Math

Up front, here is a link to my math blog that has the meat of this post. Also, there will be a PDF version attached. I’ve finally gotten around to writing another math post. There’s a mix of reasons why I’ve been writing about a lot of other things. First, there have just been other topics in the front of my mind that I felt like writing about in the moment. Second, there were not any particular pieces of math I felt inspired to write about. And third, preparation of a math post is significantly more work for me, as I care a lot about being accurate, precise, and clear in my presentation. ...

I was on one of my favorite subreddits today, /r/learnmath (of course I sort by New), and a question came up that ultimately came down to understanding order of operations. These always frustrate me because they stem from a poor understanding of how the order of operations work; furthermore, any problem involving order of operations can be fixed and cleared up with an additional set of parentheses. Consider this math problem below – the type I see making its way around Facebook. ...

At work the other day, we had a bit of a break to celebrated the month’s birthdays. During this time – my coworkers sitting around a large table, munching on pie and ice cream – a conversation broke out. How many holes are there in a paper towel roll: 1 or 2? Of course, there is the joke that if you claim 1, then there is only one hole of consequence in the human body: that which leads from your mouth to the escape of your digestive system. At that point you’re just talking out of your ass. ...

Updated November 23, 2022 I’m going back through my posts and recategorizing them, and noticed this one. My plan for having a secondary spot for math posts didn’t take off after I moved my blog to WordPress properly, and discovered the excellent $\KaTeX$ plugin for rendering math. Original Post I recently remembered my company, AoPS supports blog creation for their users. In particular, it has the full functionality I’m used to on their message board. In particular, they have native $\LaTeX$ support, in addition to support for the Asymptote vector graphics language. This makes writing math significantly easier on my end, and significantly easier to interact with on the reader’s end. ...

In this little tutorial, we’ll expand on what we’ve learned about sets and functions. Specifically, we’ll double-down on the claim that sets are vital to everything we do in mathematics. Functions are not just a way to describe interactions between sets: functions are sets! ...

(Edit, 10/29) This post has been ported to a math-focused blog here. The goal of the change is improved typesetting for me, and improved readability and interaction for you. See this post for more information. This is the next post in the series of posts inspired by my brief work for Mathbook. Previously, we talked about sets. Once we have this basic object, we can start to create, define and communicate relationships and patterns between sets. Mathematics uses the language of a function to describe these relationships. ...

(Edit 6/7/2020) No more Mathbook. I finally discovered the KaTeX plugin for Wordpress, allowing beautiful typesetting once again. (Edit, 10/29) This post has been ported to a math-focused blog here. The goal of the change is improved typesetting for me, and improved readability and interaction for you. See this post for more information. Quite a while ago, an endeavoring individual tried to start an open-source repository of mathematical information called Mathbook. I contributed an article, which I’ll put down in two parts on this blog. It seems that the project has died, although the website is still available. While this is a bit of a shame, I would like to give some of my own little lessons here. The creator’s idea behind Mathbook was to focus on giving people an understanding of why we do math in a certain way. This is missing from mathematical curriculum today, but it is vital to understand that when math was developed, decisions were made for specific reasons. Moving forward, I’ll occasionally add a new post here to that effect. The people in my life don’t always understand the math I learned, so this is part of my effort in showing it. ...

(Edited 6/7/2020 for improved typesetting) Here is one of my favorite “paradoxes” in mathematics. Many students learn it in a first year calculus course. It is called Gabriel’s Horn. ...

I had another busy week, so I’m taking advantage of old stuff I can recycle. A month or two ago, I was playing a Solitaire variation my parents taught me when I was younger, and I realized that it was a completely deterministic game once the deck was shuffled. That is, unlike traditional solitaire, there was no element of choice by the player. As such, it made it very easy to write a simulation of it and analyze the details. The very brief report I wrote up is here, and the simulation code (which is also linked in the report) is here. The short version, is that it is a break-even game on average, which is pretty interesting. Furthermore, the overall result is normally distributed around breaking even. I’m trying to include a more well-rounded amount of content here, since math is still very close to my heart and I’d like to only maintain one sight for everything. It will continue to be a mix of things, so that we’re all on the same page.

Suppose you are at a pizza restaurant with your friends. You all agree you want to buy pizza to maximize your pizza-per-dollar. There’s an easy way to make comparisons between pizza sizes and figure out what the best deal is. ...