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The square of a real number is always a real number.
Yes, zero is a real number. It is not a counting number, but it is an integer, a rational number, and a real number.
Of course, not only can it be a real number but it is a real number. When you take the square root times itself, the result is a number that is real.
A real number dosen't have to be a rational number as a real number can be rational or irrational i.e the root of 2 is irrational and real. So is (pi).
Any decimal number is a real number.