Is the reciprocal of infinity, zero?

[Paul Dirac]

Thanks!

 

[axmls]

The first thing I'd say is that infinity is not a number, and so it doesn't make sense to talk about its reciprocal. In the context of calculus, we can refer to a limit, namely that $$\lim_{x \to \infty} \frac{1}{x} = 0$$
That is to say, as x gets larger and larger, this term gets closer and closer to 0, but there is no real number called infinity that we can manipulate like a number.

Now, if I recall correctly, in the context of nonstandard analysis, you can technically call the reciprocal of infinity a "differential" (given certain assumptions), but this is avoided in standard analysis (the standard calculus that we use). When talking about infinities, we are typically dealing with limits.

 

[phinds]

I completely agree w/ axmls ... your question is predicated on the false assumption that infinity can be treated like a number.

EDIT: I've heard that in certain very specific methods it CAN be treated like a number, but even if that's true, yours is not one of them.

 

[micromass]

First of all: infinity is NOT a number and can not be treated as one!

That said, the problem is that there is not a unique thing that's called infinity. There are various notions of infinity, and they don't all behave in the same way. In particular, the answer to your question depends on which notion of infinity you take.

One popular thing to do is to take the set of all real numbers $\mathbb{R}$ and add $+\infty$ and $-\infty$. We then obtain the affine real line, which is denoted as $\overline{\mathbb{R}}$. In that context, it is indeed true that

$$\frac{1}{+\infty} = \frac{1}{-\infty} = 0$$

however, things like $\frac{1}{0}$ are still undefined.

More information: https://www.physicsforums.com/threads/questions-about-infinity.507003/

 

[WWGD]

The first thing I'd say is that infinity is not a number, and so it doesn't make sense to talk about its reciprocal. In the context of calculus, we can refer to a limit, namely that $$\lim_{x \to \infty} \frac{1}{x} = 0$$
That is to say, as x gets larger and larger, this term gets closer and closer to 0, but there is no real number called infinity that we can manipulate like a number.

Now, if I recall correctly, in the context of nonstandard analysis, you can technically call the reciprocal of infinity a "differential" (given certain assumptions), but this is avoided in standard analysis (the standard calculus that we use). When talking about infinities, we are typically dealing with limits.

In the extended Real numbers, infinity is an actual number. And in the non-standard Reals , there are indefinitely small and infinitely -large numbers.

 

[Raghav Gupta]

One can apply 1/∞ as 0 in many applications, like in many physics problems.
Directly equating it with zero doesn't make sense.
You can see by logic,
1/1000 = 0.001
1/1000000 = 0.000001
So as the denominator is increasing the value is approaching 0, as has been pointed by some members.

 

[Pjpic]

This may not apply, but here's a quote from Wiki:

As mentioned above, zero, the origin, requires special consideration in the circle inversion mapping. The approach is to adjoin a point at infinity designated ∞ or 1/0 .

 

[phinds]

This may not apply, but here's a quote from Wiki:

As mentioned above, zero, the origin, requires special consideration in the circle inversion mapping. The approach is to adjoin a point at infinity designated ∞ or 1/0 .

See post #4

 

[mathwonk]

as these answers and examples show you have to be a bit careful about what world you are working in. For some purposes as micromass illustrated, one can introduce two new numbers both infinitely large, called plus and minus infinity and calculate with them carefully, and in particular their reciprocals behave like zero. Then 1/0 makes no sense.

But in another setting, where one gives up ordering, as in the complex numbers, sometimes one introduces just one new point, not quite a number, called infinity and then again 1/z extends to a mapping defined also at zero and taking zero to infinity and infinity to zero. Interesting in this setting is that one now makes sense of 1/0 = infinity, but now (0.infinity) still gives trouble. it often ain't 1, e.g. i.e. now 1/0 has a unique value but (0.infinity) does not. This new set is called the Riemann sphere. In this setting again infinity is really not a number like the other complex numbers, but a point of a topological space with a few properties of a number.

I always wonder sort of what micromass means when he insists infinity is not a number and cannot be treated like one, but maybe he means it will always fail to satisfy some of the number properties, no matter how hard you try to treat it like one. I admit that in calculus I often am tempted to treat it some what like a number at least in taking limits. Of course I know what the precise definitions are but I take shortcuts with it too. I.e. if the limit of f is infinity, (precise meaning exists) then the limit of 1/f is zero and I am tempted to just think of that as lim(1/f) = 1/(limf) = 1/infinity = 0.

micromass's advice should of course be heeded by beginners and maybe others among us. It is certainly safer to take it as a starting point, and then carefully ask what partial exceptions may exist.