathematics needn't be wrong, radical or 'revolutionary', just insufferably poor.

I was speaking generally earlier but if this isn't straightforward enough for you I'd be at a loss to explain further.(without writing a book that is...on that note there are 2 great calculus books from the early 1900s by J. Edwards on the Internet Archive you might want to check)

Last edited: Jan 12, 2008
14. Jan 12, 2008

### Gib Z

This thread wasn't about a function approaching infinity, that is a normal thing easily dealt with. The original poster talked about arithmetic operations with infinity as a number. Two distinct things.

Not sensational results, just a load of dung. If the product of every element in the set of the prime numbers is equal to $4\pi^2$, then I am a cat. That product is not equal to any real number. If physicists had made some alternative way to formulate primes, then please, do not use the traditional Product operators without even giving notice that a different operator was meant.

Last edited: Jan 12, 2008
15. Jan 12, 2008

### yasiru89

A half dead Schroedinger's cat if you ask me, and the result is not due to physicists, I merely used it to illustrate that WITHIN (all the 'notice' you need had you bothered to read)certain theories(here but a little more than the meromorphic extension of the zeta function) supposedly 'outrageous' results hold true. The case of the result I presented is an analytically acceptable one, regularised using the Euler product.

As for my treatment of the issue Garfield, I simply added to what the subsequent posters were hinting towards, eg- rapidity of divergence.
I have little to add to HallsofIvy so far as arithmetic operations are concerned. Without specifics things go without saying.

16. Jan 12, 2008

### Gib Z

Not even physicists (at least at the level of those who create decent theories) have such terrible mathematics that they believe that product. Of course they know they are using some different operator. Perhaps this operators gives the right product for where the product of the usual operator exists, but its still different.

The only other poster who was hinting towards rapidity of divergence was MaWM, who is also misinterpreting the question. If you notice everyone elses posts, you'll realise we werent even talking about functions, order of divergence or anything very analytical at all.

PS. Something can't be half dead.

17. Jan 14, 2008

### Hippasos

Well, in my limited experience and knowledge, I would say then:

Arithmetics - as I know it - is a cat with a leg of an elephant(<-infinity) - detached or not I don't know...

Still confused...

Last edited: Jan 14, 2008
18. Jan 14, 2008

### zhentil

Easiest example I can think of: Let f(x)=x, g(x)=2x, h(x)=3x. As x goes to infinity, all three go to infinity as well, right? But if you try to do arithmetic with each "infinity," you get different answers (i.e. a contradiction). If we tried to calculate 3+2, and the answer was different than 3+1+1, we would have to be more careful about our addition rules. So it is with infinity.

19. Jan 14, 2008

### ramsey2879

No need to be confused. Just accept the fact that infinity is just a concept not a number sinced there is no such thing as a biggest or smallest number. Only if you accept that can you appreciate the beauty of mathematics. Also you can liken an infinite series that has a limit as the sum to an endless do loop in which one gets closer and closer to the answer without actually reaching it, except that you can see the what the answer should be by looking at the computer screen.

Last edited: Jan 14, 2008
20. Jan 14, 2008

### yasiru89

As was acknowledged had someone bothered to read the post. Of course in my most humble view, which I do not force upon you, it is in definition that mathematics reigns, unlike physical theories- thereby lending little except upon choosing an all too psychological conviction to one's beliefs. A conceptual clarity is very much elusive for my tastes.

That aside,
There's analytics involved everywhere(part of why I illustrated with a number-theoretic example) but if we are to take a simple point of view on the matter it should be noted that one can come up with as many systems as imaginable to include infinity as an operable quantity, however the real number field and most that follow(complex, biplex, etc.)do not succumb to such. On the other hand we have a measure of infinities like the case where functions stand in allowing a certain sense to be made of simple arithmetics on them.

Well there's the uncertainty; psychology and interpretation.