# The Echoes of the Mind (48-8) Complex Numbers

Complex Numbers

I have not the smallest confidence in any result which is essentially obtained by the use of imaginary symbols. ~ 19th century English mathematician George Airy

Early on, imaginary numbers were the unexpected product in taking the square root of a real number. Cardano was first to combine imaginary and real numbers, creating complex numbers, which he expressed in the form a + bi, where a and b are real, and i is the imaginary unit (√–1). Complex numbers form a plane. On one axis are real numbers, the other imaginary.

The man who tamed complex numbers geometrically was not a mathematician, but a surveyor: Caspar Wessel, a Norwegian . His brilliant paper on the subject was published in Danish in 1799. It made no impact in the mathematical world until rediscovered nearly a century later, in 1895. By that time others had trod the same path.

Wessel’s insight, which looks obvious when understood (hindsight bias), was to imagine complex number points within the Cartesian coordinate system, with real and imaginary number axes, as shown above.

Next, consider a point (a + bi) as being at some length (l) from the origin (0,0), and at some angle (φ) from the real number axis. This puts a complex number point in polar form.

Using the polar coordinate system, each point can be considered as a distance (l) from a fixed point (typically (0,0)) at an angle (φ from a fixed direction (e.g., the horizontal (real) axis). Hence, the point a + bi can be stated as (l, φ).

Using this scheme mathematical operations on complex numbers are greatly simplified. Multiplying by √–1 is, geometrically, simply a 90° rotation counterclockwise.

Nowadays, complex numbers have essential employment in a variety of scientific and engineering fields, including physics, fluid dynamics, electromagnetism, and signal processing. They are also used to model Nature: complex numbers employed to fathom the ultimate complexity.