Alternate title: Claudia Makes Things Way More Complicated than They Need to Be Because She Sucks
We had this bonus question on our homework for Probability today:
Suppose X has a density defined by
Let FX(x) be the cumulative distribution of X. Find the area of the region bounded by the x-axis, the y-axis, the line y = 1, and the curve y = FX(x).
And I was like, “Aw, sweet! Areas of regions! CALCULUS!”
So first, I had to find the cumulative distribution function (cdf) of X. Easy. It’s just the integral of the density fX(x) from negative infinity to a constant b. In this case:
With 2 ≤ b ≤ 3. So that’s my curve y. The area I’m looking for, therefore, is this (the red part, not the purplish part):
Now anyone with half a brain would look at this and go, “oh yeah, that’s easy. I can find the area of the rectangle formed by the two axes, the line y = 1, and the line x = 3, then find the area of the region below the curve from 2 to 3, and subtract the latter from the former to get the correct area.”
Which works. Area of rectangle = 3, area of region below FX(x) = .25, area of region of
interest = 2.75.
Or they could remember the freaking formula that was explicitly taught last week. Such areas can be calculated using:
But did I see either of those? Nooooooope.
I looked at the graph and was like, “how the hell do you find that?” I tried a few things that didn’t work, then realized that it would be a lot easier to figure out if I changed the integral from being in terms of x (or b, rather) to being in terms of y.
So then I just had to integrate. This gave me the right answer: 2.75!
Moral of the story: don’t complicate things. But if you do complicate things, you might actually end up in a scenario where you’ll use something that you were taught back in calculus I but didn’t ever suspect you’d actually use. I had appreciated learning the handy-dandy technique of changing variables, but I didn’t think I’d be in a situation where I’d apply it. Shows what I know, eh?
It was a nice refresher, at least. I’ve missed calculus.
So I pretty much constantly dream about math/numbers now. And what’s frustrating is that it always seems like I’m coming up with these super awesome theories/proofs and then by the time the morning comes I can only remember fragments of them. They’re probably trivial and nonsensical, but it’d be nice to actually see if the stuff my REM-brain’s coming up with is coherent. At least, more coherent than my waking thoughts.
There’s a way you can train yourself to remember your dreams, right?
Linear Algebra class has apparently turned into “let’s see how many stupid math puns we can come up with in a 50-minute time period.” Including figuring out the plot to a movie about null spaces.
And then I come home and do this nonsense:
(Yes, I know that’s the symbol for the empty set. It needed something, okay? In the epic movie we’re planning, the empty set saves the day anyway, so there.)
I think I have an idea for my NaNoWriMo endeavors this year now.
So here are pictures of shirts that I
That “Stoked” one? Oh my god I want it. They have women’s shirts, too, so it might have to happen. The “Extended Hospitalization” one is pretty fantastic, too.
All from here.
Someone I follow posted this awesome link to Newton’s notebooks stored in the Cambridge Digital Library (link link link!).
Now that I’ve got access to both Newton’s notes and Leibniz’ notes (thanks to checking out Dr. Wolfram’s awesome post on Leibniz’ archives), you can probably guess how freaking excited I am.
So. Graphology in itself is pretty much pseudoscience, but it’s still interesting to compare the writing styles of these two geniuses, just to see if any similarities/differences stand out. That’s allowed, right? (Screw it, I’m doing it anyway.)
A lot of Newton’s notes were written in English ‘cause…duh…he was an Englishman. From what I’ve read about Leibniz, I think he could read and write in English but not nearly as fluently as in several other languages; most of his work was in Latin, the rest in French and German. So I couldn’t find a good English excerpt from both. So let’s do Latin, just for the sake of keeping the language consistent.
Here’s a Newton page:
Look at his writing, it’s so neat! I’m no handwriting analyst or anything like that, but it looks like this section of Newton’s notes was written slowly and deliberately as if he’s just sitting there going, “yeah, I got this.” There are a few things crossed out, of course, but it looks like he took the time to carefully scratch them out and then just kept going. Slow but steady. And his numbers are so clear, too, holy crap.
The above is just a screenshot of a semi-magnified page; on the actual Cambridge site you can zoom in further and make out the English notes he made in the margin. If you look at a lot of other pages in this section of notes, Newton really seems to keep things very organized, even if it looks like he’s making scratch calculations in some parts.
