The Algebra and Geometry of Complex Numbers (Part 1)

For centuries mathematicians had found ways to describe and manipulate the perfect geometries they used to represent phenomenon of the natural world. Orbits were round. Length and amount were always positive numbers. There was no such thing as the square root of a negative number. In 1572, in his book L’Algebra, the first scrupulous description of complex numbers was given by Raphael Bombelli. In it he lays out the rules by which complex numbers may be added and multiplied. Bombelli found a practical use for complex numbers in the solving of depressed cubic equations because they let him solve such equations when negative roots were present. But because negative roots were too much of an abstraction for his day, he said of his method: “the whole matter seems to rest on sophistry rather than on truth”. In the 17th and 18th centuries advances and discoveries in mathematics were happening at an accelerating rate; however, for many, including the great René Descartes, mathematical ideas were still required to have some kind of geometric explanation or proof. These practitioners of mathematics were severely critical and dismissive of complex numbers because there was no geometric analogue for them. This fact lead Descartes, in 1637, to declare them “imaginary” stating that these types of numbers were fictitious and useless. The world would see the advent of Newton and Leibniz. It…