Math Thread of Fancy Counting

[jennegatron]

That -1/12th thing smells like BS, but then I don't know if such averaging of infinite series is really used in physics. I would think its not used in pure math at least?

Also what if you don't shift the second S in 2S?

 

There's a second video where they talk a little more about use of infinite series in physics, specifically string theory.

 

it also talks about when Mathematicians would use infinite series, and brings up the Zeta function (that's important for the Riemann hypothesis & the distribution of prime numbers.)

Real Analysis deals with infinite series, as does, I assume complex analysis.

[tberton]

That -1/12th thing smells like BS, but then I don't know if such averaging of infinite series is really used in physics. I would think its not used in pure math at least?

Also what if you don't shift the second S in 2S?

 

The reason it works is the same thing as the infinite hotel. It's an infinite series, so you can shift it around in ways that you can't shift finite series. If you didn't shift the second S in 2S, you get the sum of the two series: 2+4+6+8+10... But I don't think that would really change the -1/12 answer, because that second series doesn't have any more of an obvious single answer than the first one does.

[eot]

Both those videos have some issues. The first sum (S1) isn't equal to 1/2, it's divergent. Apparently it's convergent in a different sense, but that's not what they say. If you assume that S1 = 1/2 you can directly derive a contradiction from that, so the rest of their derivations using that result are wrong too. Secondly, the maclaurin series expansion for 1/(1-x) is only valid for x on (-1,1), and of course they plug in x=-1. It bothers me a bit because I swear the professor in the video knows it's wrong, there's no way he doesn't. He even puts 'proof' in quotes when he's done.

 

The sum doesn't converge in the usual sense, that is in the limit of n->inf.  Also, zeta(0) = -1/2, if we say that zeta(-1) = 1+2+3+4+... then zeta(0) = 1+1+1+1+... however, 1+2+3+4+5... - (1+1+1+1+...) = 1+2+3+4+5... = -1/12. It doesn't seem to me like you can interpret the zeta function as that infinite sum when s<1.

[jennegatron]

Both those videos have some issues. The first sum (S1) isn't equal to 1/2, it's divergent. Apparently it's convergent in a different sense, but that's not what they say. If you assume that S1 = 1/2 you can directly derive a contradiction from that, so the rest of their derivations using that result are wrong too. Secondly, the maclaurin series expansion for 1/(1-x) is only valid for x on (-1,1), and of course they plug in x=-1. It bothers me a bit because I swear the professor in the video knows it's wrong, there's no way he doesn't. He even puts 'proof' in quotes when he's done.

 

The sum doesn't converge in the usual sense, that is in the limit of n->inf.  Also, zeta(0) = -1/2, if we say that zeta(-1) = 1+2+3+4+... then zeta(0) = 1+1+1+1+... however, 1+2+3+4+5... - (1+1+1+1+...) = 1+2+3+4+5... = -1/12. It doesn't seem to me like you can interpret the zeta function as that infinite sum when s<1.

 

I think it's easy to get hung up on details in these videos, especially as someone who has had advanced math courses.  We know enough to trip up over things that those with less knowledge wouldn't even notice, like the bounds on x values for a Maclaurin series. To cover each of those things, they would have to add extended explanations that would just confuse most of their audience, so I think they often times opt to leave those extended explanations out. https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF

The Wikipedia article also glosses over some of those rough corners

 

Additionally the guy in the first video wrote this after making the video: http://www.nottingham.ac.uk/~ppzap4/response.html

[eot]

They made a third video, which I think is much better. The professor featured is a bit more careful with his wording (and use of equal signs!)

[Patrick R]

Ok, I barely graduated high school (a science teacher of mine literally forged me a better grade so I could graduate and protect our Catholic School's grad-rate) and don't know anything about anything, but I have been obsessed with this Sesame Street animated short for the past 24 hours and figured it might be of interest to you math nerds. Maybe it's mostly of interest because Philip Glass wrote and recorded a new piece of music specifically for this short, and he is the best. At any rate, I present to you and any 4 year olds who might be hanging around, "Geometry of a Circle".

 

https://www.youtube.com/watch?v=xH7c3F3a--U

[aoanla]

That, truly, is some fancy counting.

 

I really forgot too much of this stuff since the one semester of advanced math I took at college. No more proof by induction for this guy.

