trying to be a mile wide AND a mile deep
Theorem: Suppose a, b, c are natural numbers. If a divides b and b divides c, then a divides c. Bad Proof: Suppose the hypothesis. We can substitute a times something for b in the equation b times something equals c. The two somethings together multiply a to c so we’re done.[way too vague; might […]
Proof traits explicit/specific (non-vague) logically sound, including complete lacking irrelevant statements understandable to the reader self-contained (may assume basic things; anything else needs explicit reference to previous work or must be written out in the proof) Proof kinds Direct proof: Assume hypothesis and march to conclusion. Contradiction: If proving a single clause, assume its negation […]
From the Editor in Notices of the AMS Vol 43, Issue 10 (October 1996) was an essay on the need for mathematical literacy in the general public, beginning with Nixon’s use of the third derivative. A beginner’s guide to quadric surfaces (pdf), by me, for my calculus students and intended to supplement the textbook rather […]
Honestly, I never made use of Turing machine simulators when I taught computability theory, but they can be quite fun to play with and allow you to run programs that are far more complicated than you could ever step through with pencil and paper. The list below could be considered an update of the list […]
When I was an undergraduate, we had an entire class dedicated to making the transition from calculus-type mathematics to abstract upperclass-level mathematics. We learned basic logic and set notation and manipulation, proof structures, and induction, proving simple number theory results that didn’t require additional new concepts beyond induction. I got to teach such a course […]