I think I have a new favorite way to multiply numbers: the vertically and crosswise technique. I learned about it from George Gheverghese Joseph’s Crest of the Peacock [a book that is sadly out of print], and was recently reminded of just how cool it is.
Archive for February, 2008
Throughout this trick, Aces are 1, 2s are 2, 3s are 3, …, 10s are 10, Jacks are 11, Queens are 12, and Kings are 13. Click here for the trick and its variations.
Go watch this clip from Robot Chicken. It’s only 10 seconds long, but it’s really funny. Does anyone want to volunteer to figure out if what they’ve written on the board is legitimate? (Warning: The clip is safe for all audiences, but no guarantees are made for the rest of the site.)
As you might guess, this post builds on “Root extraction, part I“, which gave a way to visualize the traditional square root algorithm geometrically, an approach that has the advantage that each step appears natural and easily motivated.
Our goal herein is to do much the same for cube roots. The point is to find a geometric construction, ideally one well-suited to physical manipulatives, in which the steps in building the successive digits of the cube root of a number are transparent. As with the post on square roots, I make no claims to originality in what follows.
Example: Find . (more…)
I recently discussed the traditional algorithms for computing square and cube roots in my History of Math class. Our reading, on mathematics in ancient China, gave both algorithms as a set of rules for manipulating number rods. For me, it was fascinating to see past the text: the rules as given would transfer directly to an abacus/soroban calculation, and were essentially the same as the rules that prior generations of American schoolchildren would have been drilled on in school.
My students (mostly high school math teachers) found the book’s explanation of the method obscure; the key is to view the process geometrically, rather than as a mechanical set of rules for manipulating digits.
I make no claim of originality in what follows; I offer it here in part because I can’t find any lucent discussions along these lines on the web. (more…)
That’s exactly what happened on July 23, 1983. Captain Bob Pearson and First Officer Maurice Quintal were flying the aircraft, but due to mechanical difficulties the fuel gauges weren’t working. This was realized during a stopover in Montreal and the plane was still allowed to fly, but the amount of fuel had to be calculated manually. Mechanics knew that the plane would need 22,300 kilograms of fuel to fly from Montreal to its destination in Edmonton, and they were also able to determine that there were 7,682 liters of fuel in the tank at that time. The flight crew then [incorrectly] calculated that they needed to add 4,916 liters from the fuel truck. Click to read more about how a simple mix-up of pounds versus kilograms had nearly disastrous consequences.
Welcome to the 26th (and one-year anniversary) Carnival of Mathematics! (Is it odd that the one-year anniversary comes after the Silver Jubilee Edition?) Exactly one year ago tomorrow the inaugural Carnival was posted at Abstract Nonsense. Thanks to everyone in the math blogging community for keeping things going for a full year so far!
As is traditional, we begin with a few comments on the number 26. There are 26 sporadic groups in the classification of all finite simple groups, and 26 letters in the English alphabet. 26 is the only number directly between a square and a cube (proved by Fermat), and it is the smallest number which is not a palindrome, but whose square (676) is a palindrome. It is, however, palindromic in bases three (222), twelve (22). Finally, there is an absolutely astonishing system of 14 Diophantine equations in 26 variables such that the set of primes corresponds to the set of positive integer solutions of the system.
On to the good stuff.
Mike, over at Walking Randomly, writes about Music From Mathematical Constants. Following up on a post here at 360, he uses Mathematica’s SoundNote function, along with some clever list processing, to generate sequences of tones from the decimal expansions of π.
In Recounting the Rationals, part IV, Brent (from The Math Less Traveled) continues his series on the Calkin-Wilf tree, which is an elegant way to enumerate the rationals, and its amazing properties. Check out the earlier posts in the series, too.
Maria had a former student ask her for help visualizing complex analysis, so she built a great collection of images and videos for Some Techno-help for Complex Analysis at Teaching College Math Technology Blog.
“Vlorbik vents”, and at the same time wonders, about errors that students commit (such as IDF – Ignore Denominators Fallacy) in Fractions and Student Incredulity and its follow-up Oh, P.S., over at Vlorbik on Math Ed.
Denise at Let’s Play Math gives us a two-fer. In The secret of Egyptian fractions, she recounts a story (available for download as a pdf) of an archaeologist and his family learning about the Egyptian method for reading and writing fractions. As a bonus, we get That’s mathematics, a YouTube compilation of math jokes.
If you’ve ever seen mathemagican Art Benjamin perform, you may have left wondering How does Arthur Benjamin square 5 digit numbers in his head? Sol is more than happy to show you at Wild About Math.
