# Definite product of zero and infinity?

1. Apr 3, 2004

### Antonio Lao

Can the product of zero and infinity be defined as a finite quantity?

The differential calculus makes the limit approach to zero possible (existence of a derivative).

The integral calculus makes the limit approach to infinity possible (convergence of infinite series).

2. Apr 3, 2004

### matt grime

The product of something that tends to zero, say a function f as x tends to 0, and something that tends to infinity also at zero may have a limit or it may not.

x and 1/x say, the product is 1, and the limit is 1. 2x and 1/x has 2 as the limit of the product. x and 1/x^2, the product has no limit as x tends to zero.

If you just want to declare that 0*infinity is something you'll need to justify why there is number 'infinity' in your number system, or indeed what your number system is, as it ain't the reals.

3. Apr 3, 2004

### Antonio Lao

The cases you gave look like product of inverse functions. I am going to give an example of what is it I have in mind.

In the theory of gases, Boyle's law says at constant temperature, the product of pressure and volume is a constant.

If the pressure goes to infinity then the volume goes to zero to keep the product constant. Can a definition be made when the values are exactly at infinity and zero and still keep the constant?

4. Apr 3, 2004

### matt grime

What makes you think the law is still valid there? What even makes you think that isn't a nonsensical question to ask physically? Mathematically if y = 1/x for x not zero, then $$\lim_{x\to 0} xy = 1$$

5. Apr 3, 2004

### Chen

Why should the law still apply when talking about something as unphysical as "infinite pressure"?

6. Apr 3, 2004

### Antonio Lao

Thanks for all your replies. But physically, we can say such thing as zero pressure which is the same thing as there is no force. Now the volume is infinite.

7. Apr 3, 2004

### Michael D. Sewell

Zero pressure is in no way indicative of infinite volume.

8. Apr 3, 2004

### matt grime

No, definitely not - 'zero pressure' would correspond to absolute zero, there is still a volume there. The ideal gas law is not valid at these extremes.

Note, this answer was directed at Antonio, not Michael

Last edited: Apr 3, 2004
9. Apr 3, 2004

### Michael D. Sewell

Extra credit question on test:

Does a vacuum with a volume of 3 meters^3 contain three times as much Nothing as a vacuum with a volume of 1 meter^3?

10. Apr 3, 2004

### Antonio Lao

Thanks. Now to give another case where this is more paradoxical is to show that photon has zero mass.

$$m(v) = \frac {m_0}{\sqrt {1 - \frac {v^2}{C^2}}}$$

11. Apr 3, 2004

### Zurtex

I'm not a physicist but don't all vacuums contain at least something even if this be a very small amount of energy?

If not then by the way you are measuring them they must contain dimensions (at least 3 of them) and therefore are in fact not nothing but rather just space.

12. Apr 6, 2004

### Antonio Lao

Vacuum is known to be the site of infinite amount of energy. The phenomenon of vacuum fluctuation attests to this fact. The period of time that this infinite energy exists is almost zero. These are conjugate variables in quantum mechanics.

$$\delta E \delta t \geq h$$

E is energy approaches infinity. t is time approaches zero. h is Planck constant.

13. Apr 6, 2004

### matt grime

"infinite" amount of energy? Wow, stand well back from that or you might lose your eyebrows.

14. Apr 6, 2004

### Antonio Lao

This is again a case where the product of near infinity and near zero is finite.

15. Apr 6, 2004

### matt grime

Doesn't what you've written say the product "near zero and infinity" is not zero; it doesn't put an upper bound on it at all.

16. Apr 6, 2004

### Antonio Lao

The product cannot be less than Planck constant (lower bound). The upper bound is beyond quantum mechanics into classical mechanics.

17. Apr 7, 2004

### HallsofIvy

Staff Emeritus
No, mathematics is not bound by physical constraints. The moment you talk about "infinity" you are no longer talking about any mathematics that can be applied to physics. If you are talking about numbers "near infinity" and "near zero" then you cannot have a specific numerical answer because you do not have specific numerical input.

18. Apr 7, 2004

### Antonio Lao

Still I don't understand how the product of "infinity" and "zero" is finite? This happens very often in physics. To understand is the purpose of my thread.

19. Apr 7, 2004

### matt grime

The product of infinity and zero is not defined. You seem to have misinterpreted a statement about limits. It is perfectly possible for the limit as x tends to zero of f(x)g(x) to be any real number (or infinity) if f tends to zero as x tends to zero and g tends to infinity as x tends to zero. eg f(x)=kx g(x) = 1/x then f(x)g(x) tends to k as x tends to zero. Whether or not there is any meaning "at infinity" is debatable.

20. Apr 7, 2004

### Antonio Lao

I encountered two situations when I tried to find the energy of the universe whether it is zero or it is infinite.

If I am allowed to play with just two physical constants of nature, the speed of light c in vacuum and Planck's constant h, and further I assume two more variables that of acceleration a and length r, I can give two energy formulations as the following:

$$E_0 = \frac {ah}{c}$$

and

$$E_{\infty} = \frac {hc}{r}$$