Zero times infinity is defined as "indeterminate".
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No, infinity is not a number, in the mathematical sense. It is a symbol for "unlimited". As such it has uses in various theories in math and in physics. The simplest proof is adding a real value, such as 1, to infinity. Just as zero times any number is still zero, infinity plus any value will remain "infinity".
zero
Because zero multiplied by any number is always zero, but anything multiplied by infinity is infinity. Zero times infinity is being pulled both ways. Also, the definition of infinity is any number x divided by 0. When you multiply zero by infinity, the zeroes "cancel out", leaving absolutely nothing behind.
Infinity means some non-zero value divided by zero. It was commonly written as 1/0
No matter how many times you multiply zero by itself, you will always get zero.