With all of our technology, we’ve become beholden to large companies who provide the services we need to use said technology. The cable company, ISP, phone company, water department (not a company, but still a provider of a modern convenience), power company, gas company, and so on and so forth. Sometimes these companies make mistakes. They are, after all, a collection of fallible humans. With all their computer systems and automated phone trees it can be frustrating. A good release might be to mess with them. Say, write out a check (who does that any more?) with the total in the form of a mathematical expression. Their automated processing systems won’t be able to read it, and the high school dropouts working in the basement won’t have a clue. Now, suddenly, that company has to spend hundreds of dollars in manhours to decipher your check.
Or, they just reject it and charge you penalties on your next bill.
[Image Credit: I'm guessing Randall Patrick Munroe]
He should have written the text as:
zero dot zero zero two plus e to the power of i times pi plus the sum of one over two to the power of n for n from zero to infinite.
Randall Patrick MacMurphy says "What now, Bitches?" and laughs maniacally, yet approvingly.
<img src="http://i2.listal.com/image/66424/600full-one-flew-over-the-cuckoo%27s-nest-photo.jpg">
Any one care to let the mathmatically challenged here know what the solution is? Please?
/Hangs head in shame.
I think it is C-. No wait, that was my grade in math.
These are the sort of things you have to "just know", which was where I regrettably hit the wall with upper level math. That also means these are the sorts of things people spent their entire lives proving or trying to understand, while a few gifted people can just "see" it.
The first is one of Euler's many namesakes: e^(i*pi) = -1. Reading the wiki article, it actually looks like it might be useful with some of the trig I use every day.
The second is the famous half, plus half of half, plus half of half of half, and so on, which in an infinite sum Σ starting with n=1 and going to n=∞, 1/2+1/4+1/8+…..+1/∞ = 1.
In the second example, if you started with n=0, you'd have 1+1/2+1/4+1/8+…..+1/∞ = 2.
Euler's equation is actually solvable. It's based on e^(iθ) = cos(θ) + i*sin(θ), and this can be shown to be true by taking the Maclaurin Series of all three terms and doing a bunch of messy algebra. I refuse to do that in ASCII.
For sum of (1/2)^n for n=1..∞, there are a bunch of techniques for determining if the series converges to a finite number and then techniques using limits to actually find that number. I don't remember them off the top of my head, but I know which two of my college text books have the answers laid out (I can see both of them on my bookshelf from where I'm sitting).
It's a lot of crap to know (and a lot of it is little tiny details that once you've used them to prove something, you never use again). I hit the wall at tensor analysis. (I used to kind of know it, but I always feel unsure of myself doing the high-level symbol manipulations, then again, I only learned it as an aside in a couple of physics courses.)
For the second series, it's nice to imagine it as stairs that rise and run progressively less, then measure the height where the two lines along all the outer and inner corners of the stairs intersect. It's nice intuitive geometric proof and in fact the ancient Greeks knew it that way.
Took me a couple of tries to draw it right, but that is very elegant. The problem is that there are a lot of different things I can draw, and it's hard to be confident which one is definitive.
I don't know that I'd ever been shown the geometric version of that before. That's really cool, now I'm going to have to try and draw it myself. Thanks.
(Ok, just tried drawing it, that is really cool, and IIRC, it is a diagram of what the limit method is (or at least, it looks like they have to be equivalent; this is probably the kind of obvious that takes a couple hours and six chalkboards to confirm the obviousness of.).)
Randall Munroe is the guy behind XKCD.
I'm not saying it was XKCD…
…but it was XKCD.
Oh lord… I'd listened to that audio before, but I just had to do it again today. Simply priceless.