complex number
Often, we use complex numbers in physics to simplify calculations - for example, the voltages and currents in an electronic circuit have real values, but in a.c. problems, where they change sinusoidally with time, we can represent them as complex numbers and thus include the amplitude and phase of the variation in one number. It then turns out that we can do arithmetic with these numbers to work out what is going on in the circuit, which is much easier than having to solve lots of coupled differential equations to get the form of the functions. But in this case, the underlying theory still deals in real quantities, and the complex numbers are 'just' a mathematical convenience, albeit one we couldn't do without.
However, there is in principle no reason why physical theories can only use real numbers. Of course, they must successfully predict the real values of distances, times, temperatures and so on that we can measure in experiments, but the theories themselves can contain complex numbers. In quantum mechanics, the wave-function 'y(x)'is in general complex, but it is analysed in such a way that it can make predictions about real numbers that can be observed in experiments. For example, the 'probability density' is given by |y(x)|2, which is real.
also
According to the great Feynman:-
Any standard oscillation generates a standard solution,
x = C1exp{ikt} + C2exp{-ikt}
This is complex.
The complex number is actually an easy way to put more information.
and
In solving a physical problem, complex numbers are often used as a mathematical convenience. The physical answer is usually the real part of the final answer. For instance, if an answer for x is x=exp(-iwt), the physical result is cos wt.