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Irrational numbers.
The square roots of negative numbers.
Yes, except that all irrational numbers will be non-terminating, non-repeating decimals.
They are real numbers. All irrational numbers are non-terminating. Also, any rational numbers which, in its simplest form has a denominator which contains a factor other than 2 or 5 (or their powers) is non-terminating.
The answer depends on what are meant to be real numbers! If all the coefficients are real and the matrix of coefficients is non-singular, then the value of each variable is real.