Short answers, pulled from the story.
A complex number is an element of a number system that extends the real numbers by adding the imaginary unit i, which satisfies an equation no real number can. Every complex number combines a real part and an imaginary part, both of which are real numbers.
The term imaginary was coined by René Descartes in 1637 to stress the unreal nature of these quantities. Despite the name, complex numbers have a mathematical existence as firm as that of the real numbers and are fundamental tools in describing the natural world.
The Italian mathematician Gerolamo Cardano conceived of complex numbers around 1545 in his Ars Magna while solving cubic equations, though he called them as subtle as they are useless. Rafael Bombelli was the first to address them explicitly and develop the rules for complex arithmetic, and many others, including Gauss, refined the theory.
The fundamental theorem of algebra, credited to Carl Friedrich Gauss and Jean le Rond d'Alembert, states that every non-constant polynomial equation with real or complex coefficients has at least one complex solution. Because of this, the complex numbers are called an algebraically closed field.
A complex number can be plotted as a point in the complex plane, also called the Argand diagram, with the real part on the horizontal axis and the imaginary part on the vertical axis. This idea was first described by Caspar Wessel in 1799 and established in modern form by Gauss in 1831.
Complex numbers have applications in signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis. They are intrinsic to the formulations of quantum mechanics and, in electrical engineering, are combined into a single value called the impedance.
The polar form expresses a complex number through its absolute value, the distance from the origin, and its argument, the angle from the positive real axis measured in radians. When two numbers in polar form are multiplied, their absolute values multiply and their arguments add.