An Infinitely Large Napkin

"The fact that Z denotes the set of integers is a sort of mathwide standard."

I'm pretty sure I saw that in my high school in the early 90s, but it was a one-off event where we discussed ℕ, ℝ, ℤ, and ℚ, but we never used them for anything. I'm sitting here trying to remember our high-school set theory (which is getting cognitive interference from my college training on the topic), but my memory is claiming I either never had to write {x | x ∃ ℤ} in high school, or if I ever did, we blipped over it really quickly.

High school math generally implicitly takes place in "casual ℝ". I call it casual because the only time it even gets close to really hammering on the characteristics of real numbers is in the limit discussion. I certainly never heard "Dedekind cut" in high school.

I'm a computer scientist, not a mathematician (but I've taken around 25 college math courses spread over many years). Nevertheless, I do have a couple of suggestions and observations that I will address to the author that I hope is seeing this.

First, it would be a very sophisticated high school student to tackle topology and some of the other areas of abstract mathematics. I really like the topics you've picked for your book, but they do seem to require quite a bit of mathematical sophistication (e.g. Topology).

Secondly, I feel that there are a few important fields that you might consider adding to your napkin: Combinatorics, Statistics, Differential Equations, and Logic.

The usefulness and the importance of understanding statistics is pretty obvious in today's data dominated world. Statistics seems to fall outside of Mathematics at some (most?) universities, but I keep my statistics books right next to my math books.

Combinatorics is full of interesting results some esoteric (the friendship theorem) and some practical (stars and bars). The proof techniques of combinatorics are also worth studying for their own sakes (like the probabilistic method).

I've always felt a love hate relationship with Differential Equations. Theoretically, they are disappointing ("oh hey, let's try this, surprise its the solution!") but practically they are needed everywhere.

One of the best math experiences that I had in high school was a logic course that I took one summer with two other students. What fun and it always served me well in course 18.

From the current Preface:

"I initially wrote this book with talented high-school students in mind, particularly those with math-olympiad type backgrounds. Some remnants of that cultural bias can still be felt throughout the book, particularly in assorted challenge problems which are taken from mathematical competitions. However, in general I think this would be a good reference for anyone with some amount of mathematical maturity and curiosity. Examples include but certainly not limited to: math undergraduate majors, physics/CS majors, math PhD students who want to hear a little bit about fields other than their own, high school students who like math but not math contests, and unusually intelligent kittens fluent in English."

An infinitely large napkin, if made of any sort of normal matter, would have infinite mass. It's gravitational pull would propigate, at c IIRC, out from the instant of its creation across all of space/time, sucking all objects (including itself) towards it's center of mass at the speed of light (OK, just under. Literally c-0.000...1m/s). It not only would create a black hole, it would signal the end of the entire universe. The destruction of any point in space would depend simply on the distance from the napkin. For every 299,792,458 meters away from napkin center, that object would have one second of existance left. When the gravitational pull hits anything, the acceleration would be so strong, again 0-c in 0 seconds, that some crazy subatomic fusion would occur.

Any physicists out there able to flesh this out? I find the thought experiment fascinating and am sure I'm missing/misrepresenting something.