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Do you love the science and art behind numbers, variables and functions? Than this is the place for you!

I'll kick things off with what I consider to be my greatest accomplishment so far:

(d/dx)ssr(x)=ssr(x)/(x*(ln(x)+ssr(x))

Where ssr(x) is the square super-root of x- i.e. ssr(x)^ssr(x)=x.

If anybody wants to know how I derived this, I'll post a proof later.

f course, we can all appreciate the beauty of e^((pi)*i)=-1, but a lesser known one is that (e^pi)-pi=19.999099979, which is great when you want to fuck with a programmer and tell them they need to fix their floating point errors.

What's worse than divide by zero? Divide by phi:

Or… (1+√5)/2 works too. Might save you from carpal tunnel syndrome when it comes to putting it into your calculator.

I vaguely recall reading xkcd pieces about these two mathematical beauties (though I suppose the latter, being a presumed total coincidence, can't be considered beautiful). Great webcomic, by the way.

Indeed you did
And here's the second one

And yeah, definitely a great comic.

A requirement of being in this thread is having the unit circle memorized. :^)

Obligatory simplification for dummies like me:

Also, the fact that rad is often intentionally discounted for the sake of convenience, leaving just π, messed me up pretty badly in highschool. I'm fine using it now, but it still miffs me ever so slightly.

If any of you haven't tried already, I would recommend that you take a foray into the logical proof that 1+1=2 (alternatively, that 0+1=1), since it's one of the bigger mind-blowers I've been fortunate enough to come across. Mathematics is, after all, based on many axioms, and things get interesting when attempts at dissecting seemingly self-evident truths occur.

It would honestly be silly of me to assume that career mathematicians haven't heard of the Principia Mathematica, but anyone reading this thread who isn't that deep into the subject should seriously consider taking a look at it or any of its countless summaries online, even for the sake of novelty. It basically tears logic into strings with passages such as "if it is the truth, then it is true; if it is a falsehood, then it is false." It explores the nature of what addition is and everything.

"From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2"

I suppose that this is more a matter of logic than mathematics, but the two are so heavily intertwined that the subtle distinctions aren't worth fussing over on a KYM thread, IMO. Oh, but who could forget about this cool xkcd edit?

Back on topic; the proof that 1+1=2 took three hundred pages and involved almost every single logical postulate in existence. So even then, it's not truly proven – the truth of an axiom proved by utilizing purer axioms.

Simplified versions of the proof, such as this one, exist online and are really helpful. I sat down with a pen and some paper a few months ago and worked my mind through it because I was honestly losing sleep thinking about it at that point.

I'm really feeling like I'm in my element right now. Hah, get it? Oh wait… this is a mathematics general, not a chemistry general. I'll see myself out.

Mfw reading this thread while having a D in Algebra:

Oh trigonometry, it's good to see again old friend…
[Circular trigonometric functions can't be friends]

Speaking of pi, have you guys ever heard of tau number it's pretty a rad concept among mathematics:

There is absolutely no way in the three hundred and fourteen circles of hell that we're ever going to start using "tau" instead of pi. They've already tried, and the result wasn't pretty.

Here's a great article explaining why, by the way.

As an Economics undergrad, I have to take up microeconomic calculus, macroeconomic calculus, and econometrics. I failed them multiple times but that's just the way the cookie crumbles.

It's not as aesthetically pleasing as the Fibonacci Sequence or Phi, but professors obsessed with graphs will make you find these graphs appealing too.

So I just went to the UW Math Day.
While I'll admit that, even for someone who genuinely loves the subject, it wasn't exactly the most "thrilling" thing in the world. But I'll be damned if I didn't hear a lot of interesting stuff, most of all some examples of open problems. (Unfortunately, none of ones with a bounty of one million buckaroos though.) The one that I can't stop thinking about now is the question of whether an "integer box" exists, defined as a rectangular cuboid/right rectangular prism where the lengths of its edges, face diagonals and its space diagonal are all integers.
After some tinkering, I quickly discovered a way to restate it as a number theory number theory problem by essentially "building in" all the necessary geometry work of the original:
Does a Pythagorean quadruple (a,b,c,d) exist for which there also exists the Pythagorean triples (a,b,x); (a,c,y); and (b,c,z)?

From my Google searches, there is a disturbing lack of discussion of this on the net. As in, zero.

Actually… I already did. See above.

On Pi day, I have an excuse to eat all the pie I want. What do I eat on Tau day?
Checkmate Atheists.
Seriously, I've heard of this multiple times over the years and never could get my head around why Tau is better than Pi
Btw, what are everyone's thoughts on switching to a Base-12 counting system?

