The notions of Nim addition and Nim multiplication that we discussed in the first homework set are due to John Conway, who studied them in the context of ordinal numbers. The ordinals extend the natural numbers, and what we did was to only consider “an initial segment”. Recently, the excellent blog neverendingbooks by Lieven Le Bruyn has discussed Conway’s construction in detail, in a series of (so far, ten) posts that you may enjoy reading and I highly recommend:
- On2 : transfinite number hacking
- On2 : Conway’s nim-arithmetics
- On2 : extending Lenstra’s list
- The odd knights of the round table
- Seating the first few thousand Knights
- Seating the first few billion Knights
- How to play Nimbers?
- n-dimensional and transfinite Nimbers
- How to win transfinite Nimbers?
- Aaron Siegel on transfinite number hacking
In particular, the posts have links to papers and talks on related subjects.