Math on Quora

quora iconI may not have been very active on my blog recently (sorry for the three-month hiatus), but it’s not because I haven’t been actively doing math. And in fact, I’ve also found other outlets to share about math.

Have you used Quora yet?

Quora, at least in principle, is a grown-up version of yahoo answers. It’s like stackoverflow, but more philosophical and less technical. You’ll (usually) find thoughtful questions and thoughtful answers. Like most question-answer sites, you can ‘up-vote’ an answer, so the best answers generally appear at the top of the feed.

The best part about Quora is that it somehow attracts really high quality respondents, including: Ashton Kutcher, Jimmy Wales, Jermey Lin, and even Barack Obama. Many other mayors, famous athletes, CEOs, and the like, seem to darken the halls of Quora. For a list of famous folks on Quora, check out this Quora question (how meta!).

Also contributing quality answers is none other than me. It’s still a new space for me, but I’ve made my foray into Quora in a few small ways. Check out the following questions for which I’ve contributed answers, and give me some up-votes, or start a comment battle with me or something :-).

And here are a few posts where my comments appear:

I ♥ Icosahedra

Do you love icosahedra?

I do. On Sunday, I talked with a friend about an icosahedron for over an hour. Icosahedra, along with other polyhedra, are a wonderfully accessible entry point into math–and not just simple math, but deep math that gets you pretty far into geometry and topology, too! Just see my previous post about Matthew Wright’s guest lecture.)

A regular icosahedron is one of the five regular surfaces (“Platonic Solids”). It has twenty sides, all congruent, equilateral triangles. Here are three icosahedra:

icosahedron coloringsHere’s a question which is easy to ask but hard to answer:

How many ways can you color an icosahedron with one of n colors per face?

If you think the answer is n^{20}, that’s a good start–there are n choices of color for 20 faces, so you just multiply, right?–but that’s not correct. Here we’re talking about an unoriented icosahedron that is free to rotate in space. For example, do the three icosahedra above have the same coloring? It’s hard to tell, right?

Solving this problem requires taking the symmetry of the icosahedron into account. In particular, it requires a result known as Burnside’s Lemma.

For the full solution to this problem, I’ll refer you to my article, authored together with friends Matthew Wright and Brian Bargh, which appears in this month’s issue of MAA’s Math Horizons Magazine here (JSTOR access required).

I’m very excited that I’m a published author!

Math on the web

Here are two items that have been shared with me in the last 24 hours:

Item 1: Want To Be Better At Math? Use Hand Gestures! Jeremy Shere of Indiana Public Media. Check out this very short audio news that suggests that math instruction has been shown more effective with gestures. I flail around in front of my classroom all the time, so I guess that makes me a good teacher, right? I’d sure like to think so! 🙂  (HT: Tim Chase)

Item 2: How to Fall in Love With Math. Manil Suri, professor at a small school down the road from me (University of Maryland…maybe you’ve heard of it?), has a very nice piece on why math is a worthy object for our affection. It’s been heavily shared in the circles I travel–and for good reason. He reminds us that people fall susceptible to two very common errors when casually speaking about math: (1) We reduce it to arithmetic, as in “come on guys, do the math” or (2) we elevate it to something so ethereal that it’s impossible to grasp, as in “that mathematician talks and I don’t understand a word he says. I never was good at math.” Math, Suri says, is much more than arithmetic and much more accessible than people give it credit for. People can appreciate it without understanding every difficult nuance, just as they do art. (HT: Beth Budesheim)

 

 

 

Bring an end to the rationalization madness

In less than a month, we’ll be hosting the one and only James Tanton at our school. We’re so excited! I’m especially excited because he’s totally going to help me rally the troops in this fight:

He posted this a few years ago, but I only stumbled on it recently. I’ve been looking for Tanton videos to use in our classes so we can get all psyched up about his visit! Needless to say, I was loving this video :-).

For more on why I’m not such a big fan of ‘rationalizing the denominator’ see this post.

Arithmetic/Geometric Hybrid Sequences

Here’s a question that the folks who run the NCTM facebook page posed this week:

Find the next three terms of the sequence 2, 8, 4, 10, 5, 11, 5.5, …

Feel free to work it out. I’ll give you a minute.

Done?

still need more time?

..

give up?

Okay. The answer is 11.5, 5.75, 11.75.

