I learned the following trick for memorizing the value of e from my colleague, Gene White. It never fails to impress calculus students (given a wide enough definition of “impress”).
Start by carefully looking at this picture:
That’s a 20 dollar bill, so memorize “2” and put down the decimal point.
The picture on the bill is of Andrew Jackson. He was our seventh President, so put a “7” after the decimal point to get 2.7.
Jackson was elected in 1828, so put down “1828” next. Since there’s a 2 in front of the decimal place, put “1828” a second time. We’re now up to 2.718281828.
Now look at the red square over Jackson’s face. The diagonal creates two congruent right triangles with angle measures 45, 90, and 45. So, add on 459045 to get 2.718281828459045. And that’s e to 15 places.
I’m open to suggestions on how to memorize more of the digits.
Students and faculty at University Preparatory School in Redding, CA have created the world’s largest Sierpinski triangle constructed entirely out of Doritos. (Well, it’s probably the only one, but still.) It is 64 feet long and made out of 12,000 Doritos. This was done as an entry to the Doritos Crash the Superbowl contest. Watch, and be awed:
Can a 128-, 256-, etc. foot long Dorito Sierpinski triangle be far behind? I bet the parent company for Doritos would seriously consider some corporate sponsorship.
Thanks to Cory Poole, math and physics teacher at U-Prep, who sent this in. That’s a great, creative way to get students interested in math. (And you can eat it when it’s done.) There’s more on the video here.
…I got one.
The story left off with me giving up trying to justify spending $399 for the 32 GB model, even though I’d saved up for it. Cheapness is in my DNA, and I’ve never been able to spend money on anything without feeling like I should have stuck it in a savings account instead. But, one day, my wife comes home and informs me that the daughter of one of her co-workers works at the Apple Store in Indy and gets a 15% “friends and family” discount. After trading a few emails, the deal was set up, and a few days later I had my grubby hands all over it (you see just how grubby your hands really are with this thing) with $60 knocked off the price. So, you see? It pays to wait.
I’ve been using it basically nonstop for a week now, and here are my overall impressions:
- It’s incredibly thin and light, yet it also feels very sturdy, and despite having an all-shiny-aluminum back I haven’t seen any big scratches on it yet.
- The screen is just unreal. Such crispness and clarity.
- Wifi speed is quite decent, and the Safari browsing experience is just fine even on sites that show up in very tiny font.
- The built-in apps are hit and miss. Besides Safari, I’ve really liked using the Mail app (although our stupid MS Exchange server at school can’t be accessed off-campus except through a web page…), and the Maps app is absolutely killer. The Calendar app will be really useful once I figure out how to get my iCal calendars to sync with it. The YouTube app just seems really slow. Weather is OK. Calculator is cute. Stocks, Notes, Contacts, and Clock are unnecessary.
- As a straight-up music player, the whole informatics/human-computer interface aspect of the Touch is amazing. What I mean is that it’s not so much the high fidelity of the sound reproduction that blows me away but the ability to quickly browse and access songs and videos. Not a square millimeter of screen space is wasted; everything is logically laid out and easy to use. This was the same kind of feeling I had when I first used a second-generation iPod with a click wheel.
- Videos are a real treat to watch on this, and it’s been lots of fun exploring what video podcast content is available out there.
- I’ve downloaded some free apps: IM+ for instant messaging, a WordPress app for blogging (haven’t tried using it yet), Pandora (where has that been all my life?), Facebook, WeatherBug. I’ve downloaded a few more that I immediately deleted because it was crap. There seems to be a lot of good free stuff out there and a whole lot of good paid stuff and about an equal amount of crap (free and paid). I’m hopeful that the app selection will keep growing and growing so that although the crap-to-noncrap ratio might stay constant, the amount of non-crap will increase.
- I also paid $20 for the iPhone/iPod Touch version of OmniFocus, the “desktop” version of which I use religiously for GTD on the Macbook Pro. I’m still getting used to it; the main advantage is that I can synchronize tasks to and from the Macbook using MobileMe (we still have 4 months left on the subscription we got for Christmas last year). But it’s nice — rather than carting around a stack of 3×5 cards or a Moleskine for on-the-go task collection, I can just use the iPod.
- I’ve gotten surprisingly good at using the little pop-up thumb keyboard that you get whenever you have to enter text.
- The battery charges a lot faster than my old iPod (about 60-75 minutes from 0% to 100%). And if you leave the wifi off, it seems to get a lot better battery life too. But if you use the wifi, the battery life drains out fast. Not surprising.
I could go on, and I probably will, but here’s something that sums up how much of an impact this little device is having on me. I had to go to an ATM to get some cash, and when it prompted me to enter “OK”, I started tapping the screen. It took me 10 seconds or so to remember that I had to push a button instead. I wouldn’t have been surprised if I’d tried to pinch and zoom on the ATM screen. So I’m very glad to have waited until I was capable of getting exactly what I wanted, and comfortable in getting it.
