I had the pleasure of spending a couple weeks learning the basics of the surreal numbers sytem, from the Peano axioms to the basic construction of surreals. At the end of this short exploration, I looked into the world of omnific integers, which generalise the concept of integers to the surreal numbers. For instance, while the integers may contain numbers such as 5, 10, or 1000, the omnific integers include numbers such as \(\omega\), which is the supposed smallest infinite number. Omnific integers allow you to do arithmetic with infinitely large numbers, and for this reason it can be extended to consider many problems in number theory as well. One aspect that I investigated very shallowly was Pell’s equation and how its solution may differ if all integers are instead omnific.

During this short exploration, I made use of Knuth’s Surreal Numbers , which is a sort of novelette that explores the construction and discovery of surreal numbers through the lens of two individuals on a desolate island. It’s perhaps not the most rigorous “math text”, but it certainly has value and can be read like a storybook. I also looked into Conway’s (RIP) On Numbers and Games , which studies surreal numbers in its numerous aspects, from algebra to analysis to number theory, and eventually to the reason for its initial creation: games. It certainly is a rigorous text, but it is very self-contained, and if I had more time I would have been able to explore its nuances in much more detail than I have. At the end of my exploration, I gave a very brief presentation on omnific integers, and here are the slides.

(13 January, 2024)

In looking over some old stuff, I realized that I keep on referencing and returning to the idea of surreal numbers, either as a joke or as sort of false nostalgia that I have no right to feel. I realize now that I forgot to share one really cool video on the topic, so here it is: HACKENBUSH: a window to a new world of math. Enjoy!

(17 April, 2025)