A Danish cartographer named Caspar Wessel came up with an early formal treatment of complex numbers, in his work "On the Analytical Representation of Direction"

A Danish cartographer named Caspar Wessel came up with an early formal treatment of complex numbers, in his work "On the Analytical Representation of Direction" (wikipedia has a nice article about him). It was published in an obscure forum, and predates subsequent rediscovery of complex numbers by others. His formulation is IMHO beautiful, intuitive, and compelling. He did it in terms of directions on a map, replacing the "sign" of a conventional real number with a "direction" or "compass heading". So, one might say, "the nearest Starbucks is two blocks east and one block north". He was simply using what became known as the polar form of complex numbers. One can follow intuition and define reasonable notions of addition and multiplication by real values. But what of multiplying two "Directions"? Wessel derived what multiplication must mean, and went further in deriving a large number of identities involving his newly discovered "directional numbers". If you pick a specific important pair of directional numbers, the multiplicative identity (call it "1") and a number 90 degrees away from it (call it "i"), it is convenient to represent any directional number as a the sum of scalar multiples of these two numbers. Then, one considers the simple formula "(i + 1)(i - 1) = i^2 - 1". A straightforward geometrical argument demonstrates that i^2 must be equal to -1. ("Show HN": gregfjohnson.com/complex)