As Promised, Another New Lesson!

Hi All,

I’ve FINALLY got the “Bicycle” lesson up, which has been listed under “coming lessons” for far too long now.  Check it out here to learn about cyclic groups!  This is a good one, and the next one will be good as well (and should be up by early next week), so stay tuned!


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New Lesson! Finally!

Hi All,

After saying “a new lesson is coming soon” for far longer than I care to admit, I am happy to say that the next lesson, Lesson 34, which is about modular arithmetic, is FINALLY up!  Here it is.  And now I PROMISE there will still be (lots) more to come in the much nearer future.

And remember, the book is out too!  Tell your friends 🙂


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The Book Is Available!

Hi All!  Well, the time is now.  The book is available here now and will be available on Amazon shortly.  I’d love to know what you think!

Stay tuned for more lessons in the near future and volume 2 in the slightly farther future.



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Upcoming Book!

Hi All!

My apologies for a very long absence and a huge lack of new lessons!  The reason for my lack of activity is a combination of my day job as a grad student as well as the following exciting news: There’s a True Beauty Of Math book coming out!

The book includes a lot of content from the first 19 or so lessons from this site, but with a) extra/bonus lessons, b) lots more exercises, and c) full and detailed solutions to all exercises.  The first volume of what will hopefully be a very large series (extending to ideas like topological spaces and category theory, neither of which we’re even close to getting to yet in these lessons) will be available on Amazon in the coming days/weeks, so stay tuned!

And now that this super-secret project is wrapping up (well, at least Volume 1 is wrapping up), I PROMISE I’ll be getting more lessons up soon (and these will be forming a decent amount of future book volumes).

Stay tuned, and have fun math-ing! 🙂

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The Mathematical Double Standard

Consider the following possibly familiar situation.  Let us assume that you, the reader, are a musical novice, and you therefore know nothing about jazz.  You and some friends go to a jazz club, however, and there’s a trio playing (piano, bass, drums).  Throughout each tune the trio weaves its way through esoteric chord changes, constantly adapting to one another’s improvised flourishes and syncopations.  You, the novice, of course have no idea what’s going on, but assuming the musicians are at least decent the following two statements could most likely be made.  First, you’re probably enjoying the music, at least somewhat.  And second, you can appreciate and admire the talent of the musicians, the hard work that they’ve put in to accomplish what they’re doing in front of you, and you might even respect and look up to the musicians themselves for their devotion to the craft.

Now consider the following possibly less familiar situation. Suppose instead of seeing professional musicians “do what they do”, you were seeing professional mathematicians “do what they do”.  Now, this is a somewhat hard example because one rarely gets to observe a mathematician do his/her work in real time, but fortunately we don’t even need to concoct an example of this.  We merely need to reflect on what the general public’s reaction to the existence of a professional mathematician at all would be.  I believe, in general and with sufficient observational evidence to back it up, that the response of the general populace to the realization that there are professional mathematicians is “WHY?!”.

This is not a “why?” as in “why do we need people to do math?”, because I think almost everyone believes that “math is important” is a true statement (to some extent), and therefore that people understand that mathematicians should exist.  Instead, this “why?!” is more along the lines of “why would anyone CHOOSE to be a professional mathematician?”.

Note how this is in stark contrast to the reaction of seeing a professional musician.  No one questions why anyone would become a professional musician, or painter, or architect.   Conversely (almost) everyone wonders why anyone would WANT to do math.  But why is this?  After all, musicians, painters, architects, and mathematicians all a) deal with esoteric concepts that take years to fully understand and b) spend years working in solitude on these ideas.  So the answer can’t be that “math is hard” or that “mathematicians are weird”.  I would argue that being good at math is just as “hard” as being good at music, painting, and architecture (in the sense that they all take comparable amounts of time, energy, and focus to master), and that professional mathematicians are just as “weird” as their counterparts in other fields (by “weird” I actually just mean passionate and enthusiastic).

