Amaton

Won't work unfortunately. Since [math]\pi[/math] is irrational, it can't be derived by doing arithmetic on integers. Approximations can be achieved, but you'll never make an expression like such equivalent to the true value of [math]\pi[/math]. Though you can't derive it this way, there are certainly other ways to represent it, many of them quite elegant and beautiful. [math]\pi[/math] indeed has some interesting relations to the prime numbers. These are deep connections, and they're relatively advanced in content (for me at least). Here's a nice link: What are the connections between pi and prime numbers? By the way, there is a prime-counting function denoted [math]\pi(x)[/math]. It just tells you how many prime numbers there are at and below the given value. The [math]\pi(x)[/math] notation is only due from the Greek letter however, and is not a reference to the constant [math]\pi=3.14...[/math].

Okay, using the formula from x(x-y), I tried the Chelyabinsk meteor. Assuming a constant speed of approx. 0.99c and a constant rest mass of 10,000,000 kg: [math]E_k\approx1.3472\times10^{32}\,\mbox{kJ}[/math] ...according to my calculation. Now this is quite a punch!

If there's a 0/0 in a function, I don't think we can simply equate 0/0 with an existing limit at that point. Consider a function [math]f(x)=\frac{g(x)}{h(x)}[/math], where [math]g(x)[/math] and [math]h(x)[/math] are non-constant differentiable functions. If there exists a constant [math]a[/math] such that [math]f(a)[/math] results in [math]\frac{0}{0}[/math], is [math]f(a)[/math] not always undefined? Regardless of any limit that may exist?

I enjoyed your post. Interesting notion. Forget the 1 kg object. Let's think about the recent Chelyabinsk meteorite! I wonder what devastation that would cause if it was traveling at relativistically high speeds (not necessarily 0.99c but something reasonably high for cosmological phenomena).

Yep, it's big news as of now. Don't have cable this week, but I can tell since it's #1 on Google and Yahoo! News. Here's an article: Meteorite Hits Russia, Causing Panic Interesting how this paper says it injured around a thousand people, while the previous articles only balled about 400. Either more detailed reports of injuries are coming out or some news sources just want the attention.

There seems to be no information on it from a formal source, but there is some original research: Derivatives of Displacement (antiderivatives a little down the page) I agree though that it seems to hold little significance in application. What do you think? Tentative at best?

Why does this sound so elemental and mystic? It didn't clarify the question. What specific "prediction" are you referring to and by what reason is it made?

Haha, I see. Much thanks.

Semjase, your identity is a special case of a general equality that holds true for all real numbers. Working backwards... [math]x=x[/math] [math]x=\exp[\ln x][/math] [math]x=\exp[(\ln x+1)-1][/math] [math]x=\exp\left[\frac{(\ln x+1)^2}{\ln x+1}-1\right][/math] [math]x=\exp\left[\frac{2(\ln x+1)^2}{2(\ln x+1)}-1\right][/math] [math]x=\exp\left[2\times\frac{\ln^2 x+2\ln x+1}{2\ln x + 2}-1\right][/math] We can complicate it more and more if we'd like, but as you can see it's quite trivial. Another cool but less obvious identity: [math]\pi=-2i\,\ln(i)[/math] which does the same thing but in the guise of imaginary arguments.

Ah, thanks. So for example, let's say, [math]E=mc[/math] is not dimensionally correct?

I've never really been exposed to dimensional analysis, but is it really as simple as... [math]\mbox{E}=\mbox{mass}\times\left(\dfrac{\mbox{length}}{\mbox{time}}\right)^2=mc^2[/math] Of course, this is vaguely simplified, but is this the basic idea?

Parallelism is (ordinarily) a Euclidean property, and in this context seems impossible for obvious reason. However, we could consider the projective plane. So intuitively, a given pair of parallel lines would intersect at a "point at infinity" (though really, the Wiki states that there are no parallel lines in the projective plane as per the preceding condition). In any ordinary geometry, is a tri-angle not always defined to have 3-angles?

Oh, I don't even expect the majority of ordinary people to understand this, let alone have the slightest interest in it. But at least motivated high school students... who should get more exposure to the proofing context of mathematics. The only exposure I get is in dealing with simple inductive series (yawn). Also, there are lots of people interested in popularized mathematics. YouTube has several popular videos covering topics like prime numbers, transfinite cardinals, and modern geometry (etc.) Euclid's theorem is another nice topic of interest, and I thought it'd be cool to study its proofs (however beyond-me some of the more recent methods are).

