Yes.
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Every rational number does.
It must be a generalised rational number. Otherwise, if you select a rational number to multiply, then you will only prove it for that number.
Any rational number will do.
Yes, it is possible only if an irrational number is multiplied with 0.
Maybe. It depends on the number that you are multiplying. Pi multiplied by (1 / 4pi) = 1/4. A quarter is definitely rational.