I recently put a short story I wrote on Amazon. It’s called When You Come Back.
You can find a link to it here.
When I originally wrote it, mental illness was not necessarily on my mind. But, the majority of its readers have told me it resonates with them to a fairly strong degree. So, I’ve put it up for $1 and any proceeds I receive will go towards The Trevor Project, in support of mental illness assistance.
If you’re not interested in the short story, or in providing Amazon with some of the overhead that comes with buying a Kindle book, you can donate to them directly.
If you want the story in PDF form, you can contact me.
Mathbook: Introduction to Functions
(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.
Continue reading “Mathbook: Introduction to Functions”
Mathbook: Introduction to Sets
(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.
New Chapter
I started my first post-graduation job at Art of Problem Solving this past week. I was an intern here last summer, and I was lucky enough for that to lead to a job. My official role is “Curriculum Developer”. I work on developing their elementary school math curriculum, as that is their current focus. It’s an incredibly good job, with fantastically intelligent and caring people.
While it is a great company whose mission I am deeply invested in, and San Diego is a beautiful place to be, these first few days have been very tough. It was weird driving across the country, having a good time, then suddenly getting to work. I’m living on my own, in a room I’m renting (technically an AirBnB) from a nice lady. I know the area from last summer, but I’m still getting over a mental hurdle of actually going out and doing things. Although at the time I write this I’ll have only been here a few days, I’m already feeling antsy. It’s strange.
Perhaps the biggest reason for my feelings is that I spent my entire life in Minnesota. I am very rooted there, and despite many friends leaving who have also graduated, there are many people I’ve left behind. I’m leaving the comfortable world of academia to work at a place where I have no true connections. It’s a rather isolating feeling that I am working through.
Yet, this is something I correctly anticipated. I have been solidifying Operation: Have a Conversation and Comical Start as ways to keep in contact with people. I’ve reached out to people (or luckily have had them reach out to me) to stay in contact via phone calls or letters. And I also committed to myself that I would write weekly on this blog, and not worry about people reading it. It’s just a good thing to have on my schedule, both for the purpose of self-reflection, and to stop myself from falling idle after I do a day of work.
I already reached out to the San Diego Concert Band, a local community band that has fairly open policies for joining. I’ll be rehearsing with them regularly starting next week, which I am incredibly thrilled for. Although my percussion chops are not what they once were, they will improve and I will be better off for having the experience. I also plan on finding a group (or maybe just a person or two) to try and play tennis with. It’s an easy sport to play as long as you have another person, and I definitely enjoy playing it. Ideally I’d find a softball league as well, and I also have a long book list to get through.
Despite a touch of melancholy and some misgivings on traveling so far, I am excited for what is to come. Knowing that my time here is rather indefinite, I can feel more comfortable finding my place and joining new things. I have more opportunity to be involved and help myself as I go along. It’s an interesting time.
Movie Review: "Christopher Robin"
The other day my girlfriend and I went to see the film Christopher Robin, all about the titular character outgrowing his friends in the 100 acre wood, and slowly finding his way back to childhood. After leaving the movie close to tears (my girlfriend did cry multiple times), here was my one line review:
The entire movie was super predictable, but it was so well-done and moving that I didn’t even care.
Gabriel’s Horn
(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.
Abducted: A 24-Hour Musical
This past weekend I had the fantastic experience of playing drumset in a musical put together in only 24 hours. My friend Tim, along with his friend Adam, wrote the entirety of the show. We showed up at Friday on 7pm, with nobody having seen the script or music except the writers. We then performed the musical — lines memorized, music rehearsed, choreography and blocking complete — at 7pm (and 9pm) Saturday evening.
I had an extremely good time. The music was engaging and written with some interpretation allowable, as all the members of the pit were experienced in this musical scenario. We had a lot of fun putting things together quickly, and were quite successful in performing our parts within a few hours.
The show was broadly a satirical take on the characters from Scooby-Doo. In addition to the normal gang (whose names are never explicitly stated at any point in the show), there is the scapegoat Brian, who is Daphne’s current boyfriend. He is verbally abused throughout the show, with some light slapping. In addition, Scooby-Doo is just a man in a Scooby-Doo outfit (naturally), although an old Hermit we meet at the beginning addresses this fact:
Velma: Oh, that’s our anthropomorphic dog. He loves food, and hates ghosts. So, we keep him on a leash and force him to solve crimes!
Hermit (Cooper): That’s not a dog! That is clearly a man in constant pain!
Fred: (Firmly, maintaining eye contact.) It’s a dog.
The fact that the gang keeps a man in a suit on a leash, and is either indifferent to this fact or somehow unaware of its humanity, is repeated throughout the show. Every time a character (normally Brian) goes to take his leash, Scooby screams in terror. At one point, a completely silent scene opens with Scooby alone on stage. Scared by the audience, he slowly stands on both legs, moves forward, and proceeds to intently say Help me! to various members of the audience.
What I hope to show with these descriptions, as well as the plot description to follow, is the creativity and fun that Tim and Adam bring to the shows they write, as well as some interest in watching their other shows online.
Now for the plot. The aforementioned hermit opens the show, describing to Fred and Velma how he was abducted and probed by aliens. Once all the characters were quickly introduced, they were subsequently abducted. During the musical sequence, a small creature, reminiscent of the chest-bursting alien from Alien, kills Fred during the “probing” procedure. We then meet the head of the ship, Marvin.
