Complex number is a number that combines real and imaginary numbers. For example 2+3i, 1-5i, -2i etc are complex numbers where i is the imaginary unit defined as i2=-1.
Real numbers are commonly used where as complex numbers are mainly dealt by scientists, engineers and researchers. Every real life problem can be described by a special kind of mathematical model consisting group of simultaneous equations. The solution of a mathematical model provides a set of information (data consisting real and imaginary numbers) regarding the related problem. For these reasons development of complex numbers has a significant role in the areas of scientific research.
Here are some examples of complex numbers.
In electronics, the state of a circuit element is described by two real numbers (the voltage V across it and the current I flowing through it). A circuit element also may possess a capacitance C and an inductance L that (in simplistic terms) describe its tendency to resist changes in voltage and current respectively.
These are much better described by complex numbers. Rather than the circuit element's state having to be described by two different real numbers V and I, it can be described by a single complex number z = V + i I. Similarly, inductance and capacitance can be thought of as the real and imaginary parts of another single complex number w = C + i L. The laws of electricity can be expressed using complex addition and multiplication.
Another example is electromagnetism. Rather than trying to describe an electromagnetic field by two real quantities (electric field strength and magnetic field strength), it is best described as a single complex number, of which the electric and magnetic components are simply the real and imaginary parts.