Not the person you replied too, but this really helps. I was aware of the notion that oscillating particles can be modelled rectilinearly (in straight lines), but I couldn't reconcile this with the wave functions which describe them. "Waves" and "straight lines" just wouldn't click in my head. That is until I realized that the function is derived when the oscillation is measured against time. So your explanation just cleared up a lot of confusion, thanks.

I believe x(x-y) implicitly pointed this out, and here I am also. It may clear things up to denote your limits towards 0+ and 0-.Now that I think of it, one-sided limits denoted incorrectly as coefficients can terribly miskew and confuse a problem.

Ah, I understand now. Considering I have little experience in physics and was able to follow that easily, I'd say your explanations were great. Thanks a lot

Nice and clear explanation, thanks. A little confused with the last sentence. Where exactly did the released energy come from?

My take on the original proof by Euclid, which shows that there are infinitely many prime numbers. A good introductory proof I believe... Now questions regarding the nature and methodology of the proof... What demonstrations are involved in this? I'd like to think: proof by exhaustion (whole) and proof by absurdity (2.B). Can case 2B also be shown as a proof by contradiction, where it implies K is not in L (a direct contradiction of the assumption)? BTW, I'm a student. When I first encountered this theorem, this was my first real exposure to an elementary proof and I found it beautiful. I wanted to paraphrase it so that things would be a little more obvious to non-mathematicians. Although I wonder if it seems unnecessarily thorough. Thoughts?

I'm not sure where you're trying to get at. What mistakes are you referring to?

But wouldn't that be painstaking trial-by-trial testing? A conjecture can be proven based on the very definitions and properties of the objects at hand, and I'm sure irrationality proofs are not exceptions to this.

Thank you for the responses. Okay. Though I thought I've read somewhere that there is some kind of chemical combustion involved, apart from the obvious fission mechanism. Fortunately, I was able to find the following thanks to somewhat meticulous link searching... So the blast is effectively an indirect result of the energy released via fission. Which is to say the nuclear attraction reaches a maximum, resulting in its overcoming and the fission of the nucleus? Just making sure I'm getting this.

It's not a necessity in this case, and it's totally fine if you haven't. But if you're interested, you can use this as a motivation to learn the topic. Rectilinear motion in simple scenarios can be described with three functions of time: position, velocity, and acceleration -- all of which are related through the derivative. Knowing just one of these allows you to calculate either of them at any given point in time, which I think's pretty neat.

Thanks for the reply, Daedalus I was thinking "dark" in respect to both attention and understanding. Like I said, it's far from well-known in a popularized scope (as opposed to fractals, game theory, mandelbulbs, etc.). Operations beyond exponentiation are also not as established or well-formulated as the preceding iterations. But they follow the same thought train which gives us these predecessors (well, your iterated powers can be extended more easily as given the identity [math]x^{<n>}=x^{x^{(n-1)}}[/math]). But I hold doubts to the range of their applicability nonetheless. Possibly and hopefully. Maybe future technology will depend upon such operations for design and computation. One problem in developing an understanding however seems to be the increasing numerity of operations. The sequence of hyper-operations only deals with right-associative operations. Thus, tetration --> pentation --> etc. Simple. However, disregard the right-associative restriction, and it seems the numerity of operations per level increases geometrically by a factor of two.

Hey Daedalus Great work. I saw this referenced in a recent thread, and despite its inactivity as of late, I must say this is fantastic. You're pioneering a relatively dark corner of mathematics (at least from a popularized scope). Unlike fractals and such, hyper-operations are a little harder to love for layman like me. But I've always thought that a large cooperative insight into this subject might open up a lot of doors, both in the pure and applied mathematics. Though I might not be able to contribute much, I'd love to learn and discuss this topic.

I follow you at "buy 9 packs of 3 bottles, and receive 9 free bottles". But I don't have a clue as to how you follow with "28 - 9 = 19". Start over -- You decide to buy 9 packs, receiving a free bottle for each pack. 27+9=36. You now have 36 bottles, which means you've gotten more than you should have. All you need is 7 packs of three plus the free 7 that go with them, and now you have your 28 bottles. You bought 7 packs, a total of 21 bottles. So the answer is, you bought 21 bottles of coca-cola. Right?