Fishing for a presentation idea

[NRS]

Hello all. I'm a math major attending a "math in history, art and philosophy" class as an elective. For that class, we have to give a presentation on a topic of our choice. As many of the people in the room aren't math majors or had much exposure to the subject, I was thinking about giving a presentation on proofs. Perhaps why they are so important, maybe give an example of an example which appears true but isn't (Gauss's overestimation conjecture or something). But I would like to finish off the presentation with a relatively non-trivial but elementary proof. I really can't think of anything, any suggestions would be nice.

 

[scurty]

I think for a class like this, doing a proof that involves something that everyone has heard of or can relate to would be a good idea. Two thought so f the top of my head are one of the numerous Pythagorean Theorem proofs (do more than one to show that one proof is not necessarily the only way to prove something) or do something related to number theory (infinitude of primes, maybe include one of the easiest proofs like the sum of two odd numbers is even, etc). There are many beautiful proofs in higher level math, but if you get too advanced for the general audience, you will bore them or lose them in math jargon (you don't want to spend the entire presentation going over lots of terms so they understand what is happening).

I think the Pythagorean proofs are easy to see visually (at least the geometric ones) and are a good choice and any real basic number theory proofs would be easy to follow along for non-math majors.

 

[NRS]

Not bad, maybe I'll present the development of that proof, beginning with the infamous "Behold!" and end with a simple congruence proof or something. Thanks a lot, anything else is welcomed.

 

[Number Nine]

My personal favourite, among the canonical "first proof" proofs, is the proof that every number has a prime factor (and, as a consequence, that there are infinitely many primes). Honestly, everyone's heard of the Pythagorean theorem, and a proof won't really "expand many people's horizons" as far as mathematics goes. I like introducing people to the prime number proof because most people have no conceptualization about how you would go about proving something about every whole number in existence, so the proof offers a taste of mathematics that they've probably never even imagined before.

Alternately, instead of giving a rigorous proof, you could take a discipline like abstract algebra and give a conceptual understanding of the "algebraic" approach to tackling a problem (i.e. stripping away extraneous detail and focusing on the barest structure of the thing). A very good example would be bracelet counting with Burnside's theorem. You could talk about how the whole problem basically reduces to finding collections of bracelets that can be obtained from each other by rotations and reflections (i.e. the orbits generated by the action of the dihedral group on the set of bracelets). Or, you could talk about how things like addition and multiplication in the reals can be considered as binary operations, and that once you see it that way you see all sorts of similarities between things like the integers, permutations, and symmetries of geometric figures (i.e. the group structure).