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.”
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!
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!):
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!
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:
Do you ever get the feeling that Lego Bricks are becoming more expensive? When we were kids, boy, it felt like they were cheaper, right? I mean, the biggest sets were $150 at most. I have a HUGE Lego collection, and it definitely seems like Legos back in my day were more affordable.
Trouble is, that’s not really true. It turns out that Lego bricks have actually gotten cheaper, by almost every measure you can think of (weight/number of pieces/licensed sets). Check out this incredibly thorough post on Lego Price statistics over time. The article is entitled, “What Happened with LEGO” by Andrew Sielen. It’s very thoughtfully done.
Looking for a great application of systems of linear inequalities for your Algebra 1 or 2 class? Look no further than today’s GraphJam contribution:
You might just give this picture to students and ask THEM to come up with the equations of the three lines.
There’s also a nice discussion to be had here about inverse functions, or about intersecting lines. And there might also be a good discussion about the domain of reasonableness.
Here are the three functions:
This is especially interesting because I never think of the rule as putting boundaries on a person’s dating age range. Usually people talk about it in the context of “how old of a person can I date?” not “how young of a person can I date?” Or rather, if you’re asking the second question, it’s usually phrased “how young of a person can date me?” (All of these questions relate to functions and their inverses!) But in fact, the half-your-age-plus-seven rule puts a lower and and upper bound on the ages of those you can date.
As far as reasonableness, is it fair to say that my daughter who is 1 can date someone who is between the age of -12 and 8.5? I don’t think so! I’m definitely going to be chasing off those -12 year-olds, I can already tell .
For my daughter, the domain of reasonableness might be !
Congratulations to Curtis Cooper ofGIMPS for finding the largest known prime (for now!). Are you ready? It’s a Mersenne Prime and apparently it took 39 days of computations to verify its primality. Here it is:
Happy Friday! Hope everyone has their kids registered for the AMC next week. If you haven’t already subscribed to the AMC problem-a-day from the MAA, you should! It’ll keep you sharp .
Here are a few nice things seen ’round the web recently:
The Scrambler, by Dan Meyer & co. Here, Dan challenges us to analyze a classic carnival ride, and asks us to predict where you end up at the end of the ride. And by Dan & “co”, I mean “comment” folks who have generated lots of fun solutions and applets. Dan made a great interactive version here, too.
And finally, this lengthy article “Reflections on mathematics and Democracy” by Lynn Arthur Steen is well worth the time[ht: Gene Chase]. He thoughtfully discusses the need for math education among the citizenship. Is “usefulness” to the democracy the highest goal of secondary math education? Do we aim to create quantitatively literate citizens? Or do we put them on the Calculus track and prepare them for college-level STEM careers? Does teaching “quantitative literacy” even count as Mathematics with a capital M? This is obviously something I’ve been thinking a lot about recently. Here are a few of my favorite excerpts:
Ten years ago I addressed the first question posed to this panel in Mathematics and Democracy—a collection of essays from a variety of professionals both inside and outside mathematics.4 (These essays are available for free downloading on the MAA website.) The chief message of this volume is that the mathematics taught in school bears little relationship to the mathematics needed for active citizenship. That mathematics we called quantitative literacy (QL) to contrast it with traditional school mathematics which, historically, is the mathematics students needed to prepare for calculus.
Mathematics and quantitative literacy are distinct but overlapping domains. Whereas mathematics’ power derives from its generality and abstraction, QL is anchored in specific contexts and real world data. An alternative framing of the challenge for this panel is to ask whether perhaps QL might be a more effective approach to high school mathematics for all.
…
What we forget, however, is that when NCTM initiated its standards work, most mathematics teachers did not actually believe in the goal of teaching mathematics to all high school students. Whereas now we argue about how much and what kind of mathematics to teach in high school, three decades ago debate centered on who should learn high school mathematics. At that time, the curriculum was designed to efficiently sort students into those who were capable of learning high school mathematics and those who were not. So between grades 7 and 9, somewhere between one-third and one-half of the students were placed in a course called General Math—an enervating, pointless review of arithmetic.
…
Another decade has passed, and our ambitions are now much higher: a common core for all, with everyone emerging from high school ready for college. In one generation, the political view of high school mathematics has progressed from something only some need (or can) learn to a core subject in which all students can and must become proficient. That’s quite a rapid change in ends, which has been matched by a major change in means. The very idea of a common curriculum enforced with common assessments was all but unthinkable back in the 1980s.
This presentation has been around for a few years, but I’ve never highlighted it here on my blog. It’s a Prezi presentation by Alison Blank and it’s called Math is not linear.
It brings up an excellent point that math is often talked about as being a sequence with one class coming before another, when in fact math isn’t always like this. It is absolutely true that there’s room for prerequisites (it’s not a great idea to take Calculus 3 before Calculus 1), but much of mathematics can be approached at any time.
math courses–perhaps more like a tree then a line?
Math majors realize this when they get to college. Once you take a few basic classes like Calculus, Linear Algebra, and Statistics, you can take almost any other course you want. Occasionally there are other prerequisites (a two-course sequence in Differential Equations might require students to take them in order). But generally, you can order your college math courses in many different ways.
I’ve noticed this in grad school too. Here’s the order I’ve taken my classes so far with my comments:
Differential Equations (took it in undergrad too, which helped!)
Cryptography (might have been good to do abstract algebra first, but we learned the practical things we needed to know along the way)
Abstract Algebra (when we talked about cryptography and elliptic curves, it was total review!)
Real Analysis (took it in undergrad, but boy I understood it a lot better the second time around)
Statistics (took it in undergrad, but I loved learning it from a more theoretical, Calculus-based approach)
Queueing Theory (stat required, for good reason; just started this course, so I can’t tell you much more)
Did I have to take these courses in this order? Not necessarily. I’ve found that cross-references are made between courses constantly. In my queuing theory class this past week, we solved a differential equation. So there!
Could we do this in high school? To some extent, yes. I think it’s still important to have prerequisites, however. And to get to Calculus, you have to take a *somewhat* linear path. (Do you agree that Calculus is the pinnacle of high school mathematics?) Along the way, though, we should indulge in lots of mathematical tangents, as the Prezi suggests.
In some ways we already allow our students a bit of latitude. We offer AP Statistics, which can can be taken almost anytime after Algebra 2. And we offer Higher Level math, which gives a nice taste of college-level math.
As Alison Blank suggests, could we teach a little topology to high school students? Certainly we could. Weren’t we all interested in the advanced math topics before we actually took the class? I remember learning all the fun and interesting results from topology way before I ever read through a topology textbook. I still haven’t taken a class in topology! (But I read a lot of Munkre’s book and did a lot of the exercises, and I think that counts for something .)
The point is, I think we can talk about fun and interesting math ideas way before we’re “allowed” to talk about them!
That’s certainly what my dad did with me at home. Thanks dad!
I dedicate this post to all those math teachers who went on a mathematical tangent this week. Keep up the great work!
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