What I believe has manifested most to me over the course of my blog is the beauty of mathematics. Of course finding beauty in anything is a prerequisite to actually enjoying it, and so I talked about it in my first post. However, I talked mostly about the shallower beauties, such as the visual prettiness of a well-rendered fractal and the like, which I think most people can appreciate; but what manifests with some real effort and intrigue is this amazing sort of deeper beauty, which I think is a requirement to witness before one ever claims by their own volition that they “like math.”

Given that most people are forced to put in real effort into math for most of their young life by the school system, and yet so many claiming outright that they “hate math,” or simply write it off as something they can’t or will never do is a real failing of public education. The root of the problem is misplacement of emphasis.

I remember as early as first or second grade being given a large and dauntingly official looking sheet of paper crawling with countless little addition and subtraction problems to be graded based on how many were completed within the required time. It is this emphasis on a certain species of intelligence, of which it seems you are either gifted by some higher power or you aren’t, that turns off so many kids from the beginning. Math isn’t about being a meaty computer. You may recall the famous, “you won’t always have a calculator on you at all times,” that escapes the lips of most elementary teachers at some point or another, and they’re not wrong, being able to do arithmetic on paper is a useful skill for whenever you’re in the specific emergency of having to calculate a tip or count your sheep when your smartphone is dead, forgotten, or by some other extraordinary happenstance is not in your pocket; but beyond that the beauty of math must be introduced concurrently. This does not happen because math is taught in a very technical way. It is taught in the same way as lab procedure or car mechanics as if your are actually going to be out and requiring knowledge for these problems on the spot. But the other infamous question: “when are we actually going to use this?” still rings true. In most cases kids won’t use everything they learn in math class, but this comes back to the distinction of ‘knowing’ and *knowing* discussed a bit in the book from my last blog post.

*Knowing* is what is expected in schools today, understanding all details of a problem in order to be prepared to tackle it in any unexpected form on a test. Whereas ‘knowing’ is more like knowing about something, to know enough about something to see why it’s necessary, interesting, or beautiful; and to be able to *know* it quickly if ever necessary. *Knowing* is certainly a good thing if you’re a mathematician, where it’s important to not have to look things up when concentrating on work, or to have new insights from intuition; but it is this *knowing* that is turning so many kids off from becoming mathematicians in the first place, so let’s help them ‘know’ math enough to love it, for there’s so much more to math.

As someone who has already accomplished the last part there is yet a dilemma I face that has become apparent while doing this project, fueled by this bit of illustration: do I want to become a mathematician? Obviously the whole of human knowledge is so tremendously vast that any person hoping to acquire it all in one lifetime is beyond a dream, but do I really want to focus so much on *knowing* one small area of math so as to disregard all other areas of math and indeed human knowledge? For the tens of thousands of mathematicians internationally and throughout history there are so few famed greats whose work transcends the rest (but is of course supported and inspired by those in their field), yet I do not believe myself to be so special. I have no confidence in myself to be even a background mathematician, and the idea of fighting for tenure and constantly publishing papers to stay employed has no appeal to me. In fact, I have no desire for a specialty degree at all, and as someone who has gone to a “college preparatory school” their whole life, the notion might even seem blasphemous. But if it means not having a nice job, or house, or a lot of money (and, simultaneously, debt) it does not concern me; my only desire is to learn. I believe myself capable of following the canon of work from each field required to be a well-rounded man, and ‘knowing’ math is enough for me. But I digress. I hope to figure out the answers to my questions before my senior year is over.

I mentioned the “famed greats” and “the canon of work from each field.” Another thing I came to realize while self-teaching is what being educated in a subject really means. For every subject there is a history of prominent events: realizations leading to realizations, discussions to discoveries, etc. And I believe understanding the key questions behind the results, having a full context, is where the best learning takes place. I’m not talking about just history here. I don’t want to be a math historian either. I’m talking about full understanding; not just the loose but exclusive emphasis on computation, memorization, and rough guidelines that plagues general education.

Another section that goes hand-in-hand with full understanding of every subject is its meta-canon, or the philosophy of itself. I especially appreciate the philosophy of math. I believe the love for math lies tangled in its web of philosophic and meta thought, looking down on its surface beauty, and in the most important thing that I have learned: that math is metaphysical—and a full understanding of life requires knowledge of math.

I leave you with this nice video. Feel free to play some inspirational music over it, too.

