trying to be a mile wide AND a mile deep
I had the good fortune as an undergraduate to have a “bridge class” in my math curriculum. We learned basic logic, set manipulation, formal functions and relations, proof structure and induction. In that class our instructor had us read an article by Reuben Hirsch called “Math Lingo vs. Plain English: Double Entendre” (published in the […]
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 […]
In case it wasn’t clear from musings on series, these posts are collections of themed material that individually don’t make full posts. 1. There are many ways for a limit to fail to exist. It could be infinite, oscillate within a finite interval, completely devolve into radio static, or simply give you different values when […]
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 […]