Yes it is a real number. Real numbers exist on a number line, so it doesn't matter big they are, they will always fit.
Complex numbers on the other hand will never have a place on that number line.
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The square roots (not sqare route!) of 99999999999999999999999 are ±316,227,766,016.8 approx.
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).