system
Given the following post, respond as if you're commenting on Reddit:
human
Why is ∞* 0 ≠0

It looks like a simple math. I mean, I know infinity is some number very very big, but regardless of the magnitude of infinity, I would assume if I multiply that number with 0, then I would get 0.
gpt
As many other comments have stated, infinity is not a number, it is an idea.

you may ask yourself, is there an integer number which is bigger than all other integers?  The answer is no.  If you take your integer x, and you add 1 to it, you get x+1, which is always an integer bigger than x.  Thus there is no maximal integer.

Since the set of integers is contained in the set of real numbers, there is also no such thing as a maximal real number.

In fact, as a "number" infinity cannot be well defined.  If you think about all possible integers, the number of them is called "countably infinite".  If you think about the bigger set of all real numbers (including irrationals like sqrt(2) and pi), the amount of them is even bigger, called "uncountably infinite".  

These two levels of infinity are different, but still both maintain the property that they are "bigger" than any number you can think of.

Similarly, the number zero is often misunderstood.  Many times it is the constant value which is neither negative nor positive.  Many other times, zero is the result of a limit, which is the value which a sequence converges to after "infinitely many terms"

In this sense, we can define "zero" times "infinity" as the product of the output of two sequences which respectively converge to these ideas. Thus, we can pick a few different examples of sequences who have products converging to any output you desire:

Example 1: a_n= 1/n , b_n = n^2
it is easy to see that a_n -> 0 and b_n -> infinity 
but a_n *b_n = (1/n)(n^2)=n, which also goes to infinity

Example 2: a_n = 1/n^2, b_n=n
again we see a_n -> 0 and b_n -> infinity
but a_n*b_n = (1/n^2)(n)= 1/n, which goes to zero

Example 3: Let r be a real number and a_n=r/n and b_n=n
then a_n ->0 and b_n ->infinity
but a_n*b_n = (r/n)(n)=r, which is exactly our real number

thus we can pick different sequences who have limits zero and infinity but whos product ends up being any value we want.

Again this is because infinity is NOT a number, and is therefore NOT precisely defined.  thus, the concept of multiplication has no real meaning for it, and we can only consider it was the result of taking limits.

remember, multiplication is just a binary operation, which maps the xy-plane onto the real line, and is therefore not defined on its boundary (which includes such cases as "infinity * real number" and "infinity * infinity" )