]]>Well fortunately I read the heck out of this thing so I’m going to talk about it briefly. Last time I mentioned how surprised I was to find this book by chance in a bookstore. It’s by an author I’ve read very much about, and I had no idea he produced any text on mathematics. (Indeed, I found out about another very famous author who has strong emphasis on math in his fiction work: Thomas Pynchon.) I plan to read Wallace’s other work in the near future. There is much agreement about the impenetrable nature of his fiction work, so I consider this nonfiction work, where his goal is to help me understand, a good introduction to his style. Indeed, I found the very first section an absolutely delightful prose on abstraction and epistemology that everyone should read. The rest of the book reads as though it’s constantly unsure of itself, even apologizing on occasion as well, though it can get annoying. David Foster Wallace talked a bit about this in an interview I watched about his book. He described writing the book as “trying to take somebody up on an elevator at the same time you’re building the shaft.”¹ But I enjoyed DFW’s unrestrained use of footnotes², acronyms, and bonus factoids. And something I’m really glad that David made sure of was emphasis on context. I think this is a huge gap in math education when it’s not there, because the context of the development and discovery is what makes math beautiful; not just suddenly going from Zeno’s paradox to a/(1-r) as David said. Where’s the learning in that?

I’m not completely sure where to go next after finishing this book, but I will be scouring the sources in the bibliography, where I’ve already found the names of several authors that I’ve mentioned in my blog already. So I think this was a good leaping off point.

I find there are a very limited number of math topics that authors can present to a nonprofessional audience, a very popular one being the topic of this book: infinity. Indeed, the nature of infinity is not intuitive and can be mind blowing to the unexposed. Though there is more known about it than you’d initially think, there still remains much mystery, which makes it even more enticing. It is a popular topic for math content creators on YouTube especially. Recently Vsauce has been producing a lot of decent math content, most of it on, believe it or not, infinity. Links: 1 2 3. Another video: 4. So there you have it, I recommend learning a bit about infinity, it’s a mind bending topic for sure.

¹The Interview (YouTube version) accessed 5/10/16. I really enjoyed the interview for giving both a greater context on the book and the author.

²In tribute

]]>As it happens there are very few books that would meet the requirements (i.e. fiction) of the mandatory blog novel that are about pure mathematics. Strange, I guess it’s not an *interesting* enough topic to write a story around. Indeed, I was going to choose *A Beautiful Mind*, but it is unfortunately biographical (much more so than the movie). I could cop-out and choose one of the novels that don’t really relate so much to mathematics but have a mathematician protagonist (one of which can be found below). Or perhaps I could’ve used one of the several educational texts I’m reading. But no, I have chosen the classic *Flatland**: a Romance of Many Dimensions*. A novella that is not only *about *mathematics, but features a square as its main character. Well, that is somewhat a stretch of the truth, the main character doesn’t *do *mathematics the whole time, but it is the most mathematical novel I could find.

To give a short summary, the novel comes across as a fun author’s experiment, at least, that’s how I interpreted it. The novel is set in a 2-dimensional plane, featuring two dimensional characters like triangles, squares, and all of your favorites. The first half comes across as a sort of discourse on how things function in the 2D world; that’s where the author’s experiment comes in: he gets to set how everything works and see how the story builds around it. I think it would be fun to write a novel around my own rules for a 2D land. Some of the more interesting points are how the creatures see and function as a society. Us spacelanders, in three dimensions, can only see in a 2D plane, so naturally the flatlanders, in two dimensions, can only see a 1D line. This causes many difficulties, and the techniques of sight in their world are some of the main causes of class separation. Indeed, the novel features some unexpected themes including: class struggle and discrimination. What shape you are born into determines your class, and there is a natural progression over generations i.e. a square’s son is a pentagon whose son is a hexagon etc. Take a look at the neat diagram I made.

You may notice two things:

**First, **that I mentioned the woman in the story are stupid. Indeed, they are; they have the short term memory of a relatively smart fish. They also have to make a “peace cry,” a loud noise to alert nearby citizens of their presence, since they are hard to see (being so thin) and dangerous because of their point. They are looked down upon by the “rational” men as emotional beings. The author’s seeming misogyny is addressed humorously as a response to critics of the issue in the second edition author’s notes. From my own reading I don’t think the author is a misogynist (relative to his era), but he just decided the males in the story are, and thus the narrator too.