I think this thread cements the fact that I probably have a phobia of math. Well, I may be slightly exaggerating, but I'm abysmal at it. I actually used to be crazy good at math and was a few grade levels ahead of everyone else in my math classes for a while so that I just breezed through it, but slowly over time math caught up with me and I've just been falling behind ever since. Pre-algebra was the first time I really struggled with it and then Algebra 1 was pretty easy for me because it's basically an extension of pre so it's mostly the same stuff, but then everything went downhill when I started Geometry. In Geometry and Algebra 2 I struggled to maintain a C and now I'm in Stats and I struggle not to fail. Even though the homework's just completion in my current stats class I often don't do it just because I don't even know where to start. I've found myself leaving pages blank on the free response of nearly every test. It's nasty. Math and I just cannot get along and couldn't for a few years now. And the saddest part is that my main goal is to become a scientist. I'm really good at science and beat pretty much everyone my age at remembering the facts and the thinking required for science and stuff but math is the only thing standing in the way of those dreams for me, and it's a pretty big wall.

You guys seem to love this Satanic shit for some reason I'll never understand and also be good at it. More power to you all I guess. Have fun with your maths or whatever!

What field interests you? Not all sciences require this level of math. Physicists and Engineers yeah, but Paleontologists and Biologists could scoot by

So are you eventually going to do Calculus? I'd say that's pretty integral to a good education!

But in all seriousness, if you are I could maybe help you out with some advanced algebra concepts that you struggle with in order to be better prepared. Hell, even if you aren't, the offer's still there- it may end up being helpful down the line. You'd be surprised how many issues in understanding math are caused by very simple and easy to correct misconceptions.

As for Statistics… meh. Even though I did ace the class last year (by doing virtually no homework but overriding that with a 5 on the AP test, ha ha), it's really not my thing. Sure there is some "math", but it's also a whole fucking lot of memorizing terms and methods that are about as far away as you can get from theoretical mathematics while still technically being within the subject. And yeah, it's obnoxious, so I certainly don't blame you. Is your teacher any good?

@Borike

I'm planning on doing meteorology which I'm guessing will require a decent amount of math. Ughhhhhh.

@.9999

Yeah I might look for some help when I'm in calculus and stuff and at least algebraic stuff I can understand it with a good teacher or some time looking at it. The problem with stats is that like you said it's so un-math-like and super subjective. Like before I took stats I've never thought multiple answers on the multiple choice could all be right. Math is supposed to always have one answer. I'd expect something like that from a reading test or something but not math. And a lot of it is just memorizing giant page long proofs and plugging in formulas terms and stuff depending on the context and knowing which of the 12 or something different formats of proofs you need to use in each situation. I deeply hate it.

A huge problem I have with math that I have a hard time controlling is that I swear I have narcolepsy whenever I'm learning math, at least for about 2 years. I'm fine staying awake in all other subjects except for this one. While I am exaggerating about the narcolepsy thing, it's really similar because even if I'm really well-rested and had a good breakfast and all that stuff that gives you energy my brain just turns off and I slump over against my will even if I desperately want to learn a concept. Most of my learning is just staring at the homework questions and trying not to cry. To address what you said at the end, I feel like this is partially my fault and also the fault of the teachers. The teachers I've had for 2 years have been crazy dull. My algebra 2 teacher talked monotonously in a Boston accent and slurred all his words together. He was a good teacher and explained things well, but he just couldn't keep me awake even at the end of the day with 9 hours of sleep and a full stomach. My stats teacher this year is sort of the same. She teaches things okay-ish but I'm usually too far gone to even know. She talks with almost like she doesn't want to be there and adds nothing but read notes to us. Just this week instead of taking notes we did notes disguised as a game, which kept me awake all class and I learned the subject (well not completely because I didn't know much about the concepts that lead up to that one) I feel like I just need a really special teacher who can explain things well and keep me awake which is hard as hell to do. I feel like the only way for me to do relatively good at this point is to get a good tutor or something.

And .9999, are you planning on majoring in math at all? It seems like you understand it well and have a passion for it. It'd suit you well. Math jobs usually pay well too because no one wants to do them.

Yep, I'm currently staring down an Applied Mathematics major with a minor in Physics (or even a double major if I can manage it) at the Illinois Institute of Technology pending the news from the other couple colleges I applied to. What's cool about that, according to the guy I talked to who's in his fourth year in the program, is how many different directions you can branch out into. I'm definitely more interested in the theoretical side, and there's plenty of room for that possibility, but that's way less likely to land you a solid job right upon graduation. In terms of that, IIT is really damn good. Or, so I've heard anyway.

So, yeah, shoot me a message if you want a bit of informal "tutoring".

@0.9999…=1

The one that I can’t stop thinking about now is the question of whether an “integer box” exists… From my Google searches, there is a disturbing lack of discussion of this on the net. As in, zero.

For those interested, the official name of this unsolved mathematical problem is referred to as the perfect cuboid. It is essentially an extended version of Euler's brick, the difference being that Euler's brick demands a cuboid with integer edges and face diagonals, while the perfect cuboid ramps up the difficulty even further by adding in integer space diagonals as a condition. Or perhaps I'm wrong regarding the added difficulty; it's perfectly within the bounds of possibility that any functioning formula developed to find the relationship between integer edges and face diagonals (assuming that such a formula exists), once discovered, would also reveal the nature of integer space diagonals.