 

Although I recently ran into a problem in my amateur game dev stuff where it was nice to be able to call back some of this stuff. I wanted to spawn a ball on a 2D plane that would move in a random direction, so I assigned a random value between 1 and -1 to x and y of its spawn vector. But then it spawns at different speeds, of course, the resulting vector would be much shorter if the values end up being close to 0. And I just had to normalize it to prevent that.

 

The eventual process was: add together simple math functions to perform a complex thing -> realize there is a prebuilt function for the complex thing -> use that instead.

 

Why wouldn't you just generate a random r and theta, and then derive [x,y] = [rsin(theta), rcos(theta)] ? It's much easier to do random directions easily if you work in polar coordinates.

[Deadpan]

That's the first thing I wrote that vaguely achieved what I had in mind, so I'm liable to go back on the code later on, but right now I don't see how this is any easier, unless the prebuilt math code for those trigonometric functions is far more efficient than the one for normalizing.

[aoanla]

Well, sine and cosine are, at least on x86 cpus, rather efficiently implemented nowadays, so I don't think there's much of an efficiency loss, if any. Plus, the code is actually easier to understand, it's very clear what you mean by rsin(theta), while normalising on values in (x,y) is a little less obvious.

 

It also changes the probability distribution of different vectors, I think, so it depends on how you want your distribution to be uniform.

 

[More specifically, if you want a uniform direction, with unit length, then for almost all good PRNGs, it will be faster to generate a single deviate, theta, and then calculate sin(theta), cos(theta) than it will to generate two deviates, (x,y), and return a normalised vector derived from them.]

[Nappi]

Why wouldn't you just generate a random r and theta, and then derive [x,y] = [rsin(theta), rcos(theta)] ? It's much easier to do random directions easily if you work in polar coordinates.

Sounds like Deadpan does not want the speed to be random, so random r is not required, but in case anyone needs to generate uniformly distributed points inside the circle, the way to go is [x,y] = [r*sqrt(c1)sin(2pi*c2), r*sqrt(c1)cos(2pi*c2)] where c1 and c2 are uniformly distributed random numbers between (0,1). Without the square root, the center of the sphere will be more populated than the edge.

[aoanla]

Well, indeed, it depends on what kind of "uniform" you want, as I mentioned. 

(I think for velocity distribution, it makes more sense to have uniformity in r (that is, speed), rather than uniform distribution in the unit circle itself.)

[Nappi]

Oops, missed that part. And agreed on the speed being uniform making sense.

[Deadpan]

If I can improve the speed or reliability of the function by coding it differently, I might look into that when I get around to it. I don't think this way of doing it is easier to understand, for me at least, than the thing I intuitively came up with though.

[itsamoose]

Do any of the math wizards here know of any cool number/inequality relationships like we see in Rock-Paper-Scissors? I always geek out when I learn about things like tessellation, prime spirals and finding cool patterns in numbers.

[tberton]

Not sure if it's exactly what you're looking for, but these

[SecretAsianMan]

This is only partially related to math but I can't not post this.

 

So Terrence Howard seems to think that 1x1 does not equal 1 but in fact equals 2.  He's apparently created an entire system he calls 'Terryology'.  His reasoning is

 

“If one times one equals one that means that two is of no value because one times itself has no effect. One times one equals two because the square root of four is two, so what's the square root of two? Should be one, but we're told it's two, and that cannot be.”

 

He also claims to have reasoned out why bubbles make spheres instead of triangles or squares and that his logic would blow the mind of Pythagoras or Einstein if they were present.

[Spenny]

In my new system Tetrodynamics I present the fact that 1 + 1 does not equal 2, but in fact equates to window.

[tberton]

Nobody's ever told that the square root of 2 is 2. It's ~1.414.

[Twig]

This article reads like something from The Onion.

 

What just happened.

[brkl]

https://docs.google.com/viewer?url=patentimages.storage.googleapis.com/pdfs/US20150079872.pdf

 

Dude is crazy, but they sure look sweet:

http://t.co/JPxBEjvvfU

[tberton]

The funny thing about this is, is that while Howard's system makes no sense, there definitely are other systems of counting and arithmetic. The p-adic numbers, for instance, are super cool.

[jennegatron]

This explains the tweet I saw yesterday of "y = mx + terrence"

[Ninety-Three]

This article reads like something from The Onion.

 

What it reads like is Time Cube.

 

Just like Time Cube, I understand a bit of it, and I feel slightly ashamed of that fact.