Polymathematics

Patting Myself on the Back

I can't resist doing it.  I don't think I've ever felt like I've contributed something new to the body of knowledge of mathematics before, but I think I might have now.  This post modifies an obscure proof I found in a PDF hidden on a long web page (linked in the article) so that it doesn't need trigonometry.  The proof is one from triangle geometry (my recent favorite topic), and I haven't found any proof of this fact that's as simple either on the web or in any books.  I will happily give credit here if I'm wrong, but I suspect that my proof is among the simplest to follow (for non-triangle-geometry-experts) that exists for the theorem.

Pat, pat, pat.

Never As Much as I Think

I really never get as much done in the summer as I think I will.  I am still working on several posts for the AHSMP, but I've added a few to my "Math, Eloquently" blog for teaching the basics.  This will likely be the last time I link to it in a post, so if you think the explanations are good, feel free to put it on your RSS feed or link to it on your own blog;  I'd appreciate the traffic.  Thanks...more stuff coming...I hope.

The First Teaching Post

Okay, all you math fans...I put up my first teaching post.  If you're a teacher, you'll recognize the need for a good explanation.  So very, very, very (did I mention VERY?) many students get this wrong.  I'm open to suggestions on improving my style for this project, so please comment (praise or suggestions) here if you feel the urge.

Two Years and Big Changes

Whew.  How are you all doing?  It's been a while, I realize.

I'm making some changes to my blogging.  I will now maintain not just one, but three blogs.  This one will be solely for more advanced and interesting mathematical content.  In particular, I plan to continue expanding the AHSMP, but I will also continue to post other mathematical content of interest.  My math posts get by far the most hits, and I didn't want to continue muddying the mathematical waters with personal content.  New blogging about teaching and my personal life will now appear on my first new blog, Math Spectrometer (so named because I'm identifying the elements of my math life—quite the geeky pun, I know;  my father the physicist will be proud).  Previous posts will not be deleted from this one, though.  Also appearing on the new blog are my "greatest hits";  namely political, personal, and teaching-related posts that have either gotten a lot of traffic or are my personal favorites.  Finally, my third blog (which I haven't started yet...UPDATE:  it's here, but under construction!) will contain a) explanations of more basic high school math concepts (most that I find online are frankly not that good) and b) links to the book chapters I will someday write that will be available for sale via print-on-demand publishing.  My experience with math textbooks so far has been somewhat disappointing;  I may be kidding myself, but I think I can do better.  The reviews of my math writing on this site have been pretty good, so we'll see how that goes.

On a different note, this week marks the 2 year anniversary of the post that put me on the blogging map.  It's the one about how .999... equals 1.  When it was picked up on the front page of Digg, I was getting many hits (over 25,000 on one day).  During that time, I was on a road trip with my wife, and we met some of her friends whom I told about the post—my jaw dropped when one of them said, "Wait, that was you?"

The post inspired over 1000 comments, many doing their best to discredit the mathematical fact, and many more trying to convince those disbelievers that I was right (a shout out to Monimonika who has kept up with almost all the naysayers:  thanks, and you're off the hook now:  see below).  It inspired several other explanatory posts and a rant or two from me (you can link to them from the main post).  I have seen many, many discussion boards on which the topic appeared, and on which the inevitable arguments broke out, many of which made attempts to settle them by linking to my post.

Perhaps most tellingly, as of today, Googling the phrase ".9 repeating equals 1" (among other phrases) lists my page as the number one hit—above Wikipedia's entry on the topic!

Occasionally, there have been people who were convinced by the arguments after initially disbelieving the facts.  More often, comments by disbelievers on discussion boards amounted to "that guy is an idiot".  They will, I'm sure, continue to argue there.  But I've made the decision that the discussion on my site is over.  I'm closing all posts in that thread to comments, and comments about it on any other thread will be deleted.

So...I hope to see you all here for more math and at one or both of my new blogs (links on the sidebar at some point soon).

Details, Details.

