07 Mar 07 More math, shift in blog direction
I’m planning to drive this blog in a slightly different direction. That is, I want to narrow down what I write about here to a few categories that interest me most, instead of shooting off in all directions with no depth in any of them. Of course, I still reserve the right to write about any topic at any time, but…
Anyhoo, the average reader should get what I’m saying.
In that vein, here’s a little bit more math geekery; Euler’s formula. This is not new in any way, or even something that isn’t blogged on often (it is, and by people who are far more knowledgeable, talented, and better at writing about math) and in true blog tradition, I’m swiping it wholesale from someone else. Go there to see a better explanation for all this.
[tex]e^{i\phi} = cos(\phi) + i sin(\phi)[/tex]
Let [tex]\phi = \pi[/tex], and you end up with
[tex]e^{i\pi} = – 1[/tex]
which leads to a striking equation that links together the most commonly used constants in mathematics:
[tex]e^{i\pi} + 1 = 0[/tex]
Interestingly, if you let [tex]\phi = \frac{\pi}{2}[/tex], then you get
[tex]e^{i\frac{\pi}{2}} = i[/tex]
Raising both sides to the power [tex]i[/tex], you get
[tex]e^{-\frac{\pi}{2}} = i^i[/tex]
Which can be calculated as [tex]i^i = 0.2078795763\cdots[/tex]
Tags: math
07 Mar 07 Joi Ito’s Web Math Joke
As Joi Ito said, too funny not to blog. I know it’s old, but I still laughed out loud:
After explaining to students through various lessons and examples that
[tex]\displaystyle \lim_{x\to 8} \frac{1}{x-8} = \infty[/tex].
I tried to check if she really understood that, so I gave her a different example; the result of which was:
[tex]\displaystyle \lim_{x\to 5} \frac{1}{x-5} = \rotatebox{90}{5}[/tex].
Hah!
Tags: math
04 Mar 07 Latex on Wordpress.com
I saw the recent announcement that wordpress.com now supports native latex equations. The posted examples are pretty sweet, and the syntax is just fine: $latex
I thought to myself, “boy, I sure would like to have that on my blog. I bet it’s just a wordpress plugin.” That’s apparently not the case for wordpress.com. I don’t know what they’re using. However, I did find the LatexRender plugin by Steve Mayer. Even better though, I found Gunnlauger’s installation script over at Fugato. Why all wordpress plugins don’t have some automated install mechanism, I’ll never guess. Downloading zips and extracting them really sucks, but this script is handy.
Let’s test the results by typesetting some of Euler’s work:
| [tex]\displaystyle e^{{\it i}\theta}[/tex] | [tex]\displaystyle =[/tex] | [tex]\displaystyle cos \theta + i sin \theta[/tex] |
| [tex]\displaystyle e[/tex] | [tex]\displaystyle =[/tex] | [tex]\displaystyle \sum\limits_{n=0}^\infty {\frac{1}{n!}}[/tex] |
| [tex]\displaystyle =[/tex] | [tex]\displaystyle \lim\limits_{n\to\infty} {\left(\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \cdots + \frac{1}{n!} \right)}[/tex] | |
| [tex]\displaystyle J[/tex] | [tex]\displaystyle =[/tex] | [tex]\displaystyle \int_{a}^{b} F\left(t, f(t), f^{\prime}(t)\right) d{\sl t}[/tex] |
Pretty slick, I have to say. It’s not perfect though; because of the lack of a good standard for typesetting math on the web, these things still have to be done with images. MathML is pretty neat and I think it’s probably the future, but not every browser treats it the same (or at all, for that matter). It’s just not ready yet. Also, not every [tex]\LaTeX[/tex] technique will work. For the above, I tried using the “tabular” environment, but it didn’t come out right because the ampersands got converted to character codes – I eventually had to use tables to get the spacing I wanted.
Anyway, all in all, this plugin is pretty cool. It even adds a ‘tex’ button to the post editing page.
Tags: math
25 Sep 06 Poe, E.: Near a Raven
Hat-tip to Ben Schumacher via John Derbyshire.
Tags: math
21 Jan 06 Math degree?
As an undergrad, I majored in computer science and minored in mathematics and electrical engineering. As a grad, I took a few math courses.
Now, only months after completing my graduate degree, I’m thinking that if I go back for another, I might want to do a master of science program in mathematics or applied mathematics. I really would like to become much deeper mathematically.
Further still, I would also like to go deeper in computer science and electrical engineering, and therein lies my problem; I have a youthful infatuation with knowledge. What is a finite-minded man with a finite span of time to do with a seemingly infinite world of knowledge at his fingertips? Right now I’m re-studying DSP material. I had an undergraduate course on this topic and I scored high marks in it, but after 3 years, I’ve forgotten most of it. I want to re-learn it more thoroughly than I learned it the first time, so it will take a long time. After that, though, what? Wavelets? Complexity theory? Real & Complex analysis? Graph theory? Brush up on advanced calculus? Matrix theory? Number theory? NP theory? Compiler construction? Statistical signal processing? Aggh. It’s all too much. I need a jack in the back of my head like they had in “The Matrix”.
In any case, I’m not sure how I would actually go about it. Would I try to do part-time schoolwork while I am employed at $work, and slowly chip away at a degree one course at a time? Would I do online work, or try to make it to classes? If I were to quit my job to do schoolwork full-time, when would I do it (I’m certainly in no financial position to do such a thing any time in the near future)?
I follow this thought process to its conclusion all the time, and it’s always the same; I pursue self-study as devotedly as possible, and hope for a golden opportunity to go back someday.
Tags: math
