Category: Math

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.

But, being a company full of nerds and pedants, those of us who were sensible had to claim 1; after all, a paper towel has width to it and is thus homeomorphic to a torus (think, bagel).

Topology is a fun subject. Imagine taking geometry, and saying “I don’t really care about angles, or lengths, or these things that make geometry what it is. I mainly just care if I can stretch something to make it into something else, and how many holes there are in the shape.” That’s the base level of topology. An introductory topology course teaches you about homeomorphisms, which means you can continuously stretch, pull, push, or generally manipulate an object without cutting or tearing it.

Imagine you had a very flexible, putty-like bagel. If you mad the bagel very thin, skinny, and tall, you can imagine forming yourself a paper towel roll. This is what we mean when we say they are homeomorphic. We don’t put an extra hole anywhere, we just squish and shape things to our will.

Some people may ask why this is useful, particularly when you learn a coffee mug is also homeomorphic to a torus. The easiest explanation is math is about abstractions and patterns. Sure, a cube and a non-cubic rectangular prism have some differences, insofar as the rectangular prism doesn’t have three equal edge lengths. A cube and a triangular prism seem even further removed. Yet, any prism, or more generally polyhedron (which includes pyramids as well!), are all homeomorphic to a sphere. These three dimensional figures share certain properties, most notably that they are solid objects with no holes in them. And there’s something to be said for that!

While topology goes well beyond this idea, it gives a good grasp at what mathematicians are interested in. They noticed that we can generalize some of these properties, and map between objects using these homeomorphisms. So, instead of topology being about classes of objects which are the same in some small way, it is really about how we relate these similar objects. Topology is about showing the similarities and patterns between objects. It’s interesting that with a consistent set of rules, we can say “cube is to sphere as bagel is to paper towel roll.” We notice this pattern, and figure out how to make it rigorous and useful.

This is an under-appreciated aspect of math. It is not about rules that have no reason, or about abstraction for no reason. It is about finding a pattern, and seeing how other objects follow a similar pattern. This is what make mathematics a very fun subject, yet it is a mindset very few people are able to have. I hope this can change in the future.

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!
Continue reading “Mathbook: Functions as 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.

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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.
Continue reading “Mathbook: Introduction to Functions”

(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.

Introduction to Sets

We will learn the basics of how sets are used in mathematics. It is important to understand basic arithmetic before diving in, but nothing else.

Understanding Sets

In any field of mathematics, it is important to be able to deal with objects and structures. At the lowest level of mathematical objects and structures are sets. Most simply, a set is a collection of objects. We can think of the set of all flowers in Hawaii, or the set of whole numbers between 10 and 37. Typically, we use curly brackets (braces) to denote a set, such as \{1, 2, 3\}. If we are using the same set many times in a row, or talking about a set that cannot easily be written down, we can use some other symbol. Throughout this tutorial we will let S be the set \{1, 2, 3\}, and H be the set of all flowers in Hawaii.

There are certain rules and terms used with sets that allow mathematicians to be consistent when using and talking about sets. For example, we want to know what to call the objects in our sets in general, and how we can write sets.

DEFINITION

Element: Each object or member of a set is called an element of the set. Each element can only occur once in a set. \{1,2,1,3\} is not a valid set since the 1 occurs twice. In addition, the order of elements in a set does not matter. S = \{1,2,3\} = \{3,1,2\}.
To say an element s is in the set S, we write s \in S.

It is also natural to discuss how many elements are in a set.

DEFINITION

Cardinality: The number of elements in a set is called the cardinality of the set. The cardinality is often denoted by putting vertical bars around the set. For example, since S has three elements, we write |S| = 3.

For this tutorial, we will only be looking at sets with finite cardinality; this means we will always be able to list and count every element in the set. Future tutorials may explore larger sets, which becomes an even more powerful (and fun!) mathematical tool.
Often we want to look at some of the elements in a set, but not all of them. For example, we might want the elements of H which are red flowers. This is a very common pattern in mathematics: given an object or structure, how can we look at smaller objects that have a similar structure?

DEFINITION

Subset: If every element in some set A is in a set B, we say that A is a subset of B, and we write A\subset B. For example, the set \{1,3\} is a subset of S.

When doing mathematics, it is good practice to look at the simplest example of any object you are interested in exploring. When it comes to sets, it becomes natural to ask “What if my set has no elements?”

DEFINITION

Empty Set: The empty set is defined to be the set which has no elements. The most common notation is \emptyset, but you may see \{\}, especially in older math texts. This second notation emphasizes it is a set with no elements.

Almost everything you see and do has sets hiding in the background. They are a universal way of communicating mathematical ideas, and are thus very important to understand.
In a future post, I’ll give the second half of this post: Understanding Functions. Once you have established your structure (the set), and have explored some basic ideas (subsets, the empty set), it is important to discuss how you can have two sets interact. The simplest way to have two sets interact is via a function.

(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.

Continue reading “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.
Continue reading “Purchasing Pizza”