The number of details to take care of during this, my supposed spring break, is astounding.  But I can see the light at the end of the tunnel, and I just found a remarkable simple proof of the existence of the 9-point circle (or Feuerbach's circle) that I will truly try to post before school starts again.  I've been looking for a proof of that for some time now, and I found one involving complex numbers, but the one I'm going to post (here it is!) is quite surprisingly basic, using relatively simple theorems.

I have, however, been keeping up with the blog by reading comments.  Amazing, I'm still getting comments on my .999...=1 post after 2 years.  Most of the comments are just the same old arguments that I address in the follow-up posts.  I owe a tip of the hat, though, to Monimonika...a commenter who has staunchly defended the truth of that fact against every inane commenter.  I don't have that kind of patience.  Way to go, and thanks, Monika.

Okay...pay some bills...clean up a little...then maybe I can start on that proof.  I can't understand why I haven't found any other proofs of this online that are as simple.

Why I Support Barack Obama

I know, I know.  I've promised more math posts and failed to deliver.  The truth is, it takes quite a while to write a math blog post because I have to design all the diagrams and equations and convert them to the right format to upload.  So more math posts, I promise, I promise.  Eventually.

But as much as I like math, a much more weighty issue is rising to prominence:  We will elect a new president in about 9.5 months, and that person will take office in less than a year.  I've made no secret about my disdain for the current administration, and for George W. Bush, in particular.  I've also mentioned my admiration for Obama before.  But now I'm more convinced than ever that he is the right choice by far to be our next president.  I know my readership is not very high, and most people land here for my math pages.  But I think my reasons are good, and I would feel uneasy if I didn't do something here to tell you what those reasons are, especially with only a week to go before "superduper" Tuesday.  Please feel free to link to this post if you agree with me (or if you disagree, for that matter).  If you must steal my words without attribution, at least they might help elect the man.

1. This is the reason that I hear the least, but it's among the most important to me.  The president, in addition to being our leader, is our representative—the public face we show the world.  Our modern world is no longer one which appreciates a superpower throwing its rich, white, establishment weight around.  Europeans, South Americans, Middle Eastern countries, and sworn terrorist enemies resent that to varying degrees about us.  Electing a young, multi-ethnic president will present a much different face:  one that says, "We've come to realize that our strength comes from our diversity.  Our economic strength resides in the hard work of our immigrants (both willing and unwilling) in our present and our past, and we all have a stake in our success.  There's more to this country than you are accustomed to seeing."
2. Let's face it, the resentment I feel towards George Bush and the resentment a lot of Republicans feel towards Clinton (both of them, actually, but Hillary, in particular) are not helpful emotions in unifying this country to face the world's threats (terrorism, genocide, environmental degradation, shortages of natural resources like oil and water).  Obama is probably the least divisive candidate on either side.  He attracts very liberal Democrats (Like most of the Kennedys!  C'mon, that's gotta count for something!  They're saying:  "If you voted for John, or if you planned to vote for Robert, please vote for Obama.") and even conservative Republicans who are attracted to his integrity and his willingness to seek common ground.  He isn't perfect, of course, no candidate is.  But he offers the best hope of emerging from the D./R., RedState/BlueState, Fundamentalist/Secularist stalemate corner that our politics of division (thank you, Karl Rove and Lee Atwater) has painted us into.  Clinton is the opposite.  She's so divisive that Republicans will come out of the woodworks to vote against her in the general election.  And even if (somehow) she really were elected, it would just perpetuate the antagonism, and that has just got to stop.
3. It's a little clichéd, but true:  a president of color will surely send a strong signal that people of color (and, frankly, all people not born into White, male, Christian privilege) need no longer assume that this country won't let them succeed.  It won't be an end to racism, of course, far from it.  And I'm even a bit worried that conservatives will use an Obama presidency to claim that "Racism is over! We don't need affirmative action or any of that anymore, see!"  But despite that possibility, the benefits outweigh the risks.  At the very least, an Obama candidacy will emphasize the political importance of people of color, even if loses.
4. He's smart (extraordinarily so, I think) and confident about it.  This means that he won't, out of insecurity, appoint cronies and yesmen to important positions around him.  Bush didn't pick grown-ups (Gates, Petraeus) for important posts until late in his presidency, and Clinton has too many favors from her husband's administration to return to make wise, considered appointments.  I have confidence that Obama wouldn't hesitate even to name people he disagrees with to cabinet positions if he respects their accomplishments and thinks they'll do a good job.
5. He clearly inspires young people to be involved in politics.  The supposed apathy of genX and genY has worried older and "wiser" generations for a long time now.  But Obama is inspiring record numbers of young people to be involved and vote.  While the Baby Boomer generation might not be happy about losing power to a younger generation, the change has to happen sometime, and it should happen by inspiration, not by default.
6. I believe he has real integrity.  For one thing, he has tried to run a decent campaign.  Some tactics that could be considered underhanded are probably impossible to avoid, but I really believe he is committed to changing how political campaigns are designed.  For another, he talks about sacrifice and the need to for everyone to change—his speeches are inspiring without being all warmfuzzyfeelgood all the time.  For a third, he has real experience in the trenches of community organizing, which he undertook because he believed in it, not because he thought it would bring him the presidency (kindergarten aspirations, notwithstanding).