**Second: **the chromatist irregular. Chromatism, the act of coloring one’s edges, and irregularity are subjects of great moral controversy in the story. Regularity, the equality of all edges, is highly important—even a popular salutation in Flatland is “attend your configuration.” There are also completely irregular, or scalene, triangles, which function as the sort of peasant class.

The climax of the story is when our protagonist is confronted by a three-dimensional sphere. The sphere enters Flatland, much to the square’s confusion, because he can only see a 2D slice of the 3D figure, a circle, in his land. The sphere can move up and down in the third dimension, which corresponds to it changing sizes in Flatland, something incomprehensible. This is demonstrated at 1:25 in this neat video. Eventually the sphere takes the little square into the land of 3D, Spaceland, to enlighten him, so that he may spread the prophecy of the third dimension to the Flatlanders. But, the square’s downfall begins when he contemplates dimensions *higher* than three! *Infinite dimensions!* Heresy in Spaceland and Flatland. The book has somewhat of a downer ending.

I think that part of the book’s purpose is to help imagine the fourth dimension. I certainly think it helped me, though, I am only finitely closer to an infinitely impossible task. Indeed, I resent being restricted to three spatial dimensions. Higher dimensions have innumerable cool properties and practicalities. Release me from this cage of 3D, as Flatland is a cage of 2D, and into the land of infinite dimensions!

Coincidentally, I watched *Flatland: **The Film *(available free on YouTube) some time ago, before I knew anything about the novel. The movie borders on incredibly amateur and surprisingly professional throughout. It features an occasionally patronizing textual narrator and polygon anatomy. The movie takes many liberties with the story; though I liked the book better, I do concede that there are many missing elements to make it truly great.

Higher dimensions are interesting to think about. I recommend taking a look at this album on Reddit and perhaps perusing YouTube for many more interesting demonstrations. Or maybe you would like to invert a horse.

On an unrelated note i would like to present a new book I bought.

It’s by one of my favorite authors of completely non-math related writing. I had no idea he had anything to do with math. It really is a great feeling when people you respect share common interests with you. I find that many great thinkers are also prominent figures in mathematics, such as Descartes or Leibniz. I think the book is about George Cantor, the guy who decided infinity wasn’t enough, but I’m not sure; I haven’t actually opened it yet. I find that information from all the “pop-math” books I’ve read is really starting to overlap, so I’m trying much harder to delve into serious texts, but, frustratingly, there is just not as much well-expounded information. On the back of this book

there is a quote from the author of a book I introduced in my first post.

Small math-world

On a second unrelated note I would like to show you a book I hope to read soon: *The Man Without Qualities. *It is the book featuring a mathematician protagonist I mentioned in the first paragraph. Reading the first bit of the Wikipedia page I find myself relating very much to the protagonist, and I think I can learn a lot from this book.

As usual, I’d like to pick up where I left off. Last time I said I’d want to try to learn category theory. I did some research and was absolutely bombarded by prerequisites. It came to my attention that the notion of trying to learn “category theory” from my standpoint is incredibly naive. There are many seemingly simple concepts or standards that are never discussed in school and never explicitly stated in educational texts. I honestly had no idea how to become familiar with these without being in an undergraduate or graduate university course.

To try to get my bearings I figured that the thing that will immediately throw you off if you don’t have familiarity is notation. Notation is one of the most fundamental parts of mathematics, the symbolic language, and one of the most arbitrary parts of it as well. There is no way of getting a grasp of, say, any single symbol unless the text tells you what it is. Especially since symbols are used for so many different things: functions, operations, and properties to name a few. For example take this (probably bad example since it’s from quantum mechanics): (1)

The point is there’s a lot of abstraction. Written out in words — this is much easier to understand, but if you don’t understand a single symbol then you don’t understand the whole expression. Plus, how do you go about googling symbols you can’t type? Obviously, you need some educational text to teach you about a symbol explicitly.

One notation that I see in almost every paper that really annoys me is the use of ellipses and indexing numbers, such as in the definition of a polynomial: This is overly hard to read for a concept taught in middle school.

There is also the annoyingly specific mathematical lingo that comes from math being so incredibly formal. For example a math paper might require the exact knowledge of each term in this (fake) example: *infinitely differentiable and continuous smooth analytic hairy manifold*. This creates a high knowledge ceiling for any one math paper, as well as a huge knowledge gap between papers in different areas. This is well satirized in the random math research paper generator, Mathgen (2), which I recommend taking a look at to see where I’m coming from.

