Complex Numbers

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i represents the imaginary unit, which satisfies the equation i^2 = -1. Complex numbers extend the concept of real numbers by introducing the imaginary unit, allowing for solutions to problems that cannot be solved with only real numbers.

Real and Imaginary Parts

In a complex number a + bi, the real part is a and the imaginary part is bi. The real part represents the horizontal axis on the complex plane, while the imaginary part represents the vertical axis.

Complex Plane

The complex plane is a geometric representation of complex numbers, with the real part plotted on the horizontal axis and the imaginary part on the vertical axis. The complex number a + bi can be represented as the point (a, b) on the complex plane.

Operations

Complex numbers can undergo addition, subtraction, multiplication, and division similar to real numbers. The complex conjugate of a complex number a + bi is a - bi, obtained by changing the sign of the imaginary part.

Overall, complex numbers are a powerful mathematical concept that find applications in various fields such as engineering, physics, and signal processing.