There.  Those are my reasons.  I hope someone is listening.

Not In Our Country

When I get back from vacation, I promise I will post some more math posts in the AHSMP.  But for now, you should take a look at this link.  We tend to think that that violence against women is decreasing in our country.  That law enforcement organizations have learned to take rape and domestic violence more seriously.  And perhaps they have.  But not everywhere.  Of course, this situation is a legacy of our awful treatment of American Indians, and part of the problem is that, according to the law, it's not entirely clear if this really is taking place in our country.  But for the sake of the women of South Dakota (on or off of a reservation), let's hope the law enforcement and funds are provided to help that shelter.

Please consider a donation of goods or money—for the women's sake and to send a message that this is unacceptable in our country.

Nothing To Say

When I started this blog, I had plenty to say.  I had observations on teaching, political semi-rants, cool mathematical content.  Now and then I still have some of these.  And I'm doing some interesting things at work, but telling all the details might compromise my anonymity.  But the tidbits to relate to you all just aren't coming as fast as they used to.  Not that I plan to give up the blog;  I still want to expand my Advanced High School Math Project.  And I'm sure I'll have more to say as the next presidential election approaches.

But this whole child-rearing proposition is...well...very tiring.  The baby is great:  healthy, smart, cute, determined (adjectives that have been independently confirmed, so it's not just Dad talking here).  And for the most part, I'm enjoying my time with her, even if now and then I'm totally exhausted.  By which I mean "now";  my wife and I have made a pact that we will be in bed by 9pm tonight, and we're giddy at the thought of a full night's sleep.

And this tiredness just doesn't leave much room for clever observation or for a whole lot of professional introspection or for finding obscure proofs online and rewriting them in high-school-student-accessible form or for political outrage.  So, forgive me for the lapse in new material.  Look for Ceva's Theorem next in the AHSMP so that my co-writer and I can start talking about lesser-taught triangle centers.

And now...a (hopefully) blissful, long night's sleep for a 1.3 drooplid math teacher.

Initial 1's

A friend of mine has been thinking about the well-known fact that many lists of seemingly random numbers (addresses, physical constants, populations) contain far disproportionately many initial 1's (compared to other initial digits, which decrease in frequency up to 9).

He noted that many physical constants are ratios—constants of proportionality, for example.  So he calculated the probability that any two random numbers x and y between 0 and 1 (random with a flat distribution) have a ratio y/x that starts with a 1 (meaning its first non-zero digit:  .00013 counts as an initial 1).

Cool answer:  1/3

The proof is left as an exercise (unless there's an outcry for it).  Just a little tidbit to tide you over, since I haven't been posting much.

Have I mentioned?

...that regular blogging with a baby during the school year is almost impossible.

Today I watched 58 tests getting spit out of the copy machine.  The first are being written right now as I'm typing.  I will have to examine every page (5 or 6) of every test in the next week or so.  Wish me luck.

(P.S.  I'm now regularly getting significant numbers hits on some of my pages in the Advanced High School Math Project (see sidebar), and I'm convinced it's a worthwhile undertaking.  I'm still looking for submissions, especially during this time that my posts can't be as frequent.)