# Multiplication of infinity and zero

1. Feb 18, 2010

### samieee

What will happen if we multiply infinity with zero? how to describe this situation?

2. Feb 18, 2010

### CompuChip

Nothing happens, because we don't do that.

In fact, you can't multiply "infinity" with anything at all, because it's not a number. It's just a mathematical concept that we "invented" because it is convenient.

3. Feb 18, 2010

### mnb96

Just in case you made that question because you were evaluating some function, you might want to remember that $$\infty\cdot 0$$ is an http://en.wikipedia.org/wiki/Indeterminate_form" [Broken].
The limit of the function to that point might still exist and be finite.

Last edited by a moderator: May 4, 2017
4. Feb 18, 2010

### Hurkyl

Staff Emeritus
It depends on the number system, of course.

However, in all commonly used number systems that have a number called "infinity" and a number called "zero" and a product, one is not allowed to multiply infinity by zero -- so talking about the result of that product is in the same class of nonsense as other English questions like "How much does blue weigh?"

5. Feb 18, 2010

### CompuChip

[offtopic]

About 4.5 x 10-36 kg (taking 475 nm for the wavelength).

[/offtopic]

6. Feb 18, 2010

### torquil

[offtopic]
Right on!

However, that is only the effective relativistic mass (E/c^2) of one blue photon. The concept "blue" doesn't depend on the number of blue photons, so Hurkyl was right that it doesn't make sense

[/offtopic]

7. Feb 18, 2010

### KnowPhysics

when you say zero it means nothing (does not exist). if you multiply how many ever times of some thing which is not existing, it will be still nothing (zero).

8. Feb 18, 2010

### mnb96

That is true (and intuitive) only if you multiply a finite number with zero.
You are not necessarily allowed to treat $$\infty$$ as a simple finite number, otherwise you are basically suggesting that $$\infty \cdot 0 = 0$$ is always valid, which is not true.

9. Feb 18, 2010

### arildno

Mmm..nope.

Consider the following example:

x*(1/x)=1, irrespective of the (non-zero) value of x.

As you let "x" go to infinity, the first factor goes to infinity, whereas the second to 0.

Yet the product equals 1, nonetheless..

10. Feb 18, 2010

### KnowPhysics

but still (1/x) is not zero. regardless of x value. it could be negligible but it is not zero.

11. Feb 18, 2010

### CompuChip

And if you do that, you can also consider
x*(2/x) = 2
x*((-1/12)/x) = -1/12
x*(a/x) = a for all numbers a

and thus "show" that $0 \cdot \infty$ is equal to any number you please. Which leads to a contradiction (of course, we want a multiplication between two numbers to always give the same result).

12. Feb 18, 2010

### Hurkyl

Staff Emeritus
Huh? What does this have to do with anything?

You might have made sense if you were talking about cardinal numbers*... but there is no cardinal number called "infinity". However, each infinite cardinal $\alpha$ does indeed have the property that $\alpha 0 = 0$.

But this is only for the cardinal number system, and has no bearing on what may be true, false, or nonsense in other number systems.

*: I find it unlikely this is what you were getting at, though -- I expect you didn't have any number system in mind, but instead a particular physical interpretation, and you were trying to apply that interpretation in a domain where it doesn't even apply. (i.e. it doesn't even make enough sense to ask if the interpretation is valid or invalid)

13. Feb 19, 2010

### KnowPhysics

yes, Math is not my Domain. so you are saying zero is not a null value. please can you clarify why only with infinity we should not consider zero as a finite (Null) value. please if you can refer any book or link which can explain this it will be great because my mind is disturbing me.

14. Feb 19, 2010

### Hurkyl

Staff Emeritus
Infinity is a red herring. It is a fact that zero is a null value for the operation of multiplication of real numbers. If you are interested in a different multiplication operation -- e.g. the multiplication of extended real numbers -- then anything could happen.

Most number systems have various motivations: the motivation of the arithmetic of extended real numbers is to be a continuous extension of arithmetic of real numbers. Multiplication cannot be continuously extended to 0 * +infinity or 0 * -infinity. Therefore, the arithmetic of extended real numbers is defined so that 0 * +infinity is nonsense.

15. Feb 19, 2010

### KnowPhysics

Still i am not clear why they use infinite if it doesn't make sense. like if you divide still it is infinite. if you subtract still it is infinite. Any way Thank you for explaining. i will also study little bit on this for my curiosity to understand it.

16. Feb 19, 2010

### Hurkyl

Staff Emeritus
Because it's useful.

The fact 0 * +infinity is nonsense does not diminish the utility of the extended real numbers for many tasks -- such as simplifying calculus. In fact, that 0 * +infinity is nonsense is actually a feature -- things would be very awkward if things were defined differently so that that product has a value.

And, just to reiterate the other point I've been trying to make, other number systems have their own considerations. For example, the cardinal numbers have to do with "counting" -- or more precisely of bijections between sets. The product of cardinal numbers relates directly to the Cartesian product of sets. 0x=0 is an identity of cardinal numbers, because $\emptyset \times S = \emptyset$ is an identity of sets.

17. Feb 19, 2010

### Staff: Mentor

Not necessarily, at least in the context of indeterminate forms, which were mentioned elsewhere in this thread.
Several indeterminate forms involve infinity.
$$\left[\frac{\infty}{\infty}\right]$$

$$\left[\infty - \infty\right]$$

$$\left[1^{\infty}\right]$$

Keep in mind that these are only forms that are used to represent different limit types. They usually appear in brackets to emphasize the fact that they don't represent actual arithmetic problems. We're not actually going to try to carry out the division, subtraction, and so on.

For each one of these forms or categories, as many limit problems as you like can be found, each with a different limit that can be any arbitrary real number, as well as positive or negative infinity.