Non-Field Structure of the Reals, Projective System Preferred

In fact, the real numbers do not have the structure of a field. Rather, they are
the limit of a projective system. Thus, the real numbers are more accurately
viewed as a completion of the rational numbers. This means that any real
number can be expressed as a limit of rational numbers, and the operations of
addition, subtraction, multiplication, and division on real numbers can all be
approximated and performed through these rational numbers.