To familiarize myself I took a look at a few sources which list out expressions and their meaning (3, 4), skimmed a paper on notation history (5) (Interesting fact from that: instead of using parenthesis to indicate order of operations, old mathematicians used a vertical line over the grouped sections called a vinculum), and read the Wikipedia articles on a few seemingly simple concepts (6, 7). Reading these may have helped a little bit, but what I really needed to do was find a source that does introduce common concepts.

I looked for recommendations online and discovered that, instead of category theory, what I was really trying to learn about was abstract algebra. I found a book on it by a famous expositor, Serge Lang’s Algebra (8). This is a book intended as a curriculum for a graduate course so I’m not expecting to be able to grasp it completely; indeed, I’ve ran into the problems I mentioned above many times on the first few pages. The author also has an undergraduate version (9), and a book on high school mathematics that I’m sure would fill in any gaps in my knowledge and prepare me to read his other texts (10). Unfortunately, neither of these are in the public domain so it will be hard to read them legally for free, one of the big failings of self-teaching online.

A big area of applied math (and probably the only area of applied math I care about) is computer graphics. In my first post I mentioned the aesthetic of fractals. But I would like to bring your attention to something much prettier: 3D fractals.

I found this video (11) very enjoyable, as well as the other computer graphics work on the author’s channel (12).

I also had a lot fun wasting time playing with this 2D light transport simulation (13). If you scroll down you can see how complicated (and full of annoying, esoteric notation) the math gets.

This led me to a neat discovery called the steridian (14), which is a generalization of the 2D radian to 3D. Generalizations have become a more important aspect of math than they used to, and especially to me, because they often lead to interesting logistics and better intuition. Often times in math class I’ll question how a concept can be carried into higher dimensions or something similar e.g. How does trigonometry work in 3D? Is there a 3D sine wave? How do polar coordinates work in 3D space? (answer: cylindrical or spherical coordinates). What happens if you do repeated exponentiation? (answer: hyperoperations, which I talked about in my last post). What happens if you add more foci to a conic? I am often disappointed by the lack of available information on these (15). These questions are researched well in various papers, but they remain inaccessible.

I think high school math education would be superior if information on generalizations were more incorporated, as understanding the whole of a concept leads to better intuition. Indeed, the whole of high school education often treats certain topics like *forbidden knowledge*, to be taught later on or not at all, in order to avoid overwhelming the students or create a natural progression. But I find this very annoying because I like to play with a concept and see where it progresses to by myself, and being forbidden from certain parts of a concept make this more difficult. I rarely find myself overwhelmed when a teacher decides to leak a little bit of this information, and I don’t think others are either. That is why I don’t like the pace of high school and instead try to teach myself when my time’s not completely consumed by high school.

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I like to make excuses, but I haven’t stopped. I’m about half way through this book:

I think it is a good book by a good author, and I plan to read some of his other work. In fact I talked someone who had him as a professor on a forum the other day, it was a surprising coincidence. It is a general math book for everyday audiences but it goes into more detail than books of a similar type. I like it because it teaches some fun math history and fills in many little gaps in my knowledge.

Another thing I wanted to talk about is hyper-operations, something that I found very interesting. It comes from the basic idea that multiplication is repeated addition, and exponentiation is repeated multiplication etc. But what about what comes after exponentiation? Before addition? Well, the answers, respectively, are tetration and succession. They’re very simple questions that have been studied since Euler in the 18th century,(1) and I wish it was brought to my attention sooner. But beyond the concept it gets a little more difficult. Indeed, neither tetration nor its successive operations are very well studied. There are many open problems, and this is due to the fact that they require so much computing power. It is not known if is an integer or not. It seems obvious that it would be irrational, but we just can’t know because the number is just so huge that it’s impossible to look at.(2) Hyper-operations are very exciting because it’s likely they’ll be an important area of new study in the future.

My resources: Tetration.org, A neat forum document about notation, since there isn’t really any standards yet, Wiki articles for a quick overview: one and two

Another nice thing I found is this lady’s YouTube channel. She’s going through one of the oldest and most important books in mathematics, Euclid’s *Elements*, piece-by-piece in nice little videos. I watched quite a few of them and while Geometry is not one of my favorite areas of math it’s still very important and interesting. Constructions can be very fun.