Considering the fact that the perfect cuboid problem is mathematically equivalent to the Diophantine Equation, however, it is highly likely that a mathematician looking for answers would seek to develop a theory for the latter and not the former.

This ties in with the common belief that the reason why the perfect cuboid problem hasn't been solved is because the people with the equipment, knowledge, and computing power necessary to tackle such a question simply can't be arsed to. To put it bluntly, the perfect cuboid has little practical application or real-world value, or any reward (outside of perhaps a personal feeling of accomplishment) attached to it. The sheer processing power that's currently being dedicated to finding prime numbers could quite likely unravel the cuboid fairly quickly with a bit of fine tuning, but unlike the cuboid, there's financial incentive to prime number-hunting.

@Sam

I'm currently prepping Twisty for his upcoming engineering classes to the best of my ability, and it would be a pleasure to help you as well. Algebra is highly formulaic and calculus is even more so; once you learn the underlying principles, you can't go wrong. I'm sure that 0.9999…=1 would do an excellent job, but don't be hesitant to drop me a line here or in the IRC or wherever.

You don't love maths? I'll make you love maths.

I just wrote a Java program that calculates the Gaussian distribution and prints out a bar graph for it. I am so very happy~

-5: 5
-4: 19
-3: 81
-2: 194
-1: 377
±0: 418
+1: 388
+2: 217
+3: 77
+4: 22
+5: 2

█
██
█████████
████████████████████
██████████████████████████████████████
██████████████████████████████████████████
███████████████████████████████████████
██████████████████████
████████
███
█

(It uses random numbers, so it's not perfect, as you can see.)

@Particle

Oh. So that's why I couldn't find it. But as for that last part you quoted me on, I was referring to my restatement of the problem.

Does a Pythagorean quadruple (a,b,c,d) exist for which there also exists the Pythagorean triples (a,b,x); (a,c,y); and (b,c,z)?

And yes, this does indeed imply a system of Diophantine equations. (There's no such thing as the Diophantine equation- I assume a typo.) And I believe the key in unlocking it may be the parametrization of all primitive Pythagorean quadruples:

Here, m, n, p, and q are non-negative integers that are coprime with an odd sum.
Essentially, the idea would be show that there is no set as described for which a^2+b^2, a^2+c^2 and b^2+c^2 are also perfect squares by proving a contradiction when the opposite is taken as an assumption. But I've probably got a long ways to go before I can figure something that complicated out. Not like I won't still try, of course.

Also, I have to strongly disagree with you about the nature of the field- mathematicians have been proving shit just because they can, without any known real-world application, for hundreds of years. Fermat's Last Theorem, anyone? (This also applies to the use of computers; for example, the one that verified the Erdős–Straus conjecture up to n≤10^14. It's 2015- trust me, there are plenty to go around.) The simple fact is, some problems are just damn hard and take a long time to figure out. Like, 358 years for example.

While I was playing around with your idea, it quickly became apparently that it is possible to disregard the cuboid altogether and define the problem solely in terms of a set of Pythagorean triples by using d^2 -- or sqrt(a^2 + b^2 + c^2) -- as a common/shared hypotenuse:

Three Pythagorean triangles, each with hypotenuse sqrt(a^2 + b^2 + c^2), and catheti: a, sqrt(b^2 + c^2); b, sqrt(a^2 + c^2); c, sqrt(a^2 + b^2).

Hence the problem becomes a question of whether such a set of triangles could exist. It's quite a neat way of boiling it down to the relationships between a, b, and c IMO. Thoughts?

The lower bound for the odd side of a potential perfect cuboid is now roughly 2.5*10^13 as of February 2015. Needless to say, that falls a bit short of the largest known prime number of 2^57,885,161 − 1. What I said about the lack of attention given to the problem stands, though perhaps I was a bit off regarding the actual reasons. AFAIK there are currently no famous supercomputers or GIMPS-style collaborative projects dedicated to brute-forcing the problem. My point is that if there were, the answer would be found a lot faster. Unless an answer doesn't actually exist, of course, but I'll leave that for someone more experienced to tackle.

EDIT: superscript formatting doesn't work in the forums, apparently.

Yeah, I'm thinking more about a classical proof like the one you posted, and… damn. Now that right there is really interesting. He used square summation in his methodology… somehow. (I'll have time to take a closer look later. I might be able to fully understand it.) Of course, there's a very good chance that he overlooked something and his proof has a flaw that will be demonstated. We'll see if it passes the exhaustive test that it will surely recieve.
What's cool is that, no matter what methodology ends up showing the non-existence of a perfect cuboid (if it happens, that is), the answer to the question I posed about Pythagorean triples and quadruples would still be answered implicitly, similarly to the fate of Fermat's Last Theorem. I would still like to see someone tackle that directly, though.

I don't know what to post and I don't know how to explain it, but here is a graphic demonstration of Fourier transform just for sake's of this thread: