A complex number is a number of the form a + bi, where a and b are real numbers and i = √(-1).
Complex numbers extend the real number system to include solutions to equations like x² + 1 = 0. The real part is a, and the imaginary part is b. The complex conjugate of a + bi is a - bi. Complex numbers can be added, subtracted, multiplied, and divided. They are represented on the complex plane (Argand diagram) and are essential in electrical engineering, quantum mechanics, control theory, and signal processing.
The concept of imaginary numbers emerged in the 16th century when Gerolamo Cardano attempted to solve cubic equations. Rafael Bombelli developed rules for manipulating imaginary numbers in 1572. The notation i for √(-1) was introduced by Leonhard Euler in 1777. Carl Friedrich Gauss formalized the complex number plane in the 19th century. Complex analysis was developed by Augustin-Louis Cauchy.