An eclectic selection from Mark at The Universe of Discourse. First, in Nonstandard adjectives in mathematics we see that while a toy fire engine is not actually a fire engine, a toy ball is indeed a ball; this doesn’t appear to happen very often in mathematics. Major screwups in mathematics: example 1 presents the first in what we hope will be a series on results that were believed (by many) to be true, in this case for 30 years, but were eventually proven false. Finally, Mark uses Ramanujan congruences to remind us all of just how awesome math can be.
Last, but certainly not least, MathNotations‘ Dave shares a Fascination with Pyramids again…, a problem appropriate for geometry students. He is also conducting a survey in Why a Poll? Why Now? A Significant Sample…, and he would really appreciate your input on the teaching of algorithms. The poll is open until February 29, so head on over and share your thoughts.
(If anyone has a late submissions they’d like to send, go ahead – I’ll gladly post an update later today if necessary.)
The next Carnival is scheduled for February 22, but needs a host. If you’d like to volunteer, head over to Carnival of Mathematics and tell Alon.
Update 2/13: The Feb 22 Carnival will be hosted at JD2718.
The 26th Carnival of Mathematics will appear here Friday or Saturday. We discovered yesterday that we were unable to receive anything sent via the official Carnival Submission form, so the only way we have to receive submissions is if you post them as a comment (here or in the earlier announcement) or if you email them to Batman at mkoetz1 (then the @ sign then) naz.edu with “Carnival” or something similar in the title.
It’s the 100th day of our blog and, not entirely coincidentally, our 100th post as well! Calls for a celebration, don’t you think? So here are some fun facts.
We started this blog within 24 hours of commenting, “It’s a good thing we don’t have a blog or we’d never get anything done.”
We initially looked at names involving blogarithm but they were all taken. Well, maybe not all, but enough that we didn’t feel nearly as original as with 360. (Of course, that was before we knew that CNN also had a blog named 360, so I think we’re not so much about uniqueness as existence. Ba dum.)
Our most popular post has been the math jokes post So, a horse walks into a bar….
Our favorite posts to write were:
- Batman: Yes, I’m a Nerd
(though he also really liked the humor post above)
- TwoPi: The best place to buy PowerBall Lotto tickets?
(but his favorite paragraph to write was the first one in Sudoko Lotto)
- Ξ: The Rubik’s Hypercube
(but I’m also thrilled whenever I find something for the Math Mistakes category)
The most common posts to come up under search engines are Is EZ-Pass used to catch speeders? (It seems like every day someone searches for a variation of “EZ-Pass and speeding”. Apparently this is a common concern.). The Friday Software Reviews and Math Mindreading also get hits regularly. Lately, too, these posts on polygons have been showing up when people search for “real life polygons” and “real life heptagons”.
Tomorrow, 2/7, is the day we celebrate e-day, in honor of the number e≈2.7182…. And celebrate we do: there are decorations, and e-related foods (browniEs, e-clairs, pi(e), etc.) and an e-day quiz. So it’s a big deal around here. And sometimes dangerous.
The date was February 7, 2005. The time was 5:00am. I’d gotten up early to cook, and this year decided to try a recipe that involved baking the brownies, then putting white chocolate chips and crushed candy canes on top. I placed the whole pan under the broiler for a few minutes for the chips to soften before spreading them around. And I’m sure all would have gone well if TwoPi hadn’t asked from the dining room, “Hey, are there any nonzero integers a, b, and c such that ?” Naturally I started to work on that question, and forgot all about the brownies.
Until the fire alarm went off.
Click for the rest of the story, plus a solution to the math problem.
Neither of these are recent, but they’re pretty entertaining. If you’ve never seen them before, you really ought to take a look. Enjoy.
A “review” of Dummit and Foote’s Abstract Algebra. (This will be funnier if you’ve taken an abstract algebra course before.)
The Amazon page for A Million Random Digits with 100,000 Normal Deviates by the RAND Corporation has nearly 100 reviews, from humorous to downright hilarious. My favorite is the one about “updating” the text to use hexadecimal digits instead of decimal, with an example of the conversion.
Remember when you first learned how to write paragraphs? For me I think it was seventh grade, and the approach we were given was to craft five sentence long paragraphs: the first sentence set out the topic (or thesis); the next three sentences gave further detail supporting the thesis; and the last sentence summarized the content of the paragraph.
By the end of seventh grade, we were starting to write essays. An essay consisted of five paragraphs: the introduction, three supporting paragraphs, and the conclusion. Each of those paragraphs were themselves five sentences long, following the same structure.
Egads, a fractal!
I was recently looking up information about women and mathematics. In 2004, for example, women earned almost half the bachelor’s and master’s degrees in mathematics and statistics, and 29% of the doctorates (from the Association of Women in Mathematics, summarizing the most recent statistics from the U.S. National Science Foundation). Another source of statistics is in the Annual Survey of the Mathematical Sciences, published in three parts each year. There are a number of ways to interpret this data, some positive and some negative. Click for more, including the 1943 article about hiring women that is the point of this post.