Is \frac {1}{\infty} equal to 0 or infinitely close to 0?

[Warr]

Is \frac {1}{\infty} = 0 , or is it just infinitely close to 0?

 

[mathman]

= is correct, the other alternative is not too meaningful, unless you get involved with non-standard analysis.

 

[master_coda]

= is correct, the other alternative is not too meaningful, unless you get involved with non-standard analysis.

Unless you're using non-standard analysis I don't think the statement \frac{1}{\infty}=0 is even meaningful.

 

[jcsd]

Well, you could say 1/x -> 0 as x tends to infinity (though that obviously doesn'; mean 1/inifnity = 0 especially when infinity isn't even a number).

 

[Warr]

Lets say there is 1 unit of something in an infinitely large area...then would you say = ? Because then that says that the unit doesn't even exist...

 

[jcsd]

No in that situation all we would be saying is that it would be meaningless to talk about the ratio of the area to the unit area.

 

[Warr]

No it isn't...Because that unit DOES exist. But by saying 1/inf = 0...we say it is non-existant. In the same way, human population with respect to time would be 0 if the above statement were true. This is not so...

 

[Hurkyl]

1/∞ doesn't have a "standard" meaning; in some systems where infinite numbers are defined, division doesn't exist. In some others, 1/∞ is some infinitessimal positive nonzero number. In others, 1/∞=0.

If you're thinking of ∞ as that "big number that sits at the positive end of the real numbers", then you probably mean to use the extended real numbers, where 1/∞ is defined to be equal to zero.

 

[Integral]

In my books when infinity is defined as an extension to the Real number line, operations on infinity are also defined, included with these definitions is:

\frac 1 \infty = 0

This is a very specific definition for a very specific application ie the real numbers. If you attempt to apply this definition out of context your results may vary.

 

[JonF]

I always thought it meant infinitely close to zero, and that’s why the delta at the end of an integral doesn’t yield zero results, because the delta doesn’t actually = zero, just something infinitely small.