The pattern is interesting. Informally, we might say “add 6, divide by 2.” This is an atypical kind of sequence, in which it seems as though we have two different rules at work in the same sequence. Let’s call this an Arithmetic/Geometric Hybrid Sequence. (Does anyone have a better name for these kinds of sequences?)

But a deeper question came out in the comments: Someone asked for the explicit rule. After a little work, I came up with one. I’ll give you my explicit rule, but you’ll have to figure out where it came from yourself:

a_n=\begin{cases}6-4\left(\frac{1}{2}\right)^{\frac{n-1}{2}}, & n \text{ odd} \\ 12-4\left(\frac{1}{2}\right)^{\frac{n-2}{2}}, & n \text{ even}\end{cases}

More generally, if we have a sequence in which we add d, then multiply by r repeatedly, beginning with a_1, the explicit rule is

a_n=\begin{cases}\frac{rd}{1-r}+\left(a_1-\frac{rd}{1-r}\right)r^{\frac{n-1}{2}}, & n \text{ odd} \\ \frac{d}{1-r}+\left(a_1-\frac{rd}{1-r}\right)r^{\frac{n-2}{2}}, & n \text{ even}\end{cases}.

And if instead we multiply first and then add, we have the following similar rule.

a_n=\begin{cases}\frac{d}{1-r}+\left(a_1-d-\frac{rd}{1-r}\right)r^{\frac{n-1}{2}}, & n \text{ odd} \\ \frac{rd}{1-r}+\left(a_1-d-\frac{rd}{1-r}\right)r^{\frac{n}{2}}, & n \text{ even}\end{cases}.

And there you have it! The explicit formulas for an Arithmetic/Geometric Hybrid Sequence:-).

(Perhaps another day I’ll show my work. For now, I leave it the reader to verify these formulas.)

Progress Toward Twin-Prime Conjecture

This nice article came through on wired today:

Unknown Mathematician Proves Surprising Property of Prime Numbers

By Erica Klarreich, Simons Science News

Image: bwright923/Flickr

On April 17, a paper arrived in the inbox of Annals of Mathematics, one of the discipline’s preeminent journals. Written by a mathematician virtually unknown to the experts in his field — a 50-something lecturer at the University of New Hampshire named Yitang Zhang — the paper claimed to have taken a huge step forward in understanding one of mathematics’ oldest problems, the twin primes conjecture.

Editors of prominent mathematics journals are used to fielding grandiose claims from obscure authors, but this paper was different. Written with crystalline clarity and a total command of the topic’s current state of the art, it was evidently a serious piece of work, and the Annals editors decided to put it on the fast track.

Just three weeks later — a blink of an eye compared to the usual pace of mathematics journals — Zhang received the referee report on his paper.

“The main results are of the first rank,” one of the referees wrote. The author had proved “a landmark theorem in the distribution of prime numbers.”

(more)

This is very exciting news, and the whole story has a fantastic David & Goliath feel–“little known mathematician delivers a crushing blow to a centuries old problem” (not a fatal blow, but a crushing one). It’s such a feel-good story, almost like Andrew Wiles and Fermat’s Last Theorem. Here’s my favorite part of the article:

…during a half-hour lull in his friend’s backyard before leaving for a concert, the solution suddenly came to him. “I immediately realized that it would work,” he said.

Just chillin’ in his friend’s backyard…and it came to him! Anyone who has worked on math problems or puzzles has had this experience, right? It seems like an experience common to all people. This has definitely happened to me lots of times–an insight hits me out of nowhere and unlocks a problem I’ve been working on for weeks. It’s one of the reasons we do mathematics!

Paradoxes, Infinity, and Probability

I’ve been loving the videos that SpikedMathGames has been posting on youtube. Check out their channel here. In particular, I’ve enjoyed Paradox Tuesday. Here’s one from a few weeks ago which really interested me (if you go to the youtube page, you’ll see I’ve been active in the comments!):

They also cover some famous paradoxes like the Unexpected Hanging Paradox or the  Barber Paradox. If you’ve never heard of these, go watch these now.

I’m especially interested in paradoxes that deal with infinity, countability, and probability. Here’s another great paradox that deals with just those issues that my friend Matthew Wright shared with me a few months ago (thanks Matthew!). It’s called the Grim Reaper paradox (can’t link to the Wikipedia article–it doesn’t yet exist), proposed in 1964 by José Amado Benardete in his book Infinity: an essay in metaphysics, and I first read about it on Alexander Pruss’s blog here, and I quote:

Say that a Grim Reaper is a being that has the following properties: It wakes up at a time between 8 and 9 am, both exclusive, and if you’re alive, it instantaneously kills you, and if you’re not alive, it doesn’t do anything. Suppose there are countably infinitely many Grim Reapers, and before they go to bed for the night, each sets his alarm for a time (not necessarily the same time as the other Reapers) strictly between 8 and 9 am. Suppose, also, that no other kind of death is available for you, and that you’re not going to be resurrected that day.

