Category Archives: Math on the web

Benford’s Law & Infinite Primes

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At the risk of sounding like a broken record: Dave Richeson has had some more great posts recently. These two in particular left me in awe of mathematics. Both results I had seen before, but not proved in these ways. These two proofs are so unique. Unfortunately, for my students, both of these particular proofs also require math typically not covered in high school. But that shouldn’t stop you from checking them out. In fact, both results have much simpler and more accessible proofs that I’ve linked to.

1. Irrational rotations of the circle and Benford’s law. Benford’s law is amazing. It predicts, for instance, that in a large list of spread out data, approximately 30.1% of the data will start with a “1”. The other digits have predictable frequency as well. This law allows analysts to detect fraud in data. Cool! Richeson proves Benford’s law for the powers of 2 (so…not a full proof of Benford’s) using irrational rotations of a circle. For more on Benford’s and a more straightforward justification, read the wikipedia article or one of the other sources Richeson links to.

2. Hillel Furstenberg’s proof of the infinitude of primes. There are infinite primes. For other proofs, see this page, especially the classic number-theoretic proof by Euclid. Euclid’s proof is definitely accessible to high school mathematicians, and it’s pretty elegant and exciting if you haven’t seen it before.

The Earth, in Perspective

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If you’re like me, you have trouble putting the heights of objects and land features in perspective–tall buildings are 1000 feet or more, tall mountains are 20,000 feet or more (though they rarely have a prominence of half that height), and airplanes fly at 30,000 feet. All of these features are nicely synthesized in the info-graphic below, thanks to ouramazingplanet.com. I especially appreciate all the under-water features that have been included. The creators of this image have another new info-graphic that deals with even larger distances here.

[Hat tip: Division by Zero]

Summer Odds and Ends

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I promise I’ve been mathematically active this summer, despite my little blogging vacation (I do get the summers off, you know!). So, in one post, let me highlight some of the nice stuff I’ve seen around the web recently:

  • The left-handed boy problem. Another post by Dave Richeson (can you tell I’m a fan of his blog?). The post is inspired by this kind of probability question:

Given a randomly selected family that has two children, one of whom is a boy, what is the probability that the other is a boy?

The post is great, but the conversation in the comments is just as great. You’ll notice that I’m a heavy participant. I really liked thinking about this class of problem. It highlights some of the most treacherous territory in mathematics, probability theory. I’ll probably end up devoting an entire post to this topic sometime in the future.

  • P ≠ NP. The Math Less Traveled blog hosted this announcement of a possible proof that P ≠ NP, by Vinay Deolalikar. It’s a Clay Mathematics Institute Millennium Problem, which has a $1 million prize attached to its (vetted) solution. Unfortunately, fatal flaws have already been found. Oh well. So it’s still an open problem and perhaps one of my students will solve it someday!
  • Stars and Stripes. Slate.com has a fun US flag generator, given any almost number of stars between 1 and 100. It’s great fun to play with the little interactive flag. And the task might provide you countless hours of entertainment, coming up with these arrangements on your own, without its help. A mathematician developed this, so that makes it appropriate for this blog :-).

And now for a bunch of books I read this summer which are all good, all of which I recommend.

  • The Mathematical Universe, by William Dunham. It’s not new, I know. But I finally finished reading it this summer and I highly recommend it. I love his lighthearted tone and all the wonderful anecdotes sprinkled throughout the book. It’s the perfect mix of fascinating history and little mathematical facts & puzzles that will make you hungry for more! Feel free to come and borrow it from me.
  • Outliers, by Malcolm Gladwell. It’s not purely mathematical–I call these books “pop research” books. Very good read. Tons of great tidbits that you can bring up in conversation, sure to fascinate your next dinner guests.
  • Freakonomics, by Steven Levitt & Stephen Dunbar. Another book that’s not really new but that I’ve only recently read. It’s also a “pop research” book–mind candy, if you will.
  • Godel, Escher, Bach, by Douglas Hofstadter.  Actually, I’m only half way through this book. But I can already tell you it’s absolutely classic; truly brilliant. Maybe someone will write another book, Godel, Escher, Bach, Hofstadter, someday. Citing books within books would be the kind of recursion that Hofstadter loves. Can’t wait to finish the book…I’m sure to post about again later, and in more detail.

An island on an island on an island

Interesting Cube Problem

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If the cube has a volume of 64, what is the area of the green parallelogram? (Assume points I and J are midpoints.)

Go ahead, work it out. Then, go here for a more in depth discussion, including a video explanation. Also, see my very simple solution in the comments on that page. (My Precalculus students should especially take note!)

And, welcome, SAT Math Blog, to the internet! Thanks for pointing us to this great problem and creating the nice diagram above.

Very Nice Java Applets

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I’m always on the look out for nice java applets (or flash, or javascript, or whatever) that help visualize tough math concepts. The best applets are not merely gimmick, but truly make the mathematics more accessible and invite you to explore, predict, and play with the concepts.

Most recently, my dad pointed me to this website.  At this site, you’ll find well over one hundred very nice applets that show things I haven’t seen around the web before. For instance, play around with your favorite polyhedra in 3d, then let the applet truncate or stellate the polyhedron, all with nice animation. And of course, you’ll want to mess with all the settings on the drop down menus, too! Additionally, you can see a mobius strip, klein bottle, cross cap, or a torus. Also, I was just recently blogging about complex functions. Some of these applets allow you to visualize with animation and color various complex functions. Having just completed a grad class in differential equations, I was also interested in these ODE applets. You may also have fun with fractals, like this one.

There are lots of other nice applet websites out there. As you know, I’m a huge fan of GeoGebra (free, open source, able to be run online without installation, easy to use, powerful). Someday maybe I’ll post more extensively about its merits. In the meantime, you can check it out here.  Many of these applets use GeoGebra. This site has a lot of nice applets too.  And here. This one’s rather elementary, but fun for educators. These are great, too. These are all sites I’ve used in class. Like I said, I think they’re more than just ‘cool’–they really do help elucidate hard-to-understand topics.

Feel free to share other applet sites you like.

More Press

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Another math blog has featured one of my recent posts. Go check out the 66th Carnival of Mathematics at the Wild About Math blog.

In reading through the carnival, I was especially interested in (as a math teacher), JSXGraph, an interactive graphing application that you can easily embed in webpages using javascript. I’ve also been meaning to point you to Dave Richeson’s recent blog post about the volume of n-Dimensional balls.  I would also be remiss to not mention the recent passing of world famous, and well-loved mathematician and puzzle-creator, Martin Gardner. The aforementioned carnival points us to a post here, which honors his memory (among hordes of other posts on math blogs around the web that have honored him recently).