So then why is there such a “mathematical double standard”?  I believe the answer is easy. Subjects like music, painting, writing, and architecture, are all COOL.  People generally know that the core ideas of these crafts are interesting, beautiful, and worthy of committing one’s life to the study of.  Thus, when a non-musician meets a professional musician, they can at least think “hey, I have no ability to understand what you really do and think about, but I think that what you do and think about is COOL and interesting, so I’ll respect and admire you for devoting your life to it”.  Let’s compare that to the thought that often happens when a non-mathematician meets a mathematician: “Hey, I have no ability to understand what you really do or think about, and I also hate math a lot, so I have no idea why you’d EVER want to purposefully devote time to the subject, and therefore you’re weird and I’m confused by your existence”.

Now I’m not saying that we need to make mathematicians replace the jocks as the cool kids in high school.  Instead what I’m saying is that we need to make math itself COOL in the following particular sense: we need to make people understand that the core ideas and thought processes in math are cool and interesting and worthy of purposefully devoting one’s time to.  THIS is the only way people will start to change their opinions about math, even if they don’t decide to pursue math seriously, just as I wouldn’t want everyone who has ever gone to a jazz club to attempt to become a professional musician.  Luckily, math really IS cool and interesting!  Just usually not the math that is actually in the curriculum.  And THIS is why we need the actual core ideas and thought processes of math—abstraction, proof, rigor—to be made more apparent.

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Why I love Math, Pt. 1

It’s been a long time since I’ve added to the blog, and I figured there would be no better way to get back in the game than simply talking about some of my reasons for loving math.  We all have different reasons for loving this subject, or we may not even love the subject at all.  But, since one of the aims of this site is to present some reasons for loving the subject, or to increase the number of such reasons, I figured I’d make my own small contribution here.

There are lots of reasons why I love this subject, and that’s why I’ve included the “Pt. 1” in the title, to imply that there will be parts X with X greater than 1 to follow.  For this first episode in the series, however, I’d like to focus not on the beauty of math, the logic involved, or the rigor, but rather the travel.

Mathematics provides the cheapest form of travel to some of the most beautiful locations the Universe has to offer, and these trips can be taken at just about any time of year.

Let me be clear though: I’m not talking about the kind of travel that the world’s best mathematicians get offered, which is to various top-ranked universities around the world and which is therefore only available to those select few semi-divine minds that put the rest of us mere mortals to shame.  Instead, I’m talking about the kind of travel that mathematics offers all of us more modest practitioners of the field—those of us who will likely not have any equations written on our tombstone.

I first fell in love with the travel that math affords when I lived in New York and spent a lot of time on subways.  Spending an hour on a crowded NYC subway in the middle of summer is often not the most enjoyable experience, so I would often take that time to take an hour-long mathematical journey.  Knowledge of math, along with a little knowledge of physics, allowed me to travel to the edge of a black hole and take a look at what’s going on, or into the workings of a single electron and ponder its behavior, or into the very beginnings of our universe and try to imagine how it looked back then, instead of worrying about the hairy man standing a little too close to me or staring at the high school couple making out just slightly too passionately to be suitable for any public arena.  Even without drawing upon any knowledge of physics, I have used math to take some great trips to higher dimensions and/or wild geometries and/or counterintuitive algebraic structures.

By taking these trips I have saved myself a lot of the frustration that comes along with waiting in traffic, or standing in line, or sitting in a boring lecture (don’t tell my undergrad anthropology professor about that, though).  I have used my mathematical travels to help me get through some of my physical travels, as plane and train rides are all made much more bearable when you can take a little journey within a journey.

Now one should be careful about traveling too much.   Cruising down the 101 at 80 mph is probably not the best time to take a trip into noncommutative geometries, and sitting across the table of a job interviewer or a first date is probably not the ideal time to travel into categorical quantum mechanics.  There’s no doubt, though, that when used properly the travel afforded by knowing some math can really come in handy in some of life’s more mundane experiences, infusing them with same fire that lit the stars (to quote David Foster Wallace)—quite literally.

