This essay describes operations on finite sets that mimic the operations of adding, multiplying, and exponentiating counting numbers and uses these operations to explain why discrete mathematicians and computer scientists take 0^0 to equal the number of functions from the empty set to itself, which is 1.
This essay tells the story of how the modern theory of vectors was gradually discovered, and covers the colorful yet often forgotten late-19th century quarrel between quaternionists and vectorialists.
- Since you (a) mentioned a working title "... Adventures of Plus and Times" and (b) have classroom experience, do you have any opinions on the following: when overloading the "high school" arithmetic operators for ring- and lattice-like structures, is it easier for people to learn (a) overloads with traditional use (ie × ↦ meet, + ↦ xor in Electrical Engineering), (b) overloads with mnemonic stories: "product makes things larger; modulo makes things smaller", or (c) any consistent overloads, because just making the abstraction leap is much more difficult than remembering any operator assignments?
I don’t have classroom experience with this kind of teaching. I may write a companion to “When 1+1=0” called “When 1+1=1”, at which point I’ll have to think harder about these issues. (Are we using a different 0 and a different 1 here? Or a different +? Or a different =?)
This essay discusses finite fields and the role they played in the early years of the computer revolution (remember paper-tape drives?) and in the ongoing Voyager mission to explore the outer solar system.
In this month’s Mathematical Enchantment, I wrote about finite fields and the role they played in the early years of the computer revolution (remember paper-tape drives?) and in the ongoing Voyager mission to explore the outer solar system.