(1)Euler, L. “De serie Lambertina Plurimisque eius insignibus proprietatibus.” *Acta Acad. Scient. Petropol. 2*, 29–51, 1783. Reprinted in Euler, L. *Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae*. Leipzig, Germany: Teubner, pp. 350–369, 1921.

(2)Charles (http://math.stackexchange.com/users/1778/charles), Why is it so difficult to determine whether or not $\Large \pi^{\pi^{\pi^\pi}}$ is an integer?, URL (version: 2016-02-11): http://math.stackexchange.com/q/853463

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I was reflecting on the cool and original reference to *Dr. Strangelove Or: How I Learned to Stop Worrying and Love the Bomb* in my last title when I realized a coincidence that might have actually made it seem clever. When I was five years old I was big into Club Penguin, a sort of visual chat room for kids with mini-games, and, being the coolest kid, my main account name was Dr. Math.

So henceforth I shall consider the previous title to be: Dr. Math Or: How I Learned to Stop Worrying and Love Math. It works better, but is still disappointing.

This shines light on my interest in math since a young age, though I sort of lost interest a few years later, and didn’t come back to it until recently.

I would also like to point out that my ramblings about how people often don’t see that math is more than just arithmetic is not some unique thought of my own. I’m sure every mathematician knows this same truth.

You can see that I am using roman numerals to index the tangents and it’s because I am tired of these:

0123456789

Please get new ones. And while you’re at it change these too:

abcdefghijklmnopqrstuvwxyz

And on that same note I would like to bring up *bases*. You may be familiar with the everyday base ten system, the digits of which are written above, but something a lot of people don’t realize is how arbitrary it is. Besides the fact that we have ten fingers and *it’s just what we’ve always used, *there’s really no *basis* for why we use them. And indeed the metric measuring system is *based* on base ten and therefor arbitrary as well. So for those not in the know I would like to bring to your attention the base 12 system, otherwise known as duodecimal or dozenal. Not only does it have two pretty new digits:

12 is also one of the early superior highly composite numbers, which means it has more divisors and is better for representing common fractions e.g. ⅓ in decimal is 0.333333… but in dozenal it’s 0;4. Also to count on your fingers in dozenal simply count the *pads* of your fingers with your thumbs.

I have joined The Dozenal Society and so should everyone. Though base 12 is interesting and slightly better than base 10, still, the farther along you go in the sequence of highly composite numbers the better. The Babylonians used a base 60 system, which is still seen in our units of time. But then there’s still the problem of having more and more symbols to keep track of, so no matter what base we choose it’s still arbitrary. I dream of one day having a perfect representation of numbers.

Whether you love or hate Sal Khan’s voice and explanations I still think it’s a great resource for self teaching. However, whenever I see this big grid:

I get the urge to fill it up. So of course I start at the bottom: with elementary school math and filling all those up without ever learning anything new…

I haven’t made any particular dents in the great barrier to mathematical knowledge since last time, mostly due to school robbing me of any ambition I have left, though I am still going slow and steady. I started Professor Leonard’s Calculus II course, and I spend a lot of my free time watching math related YouTube videos and random math lessons. I forget to save most of these but here are some [‘()’s are links to source]:

- A really good linear algebra lesson ()
- A short blog post that combines math with my interest in all things language: mathematical typesetting, particularly LaTeX. A bit of which I used in my last blog post ()
- A nice Calculus summary ()
- And a quite silly mathematical overview video, which has some very profound points, and also “The Map of Mathematistan” ()
- I started reading the Introduction to Mathematical Philosophy by Bertrand Russel, one of the greatest logicians of the 20th century. It is the “abridged” companion book to The Principia Mathematica, an incredibly famous book that influenced great work such as Gödel’s Incompleteness Theorem and ZFC, and famously took hundreds of pages to prove that 1+1=2. This one talks about the meaning of numbers and the most “simple (using ‘simple’ in a
*logical*sense) (2).”

If… there be mathematicians to whom these definitions and discussions seem to be an elaboration and complication of the simple, it may be well to remind them from the side of philosophy that here, as elsewhere, apparent simplicity may conceal a complexity which it is the business of somebody, whether philosopher or mathematician, or, like the author of this volume, both in one, to unravel (vi).

parts of mathematics ()

I also began the book Concrete Mathematics, a rather serious math textbook by some of the best computer scientists of the late 20th century. * *Though I could generally grasp the concepts, I struggled immensely with some simple demonstrations of the inductive step. I’m not sure what my experience with induction will look like with some real math, though I know it is probably a weak area that I will want to improve.

