Infinity x 0: Is the Answer Zero?

[dyn]

Hi.
Is infinity multiplied by zero defined and is it zero ?
And is infinity multiplied by any positive number , infinity ?

 

[jedishrfu]

Infinity is not a number in the normal sense of the word and hence applying any kind of arithmetic operation is undefined.

https://en.m.wikipedia.org/wiki/Infinity

You can find out more by googling on infinity and zero.

 

[Mark44]

Is infinity multiplied by zero defined

No

and is it zero?

No

And is infinity multiplied by any positive number , infinity ?

Yes

To qualify the answers above, I'm talking about limits.
For your first and second questions, here are three limits of the form $\infty \cdot 0$:
$\lim_{x \to \infty} x \cdot \frac 1 {x^2}$
$\lim_{x \to \infty} x \cdot \frac 1 {x}$
$\lim_{x \to \infty} x \cdot \frac 1 {\sqrt x}$
In all three limits the first factor "approaches" infinity while the second factor "approaches" zero.
The value of the first limit is 0; the value of the second limit is 1; the value of the third limit is $\infty$.

 

[member 587159]

Hi.
Is infinity multiplied by zero defined and is it zero ?
And is infinity multiplied by any positive number , infinity ?

It is in general not defined, but in some cases exceptions are made (e.g. measure theory).

 

[Mark44]

It is in general not defined, but in some cases exceptions are made (e.g. measure theory).

With regard to limits of the form $k \cdot \infty$, a positive constant times infinity equals infinity.
If $\lim_{x \to \infty}f(x) = \infty$, and k is a positive constant, then $\lim_{x \to \infty}k \cdot f(x) = \infty$, as well.

 

[member 587159]

With regard to limits of the form $k \cdot \infty$, a positive constant times infinity equals infinity.
If $\lim_{x \to \infty}f(x) = \infty$, and k is a positive constant, then $\lim_{x \to \infty}k \cdot f(x) = \infty$, as well.

Yes I commented on the $0.\infty$ part.