And then there’s Leibniz:
I was planning to do both samples in Latin like I just said above, but I’m snatching pictures of Leibniz’ notes from Dr. Wolfram’s post on him so there aren’t nearly as many choices as with Newton. So I figured a more appropriate comparison would be pages written by both men that contained both words and numbers. I believe Leibniz’ page is written in French, but I seriously can only make out like three words here.
I’m not sure if it’s just because of different writing tools or different ink/paper, but Leibniz looks like he pressed fairly hard (or at least as hard as you could with a quill). Also, in contrast to Newton, it looks to me like Leibniz wrote pretty rapidly. Newton’s corrections were either neat single cross-outs or carefully scribbled out so the mistake couldn’t be read. All of Leibniz’ corrections look like, “no time for error must keep writing!” *scratchscratchscratch* “ONWARD!” Even his numbers look rushed (look, it’s binary!). It almost looks like he used this page for just those calculations but then wrote around them, continuing from a previous page.
On some of the other pages Leibniz really manages to get a lot on a single page. We’re talking ITTY BITTY scrawl, a consequence of his becoming very near-sided in his 20s and it only getting worse as he got older. I’m actually not sure how good (or bad) Newton’s vision was. Of course he did stick a darning needle back behind his eye and wiggled it around (optics experiment), so…
Anyway. Just an interesting thing to see the differences/similarities in their styles.
So I was dicking around with drawing ideas this evening and eventually started thinking about something cool to do with the zodiac signs. I was scribbling Aquarius’ sign everywhere when I considered how similar it looks to the “approximately equals” sign.
Which led to this:
I wanted to find commonly-used math symbols that best matched the shape of the actual zodiac symbols to give you MATH ZODIAC, but for some of them I had to take a little artistic license. Virgo and Scorpio, I’m looking at you (yes, I just took Virgo’s “M” shape and made it an “N,” deal with it).
And for those of you screaming, “hey u cant mix mathz with pseudoscience SHAME lolz,” I say, “screw you.” Plus, now instead of saying “Taurus” when people ask you your sign, now you can reply “Universal Quantifier!” and confuse the hell out of them. Upturned A’s are cooler than bulls anyway.
Also, as I was waking up this morning, I swear I was trying to explain Euler’s Identity to my cat. Not sure how successful I was considering I was half asleep and she’s a cat.
“WTF is a Curta?” you may be asking.
I’ll tell you!
The Curta is a little handheld mechanical calculator introduced in 1948 by Austrian engineer Curt Herzstark (so I guess “Curta” was an easier name than “Herzstarka”). Real-life ones look like THIS (source for pic)…
…and were considered the best portable calculators until the digital ones started coming out in the 1970s.
Its design is based in part off of…(wait for it)…Leibniz’ Step Reckoner. It is able to do addition, subtraction, multiplication, division, and also square roots. What made it different from contemporary calculators was, though it employed a stepped drum mechanism like most others, Herzstark was able to create and patent a single drum that did the work of 10+ drums, thus making the Curta super compact.
I wanted to get a real one, but they’re like $400 now, so this simulator was a cool find. If you want to get all up in the Curta’s business but are intimidated by all the arrows/dials (it’s like a slide rule on steroids!), check out this manual.
Holy crap dudes, this is the best discussion of imaginary numbers I’ve ever heard. Listen to this, it’s really cool.
So in my continuing saga of “Let’s Make Stupid Jokes About Everything” (aka, “My Life”) and in the same vein as that Neil & Prey dream I had awhile back, I think someone should propose a detective/mystery show called Green & Stokes. It’d be like NUMB3RS crossed with Law & Order crossed with Columbo, except with exponentially more puns.
They’d work for the LAMD (Los Angeles Math Department) or something, because cities would have their own math departments in whatever universe that would allow Green and Stokes to be mathematicians AND detectives AND live during the same time period.
And the episode names could each be a pun on some other famous mathematician’s name (or other dumb puns).
- “Rolle with the Punches”
- “Out with the Old, in with the Newton”
- “Bourbaki and the Case of the Empty Set”
I DON’T EVEN KNOW.
This is why I need school to start again.
holy crap, I forgot how crappy gifs can be when they’re exported from Flash (especially when you don’t know what you’re doing), but here’s the theoretical show’s opening animation nonetheless:
Edit 2: fixed it (sort of; it’s still dumb)