To finish, let me just say that this travel is completely free, always available, and requires minimal packing time.  In fact, I would argue that by simply reading the lessons here one can gather enough miles to start taking some pretty cool trips to non-Abelian groups and new set-theoretic paradoxes.  As the lessons continue to increase in number, I hope some of you cherished readers can continue to find some new and exotic locations to travel to next time you’re at the bank waiting in line to argue about some recent credit card transactions.

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Solutions Starting to Surface

Again after much too long last, I’ve finally started working on the solutions manual for The Language of Nature, and I’ve uploaded the first few on that same page.  I’ll upload them as I do them, and I’ll try to do a few every day modulo any serious limitations that my real life presents.  I’ll simply be updating the same pdf (titled “Solutions Manual”), otherwise there will be way too many editions.  Finally, the solutions manual is also a sort of extension of the book, in that I sometimes ramble off into things that aren’t immediately relevant to the problems at hand, but are nonetheless interesting.  I hope this is welcomed and not a cause of frustration.

More lessons to come, and I also think I’m long over due for some less formal ramblings!

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First Edition of Book is Now Up!

In this post I talked about a book that I’d be publishing soon.  Well, I decided not to publish it yet, and instead I’m putting up the first edition for free here.  There’s more explanation on that page, so I’ll spare the reader as much duplicate reading as possible by ending this post here.

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From Notes to Book

Hey all, again no math, sorry, although I hope you do/did enjoy the most recent lesson on dihedral groups.  As I mentioned in my last blog post (“My Absence“), I’m currently in the process of finishing up lecture notes for an intensive three-week long high school course.  Seeing as the course starts tomorrow, I’m just about done with them.  Upon polishing them up, though, I got an idea.

I’ve received emails from readers who have read through the entire site and who want to keep reading and exploring math.  This is absolutely fantastic, and you have no idea how happy emails like that make me.  The problem is, however, that I have very few ideas for “what comes next” (i.e., what comes after the site (there’s still lots of content that I will be providing here in the future)), because quite honestly the only thing that comes next is a hardcore college math textbook.  These might be a little daunting for someone who’s fresh off my site, but there simply is nothing to fill this void (that I know about, at least).

This brings me to my idea.  Instead of just posting my lecture notes online, what I’ve done is turn them into a full-fledged book.  It’s well over a hundred pages of math (and some physics), some of which we’ve seen here already, and the vast majority of which will comprise the next few hundred lessons on the site.  The book will, of course, move much more quickly than the site, for otherwise it’d be 5,000 pages long.  It is intended to be a bridge between this site, for example, and a full on college textbook.  It will hopefully be completely done in the next couple of weeks.

Unfortunately there’s not enough time in the day to write content fast enough for some of my extra keen readers (which again, is a great thing), but I hope that this book will provide what these keen readers are looking for.  I’d love to hear what you think about all of this, and I hope you continue to read the site regardless of if you want/like/read/buy the book.  Once this is done, I’ll be pumping out lessons at my former pace, I promise!


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My absence

Hey all,

No math in this post, just an apology.  I’ve been completely out of action with the site for the last couple of weeks, and I apologize for that.  Please don’t take it as me being gone for good.  The reason for my lack of activity here is that I’ve been working–for months–on a set of lecture notes for a high school course that will be starting in about a week.  It covers everything the site does, plus a lot of what the site will cover in its first, say, 200 lessons (yes, I plan on writing that much, eventually).  The notes are obviously more dense (the course is three weeks long), but still just as accessible (although requiring a bit more work).

When I’m done writing them I plan on posting them to the site, in case there are any extra keen readers who’d like to see what some of this stuff is all about.  I’ll be done with those in a few days, and afterwards I’ll get right back to writing lessons.  My promise to you is that I’ll have two more lessons up within the next 8 days.

So sorry again for my absence, but please don’t take it as the new status quo.  I plan on sticking with this for a long, long time, and writing LOTS more lessons.


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