Shaggy accuses him of working too hard, so he takes Marvin away to “relax”. We later learn their natural high is Captain Crunch.
Daphne falls into a motherly love, with a bit of sexual tension, with the alien in Fred’s chest, much to Brian’s dismay. Velma proceeds to look for clues, until she meets another female alien with the exact same disposition and actions. Some innuendos occur.
Brian sings his heart out to the audience, beautifully I may add.
We finally learn the Hermit is on the ship. His plan was to run everybody out of town with talk about aliens, so he could have the oil deposit he discovered entirely to himself. Of course, he is now abducted by aliens so there is not much to do. Brian and Scooby were within earshot, and attempt to tell the rest of the gang.
Daphne appears, crazed, yielding a gun, rounding up the aliens and, as Marvin put it, “Yelling lines from Alien 3.” Of course, the alien in Fred’s chest talks her down, at which point another chest-bursting alien emerges from Daphne. Those two aliens go to town.
Brian explains to Velma that he solved the crime, and attempts to remove the mask (as is customary). Of course, being Brian, he takes some other alien and nearly chokes it. At this point, Scooby goes up to the Hermit, who is wearing a very obvious mask, and removes it. It is revealed the culprit is none other than D.B. Cooper.
At this point, we have “A case with no loose ends”, as Shaggy puts it. Of course we then realize the characters are still on an alien spaceship. Brian asks Marvin if they can be dropped off on Earth. Marvin declines, but makes an allowance that they can stay another day, until they are eaten.
Due to the inevitability of their demise, Brian removes Scooby’s collar. Scooby rises to his foot, removes the dog head from his head, and embraces Brian. The show ends on this scene.
It was an incredible show. Things weren’t perfect, but they were as imperfect as the musicals I did pit for that had weeks of rehearsal. It was a great experience, and while Tim and Adam have just graduated college, making another musical unlikely in the near future, I am excited to hear about whatever creative endeavors they have moving forward.
Dull Edge
The cutting edge of technology is particularly awesome these days. Cars are doing more on their own, phones are surpassing some current computers in their performance, and VR is coming into its own finally. I listen to a lot of tech podcasts, and love messing around with technology, but due to my status as a recent college graduate, I am definitely not maintaining a collection of cutting-edge devices. And that’s okay.
First, let’s talk about cars. I recently purchased a post-lease 2015 Honda Civic, LX trim (in other words, the only model more basic comes with a manual instead of a CVT.) The disparity between that car and other higher-end cars from the same year is rather shocking. Sitting in it, I feel super cool. It’s a big upgrade from the 1998 EX-L Honda Accord I had been driving. There’s a “cockpit” feeling to it, good Bluetooth connectivity, and a back-up camera. It’s relatively zippy for a cheap car, and the gas mileage gained by the CVT cannot be beat.
Then, I read a Golf R Review by Casey Liss. He is one of three car enthusiasts on the Accidental Tech Podcast. Not surprisingly, the only one I can identify with is John Siracusa, who to my knowledge has mostly driven manual Civics and Accords for his entire life. Casey though, he complained about the lack of assisted driving and automatic parking. The car needs to be fast, it needs a sunroof, and of course Carplay! This was absolutely baffling to me. I just cannot get my head wrapped around why some of these things are important. These three men have attempted to address it on their podcast, but it still does not click with me. Cars can be purchased only so often to be at all reasonable, and so one cannot even stay on the cutting edge.
The other issue is that he recently began working from home exclusively, yet sounded very hesitant to become a one-car family. Oh well, to each their own.
Now, obviously cars are a very special case of not staying on the cutting edge. I’ve only owned my own car for a few months, and it’ll be a number of years before I can even begin to think about getting another. But phones: now that’s another matter.
I’ve had my Galaxy S7 for two years now. Previously, I had a Galaxy S4 for two years, and then some random LG (I think) phone for 4 years throughout high school. In my mind, until I can afford the “every year” phone upgrade, two years is a reasonable time in these days of mostly non-replaceable phone batteries. So, with my S7 really slowing down and the battery life starting to tank, it was time to figure out what to do. Was the newest Galaxy S9 worth the incredible price tag? Did I want to save some money and get a Pixel XL or a new LG phone, each containing last year’s processors, despite them being the newest in the lineup?
I ended up choosing a Galaxy S8+. Due to the release of the S9, I got a very good deal on it, and the processing difference (and battery life) between the S7 and S8+ is much larger than the S8+ to a comparable newest generation phone. Once again, I opted to stay on the duller edge of technology. And I am happy with that decision. I tried the S9 in stores, and it truly did not impress me anymore than the S8 does. The S8 was the revolutionary phone (just like the iPhone X, and whatever comes out next will not be quite as lauded).
This is what is interesting about technology. So many people are excited to get the newest and best thing. The hype is always there, but the price-to-performance normally isn’t. I spent all of last year in school working on a 4-year old Ideapad and a 5-year old refurbished Thinkpad. They performed admirably for me, because like most people, I’m not doing much heavy-lifting.
Being on the dull edge, and looking out at what is available and what others have, can be fun. I don’t think there is anything wrong with living life on some sort of delay with technology. Perhaps as I grow older and make a bit more money, that will change. But for now, I am happy with scouring the internet for good deals, and getting what I actually need for the best price.