The complex number system extends the traditional number system by including numbers that have both a real part and an imaginary part. A complex number is typically expressed in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit, defined as the square root of -1. This system allows for solutions to equations that do not have real solutions, such as x^2 + 1 = 0. Complex numbers can be represented graphically on the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This representation helps visualize operations like addition and multiplication of complex numbers. The complex number system is widely used in various fields, including engineering, physics, and computer science, due to its ability to model oscillations, waves, and other phenomena.