Then, you’re going to be dead at 9 am, since as long as at least one Grim Reaper wakes up during that time period, you’re guaranteed to be dead. Now whether there is a paradox here depends on how the Grim Reapers individually set their alarm clocks. Suppose now that they set them in such a way that the following proposition p is true:

 

(p) for every time t later than 8 am, at least one of the Grim Reapers woke up strictly between 8 am and t.

Here’s a useful Theorem: If the Grim Reapers choose their alarm clock times independently and uniformly over the 8-9 am interval, then P(p)=1.

Now, if p is true, then no Grim Reaper kills you. For suppose that a Grim Reaper who wakes up at some time t1, later than 8 am, kills you. If p is true, there is a Grim Reaper who woke up strictly between 8 am and t1, say at t0. But if so, then you’re going to be dead right after t0, and hence the Grim Reaper who woke up at t1 is not going to do anything, since you’re dead then. Hence, if p is true, no Grim Reaper kills you. On the other hand, I’ve shown that it is certain that a Grim Reaper kills you. Hence, if p is true, then no Grim Reaper kills you and a Grim Reaper kills you, which is absurd.

Go visit his blog post for a discussion of why this seems unresolvable, and how it may actually put forward a case for time being discrete rather than continuous. Crazy thought.

There’s something deeply unsettling about this paradox and also the Unexpected Hanging paradox. Anytime we deal with probabilities and certainty, paradox seems to be lurking nearby.

I sometimes ask my students this somewhat related question–perhaps you’ve heard it too:

How many positive integers have a 3 in them? (That is, in their decimal representation. 6850104302 has a 3 but 942009947 does not.)

If you haven’t ever considered this question, take the time to do it now.

Though I actually once worked out the result using limits (like Alexander Bogomolny does marvelously here), it’s easy enough to work out the result in our heads:

First ask yourself how many digits a randomly selected integer has. The number of digits is almost certainly greater than 2, right? There are only 90 two-digit positive integers, a finite number, and there are an infinite number of integers with more than two digits. It follows that if you were to pick one at random from among all positive integers*, it would be almost certain to contain more than two digits.

The same argument could be applied to a larger number of digits. By the same logic as above, we can convince ourselves that ‘most randomly selected integers have more than a trillion digits’. It’s a bit of an incredible statement, really. We rarely ever work with the ‘most-common’ kind of numbers (the big ones!).**

What is the probability that a number with a trillion digits has a 3 in it? Well, it’s almost certain. The probability approaches 100%. If we consider ALL numbers, the probability IS 100% (or is it?). This is a real dilemma. How can we say that 100% of numbers have a 3 in them when this is clearly not true?

We’ve been pretty sloppy here, but regardless, this kind of fast-and-loose infinite probability question is unsettling.

Do you want to try taking a crack at these? Feel free to comment below.

Oh, and Happy Birthday Euler!

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Footnotes:
* Picking a number from the set of all positive integers requires the axiom of choice.
** My comment that the ‘most-common’ kind of numbers are the big ones reminds me of Ronald Graham’s quote: “The trouble with integers is that we have examined only the very small ones.  Maybe all the exciting stuff happens at really big numbers, ones we can’t even begin to think about in any very definite way.  Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed.  Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions.” Love that quote, especially considering it comes from Ronald Graham, an expert in Ramsey Theory, and the creator of one of the largest named numbers :-). The fact that we have only ever studied the most common kinds of numbers is also confirmed by the fact that most numbers are irrational. Worse, most numbers are indescribable!

Women & Math

Here’s a thoughtful TED talk from Laura Overdeck of bedtime math. I’ve highlighted this website before and I think it’s such a brilliant idea for kids! Laura was the lone girl in her astrophysics undergraduate studies, so she has a great vantage point from which to give this stellar talk (pun intended! 🙂 ).

She has lots of great points. In particular, I liked these three simple recommendations for women–and for everyone:

  1. Don’t play the lottery.
  2. Don’t say you hate math.
  3. Don’t ask others to calculate the tip.