The numbers believed to be the 45th and 46th Mersenne primes have been proven to be prime. The 45th Mersenne prime is and the 46th is .Full text of these numbers is here and here.
Of course what you are really wanting to know is how my spreadsheet models worked out for predicting the number of digits in these primes. First, the data:
- Number of digits actually in : 11,185,272
- Number of digits actually in : 12,978,189
My exponential model () was, unsurprisingly, way off — predicting a digit count of over 24.2 million for and over 35.8 million for . But the sixth-degree polynomial — printed on the scatterplot at the post linked to above — was… well, see for yourself:
- Number of digits predicted by 6th-degree polynomial model for : 11,819,349
- Number of digits predicted by 6th-degree polynomial model for : 13,056,236
So my model was off by 634,077 digits — about 6% error — for . But the difference was only 78,047 digits for , which is only about 0.6% error. That’s not too bad, if you asked me.
There’s only one piece of bad news that prevents me from publishing this amazing digit-count predicting device, and you can spot it in the graph of the model:
So evidently the number of digits in will max out around and then the digit count will begin to decrease, until somebody discovers , which will actually have no digits whatsoever. Um… no.
At the end of this post, I made a totally naive guess that the recently discovered candidate to be the , the 45th Mersenne prime, would have 10.5 million digits. There was absolutely no systematic basis for that guess, but I did suggest having an office pool for the number of digits, so what I lack in mathematical sophistication is made up for by my instinct for good nerd party games. On the other hand, Isabel at God Plays Dice predicted 14.5 million digits based on a number theoretic argument. Since I am merely a wannabe number theorist, I can’t compete with that sort of thing. But I can make up a mean Excel spreadsheet, so I figured I’d do a little data plotting and see what happened.
If you make a plot of the number of digits in , the nth Mersenne prime, going all the way back to antiquity, here’s what you get:
The horizontal axis is n and the vertical axis is the number of digits in .
Admit it — one look at this plot and you’re itching to add some trendlines. Here’s what you get when you add both an exponential trendline (perhaps the obvious choice given the shape) and a 6th-degree polynomial:
The exponential one has a higher value, but that’s perhaps misleading because of the really good fit for all those low-digit Mersenne primes that happened prior to around . We’ll take that issue up in a moment. But for now, let’s put those trendline equations to work. The exponential trendline would predict that would have a digit count of
which is obviously rather a lot more than either my prediction or Isabel’s; and if you put in into the 6th-degree polynomial, you get a digit count of 11819349, which is in the ballpark of both my rough estimate and Isabel’s estimate.
It doesn’t make much sense, though, to include all Mersenne primes, since Mersenne primes didn’t even cross the 100-digit mark until in 1952. A more accurate idea — if you can call this kind of reasoning accurate in the first place — would be to run the numbers starting at around and seeing what we get. I’ll save that for later, unless somebody wants to beat me to it.
Here’s a list of 50 Books Every Geek Should Read from InsideTech. I thought this list might go well with my request for basic reading in educational technology from a few days ago, and in fact there could probably be some overlap.
Of the books on the list, I’ve read:
For math geeks, and perhaps for general geeks, I’d add G. H. Hardy’s A Mathematician’s Apology. For higher education geeks, add on The Shadow University by Kors and Silverglate.
I think Longitude is going to go in my personal queue next.
Here’s a great illustration from George Gamow’s classic book One Two Three… Infinity which shows two things: just how big really is, when thought of as a scaling factor; and also the power of a good illustration to drive home a point about math or science. The picture shows a normal-sized astronomer observing the Milky Way galaxy when shrunk down by a factor of .
That’s a big number, folks.
Gamow’s book is one of several on my summer reading list, and there’s a reason it’s a classic. In particular, it’s chock full of cool illustrations like this that convey more information about a science concept than an hour’s worth of lecturing.
Today, my 4-year old (who goes by “L” here) is “Student of the Day” at her Montessori preschool. I’ll be spending most of the morning in school with her, hanging out with her and joining her in some of the activities they do. One of the activities we’ll do is take some time to pass around a photo/scrapbook page we put together about L and to let L do a show-and-tell of a special item for her. During that time, she’s supposed to introduce me to the class and then I’m supposed to describe what my job is.
That’s where you readers come in. How would you describe the job of mathematics professor at a small liberal arts college to a room full of 4-year olds?
What you have to work with: The kids are bright, active, know their shapes and numbers, know how to count (most of them to 100 and beyond), and know a tiny bit of basic science.
Both humorous and serious replies are welcome in the comments.