Yes, it is considered a number. In the past, people had difficulties accepting zero as a number, but nowadays it is universally accepted as such.
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Yes it is considered as a natural number.
Yes, it is true that any number multiplied by 0 is 0. But zero is not considered a "multiple" (integral product) of any number.
Some examples of zero integers are 0, -0, +0. Zero is considered an integer because it is a whole number and can be represented on the number line.
They are not. 0 (which is a whole number) is not considered a counting number(natural #).
Yes, any number/term that does not contain a variable (x, y, t, etc.) is considered a constant.