This leads me to my final point. Though I’m pretty good at being a computer and understanding mathematical concepts *that are already discovered*. I’m not so confident in my ability to do real mathematics: *making discoveries*. Indeed, mathematics is so vast knowing all of it that is known in one lifetime is, to my despair, simply out of reach. The best mathematicians are those who likely don’t know all that much outside of their field (unless you’re Terry Tao). I don’t think I’ll ever make any significant advancements in mathematics in my lifetime, but I don’t care, my goal is simply to know and love mathematics as much as I can.

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Here’s a simple answer: I was already studying math and didn’t want this project to get in the way, as that would be counterproductive, if I do say so myself.

So here’s a better question, *essential *even.

To start I’d like to chart the progression of my own interest, starting from the early years and speeding all the way to the next heading.

In order to appear conceited I will say the I’ve always been pretty decent at math. Mrs. Grant even pulled me out of class to question my aptitude. However, my love for math remained aside for many years. It wasn’t until I stepped out of the world of arithmetic and saw that I knew nothing, and it was exciting. It was the realization that I believe many people failing to make is the reason that math remains esoteric and uninteresting. It’s that having the talent to add or subtract big numbers in your head isn’t what math is about at all.

Early on it’s apparent that the reasons certain people love math are as subjective as their favorite piece of art, (Math With Bad Drawings: 39 Ways to Love Math, and I can recommend any post on that blog) but it’s not too difficult to generalize. Math isn’t just about numbers. Math is the love for patterns, being able to uncover the interesting and surprising using logic and your very own brain, beauty and truth, and minimalism and perfection.

Now, I’d like you to brace yourself for a few popular math things that get me excited about math, but going into detail on them won’t really add anything to this post. Ironically, statements like *that* are also one of the main reasons I started learning math. Blogs and articles with exciting math that is “too complex” to show a layman. I doubted that I couldn’t understand them eventually. I want to understand and get excited too! Even if out of spite, and I hope you do as well.

The go to for surprising and beautiful results. It shows how a bunch of unrelated, complicated ideas developed for thousands of years come together to produce something simple and elegant, showing how fundamental math is in nature. More

A simple expression that looks like Pythagoras’s theorem. After 358 years it was proven by one man’s 7-year-long dedication using a lot of complex and new mathematical ideas.

**A E S T H E T I C**

“There is no formal (axiomatic) system of mathematics that can be both consistent and complete”

Beautiful because in a way it crushes mathematicians’ dreams of perfection, but is still amazing and in another way only *reinforces* mathematics’s perfection.

And there’s so much more. In fact I was surprised to find a Wikipedia page on this very idea.

There is something common between every other branch of human knowledge, and that is *uncertainty*. No matter how clear or intuitive, or how much evidence we have about some fact, it is still impossible to know whether it is true, take gravity for example, if you like.

This is not so with math. Even if you’re a solipsist, you can still prove math to yourself. No matter where you are in this universe mathematical statements will still be true, math is *universal*. It is the very language reality is written.

So if you’re like me, you come to terms with the ambivalent, deterministic universe. You become depressed and search for meaning, find none, and turn to the only thing you have left, aestheticism. You find beauty in knowledge, truth, and once you see that math is all that is really true, you see that math is also the most beautiful.

Now that was somewhat pretentious, but what I am saying is that to me, math is the most important part of life. All I need is my health and math.

I really started to get serious about teaching myself math over winter break of this school year (2015-2016). I want to be able to understand complex ideas like Godel’s Incompleteness Theorem in all its beauty, but I must begin at where my knowledge ends.

First I picked up a couple of good looking books from the library. However, being the slow reader I am and with school getting in the way I’ve only finished one, The Information. And I have many more math books on my wishlist (buy me things).

I also have gone through a complete Calculus I course. Taught by my jacked math hero, Professor Leonard.

And I can confidently say that I can take a derivative.

Lastly, I began teaching myself Python using this guide, because I have not mentioned my equally large interest in programming and computer science (which is basically still just math), but more to the topic at hand: so that I can code pretty interactive math visualizations.

I plan on going through Calc II and III online, then on to the interesting stuff, probably while I’m still in Calc I in school.

Q.E.D

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