Mathematical rules are based on the defining
limits we place on the particular numerical quantities dealt with. When we
say that 1 + 1 = 2 or 3 + 4 = 7, we are implying the use of integer
quantities: the same types of numbers we all learned to count in elementary
education. What most people assume to be selfevident rules of arithmetic 
valid at all times and for all purposes  actually depend on what we define
a number to be.
For instance, when calculating quantities in AC circuits, we find that
the "real" number quantities which served us so well in DC circuit analysis
are inadequate for the task of representing AC quantities. We know that
voltages add when connected in series, but we also know that it is possible
to connect a 3volt AC source in series with a 4volt AC source and end up
with 5 volts total voltage (3 + 4 = 5)! Does this mean the inviolable and
selfevident rules of arithmetic have been violated? No, it just means that
the rules of "real" numbers do not apply to the kinds of quantities
encountered in AC circuits, where every variable has both a magnitude and a
phase. Consequently, we must use a different kind of numerical quantity, or
object, for AC circuits (complex numbers, rather than real
numbers), and along with this different system of numbers comes a different
set of rules telling us how they relate to one another.
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