Hamilton, the Quaternions, and Creativity – Nick Higham

Complicated numbers have the shape

the place is the imaginary unit. Quaternions comprise two extra imaginary models, and :

Sir William Rowan Hamilton found the quaternions in 1843 as he walked alongside the Royal Canal in Dublin, and he famously carved the formulation on a stone of Brougham Bridge. He had for a while been attempting to increase the “couplets” to “triplets” , however he realized that the requirement for the product of two such numbers couldn’t maintain. His key perception was to comprehend {that a} fourth imaginary unit, , was required and that multiplication can be noncommutative: usually.

Sir William Rowan Hamilton. Etching after J. Kirkwood. Credit score: Wellcome Library, London. Wellcome Photographs. Supply.

We have no idea precisely how Hamilton found the quaternions, although his writings present insights. A method through which he may have constructed the quaternions is by beginning with the complicated numbers and asking for every of their properties “How may this be totally different?” 4 phrases quite than two and noncommutative multiplication are the variations. This strategy of figuring out the important thing options of an issue and asking, for each, “How may this be totally different?”, is a robust course of for creativity and discovery—a course of that anybody can apply to any downside.

For extra on the invention of the quaternions and on the right way to generate concepts see the article Hamilton’s Discovery of the Quaternions and Creativity in Arithmetic by Dennis Sherwood and I within the LMS E-newsletter.