LaTeX Plugin

I’ve just installed Pavel Holoborodko’s amazing QuickLaTeX WordPress plugin, and so now I can show fancy equations like this: |y – x| < \delta

And this: w_i = \frac{2}{\left(1-\xi_i^2\right)\,\left[P'_n(\xi_i)\right]^2}

And even super complicated stuff like this:

( \langle \sqrt{ \sqrt[3]{ \langle \frac{ e ^ { |f(y) – f(x)| < \pm \epsilon } } { |x - (y + z)| } \rangle } ^ 3 } \rangle ) ^Q_m

LaTeX is pretty expressive, and these examples are kinda addictive.

|x| = \left\{ \begin{array}{ll}<br /> x & \mbox{if $x \geq 0$};\\<br /> -x & \mbox{if $x < 0$}.\end{array} \right.<br />

Matrices look pretty good too. Let’s say I wanted to explain that the \emph{characteristic polynomial} \chi(\lambda) of the 3 \times 3 matrix

\left( \begin{array}{ccc}<br /> a & b & c \\<br /> d & e & f \\<br /> g & h & i \end{array} \right)<br />
is given by the formula \chi(\lambda) = \left| \begin{array}{ccc}<br /> \lambda – a & -b & -c \\<br /> -d & \lambda – e & -f \\<br /> -g & -h & \lambda – i \end{array} \right|<br />

Looks awesome. Have a look at David R. Wilkins’ Getting Started with LaTeX for some excellent examples, including some of the ones above.

Powered by QuickLaTeX.com” and QuickLaTeX as a WordPress plugin.

June 5, 2010 | Filed Under Blog, math

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WordPress Hijacking

I want to appologize to anyone who visited this blog over the last few days and was re-directed to Chinese spam advertising sites. My WordPress PHP theme files were hacked. I’d have noticed sooner except I’m on the road. It’s related to the IframeRef.gen exploit.

Again, sorry. It should be all fixed now, and should not have harmed your machine or affected it in any way after you closed the spammy tab.

Thanks for reading! I’ve gotta start posting here again.

February 5, 2010 | Filed Under Web tools

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Freeman Dyson and 1/19

I just read this article about Freeman Dyson and a math puzzle that asks you to find a number that is doubled when you tear off the rightmost digit and stick it on the left. For example, tearing the 2 off 12 and sticking it in front gets you 21, which isn’t 2×12, so 12 isn’t right. Moving the 1 in 7654321 gets you 1765432 which also dosn’t double it, so that’s not the number either.

Anyway, it turns out the number we’re looking for is 18 digits long. It’s 052,631,578,947,368,421 which doubles to 105,263,157,894,736,842. (You’re allowed to stick a zero in front.)

Well, my son loves this kind of base-10 number tomfoolery, so I asked him if he could figure it out. And he reeled off the answer without pause. All 18 digits.

“Uhh, how’d ya know?” I asked.

He said it’s because 1/19 is 0.052631578947368421… (repeated forever), and 2/19 is 0.105263157894736842… (just move the 1). And of course 2/19 is twice 1/19. So if you know your N/19 repeating decimals, this is apparently easy peasy. And all the N/19′s are buried in that infinite sequence. You just have to start at different places. 3/19 is 0.157894736842105263…, 4/19 is 0.210526315789473684…, 5/19 is 0.263157894736842105…, etc.

So the next time someone asks “what’s a number that’s quadrupled when you tear the last two digits off the right and stick them on the left” you can say “doy, it’s 052,631,578,947,368,421 of course” and then roll your eyes. Because you’ve seen 1/19 and 4/19 written out as decimals.

Anyway, my son also told me there’s something similar for 1/29, except you triple the number by tearing off the rightmost digit and putting it on the left. I think it’s 28 digits long.

And also for 1/39, except you quadruple (x4) the number. And with 1/49 you x5 the number, and 1/59 gets you x6. Even 1/99 (0.0101010101…) works (x10 gives you 0.1010101010…).

As for 1/109, I don’t know, I’ll bet there’s some kind of x11 trick there. My son’s in bed now, but I’ll ask him tomorrow when he gets up.

April 11, 2009 | Filed Under Family, math

8 Comments 

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