Calculus by Michael Spivak is really, really good if you want a thorough understanding of the concepts. The problems are all useful and challenging, and you learn a lot more than calculus in the process.
However, I've heard it's quite hard compared to other calculus books, and you won't like it...
With absolute values, only the magnitude of the change matters. It doesn't matter if the change is positive or negative. So H must change more than TS.
I calculated the first 8 and put them in to OEIS, and got: oeis.org/A072752.
What you're after is not the gaps, but the difference, so it's one more than the terms in the sequence I linked to.
I'm not sure about an efficient algorithm, my jumbled together program could only do 8 before taking...
So it seems to be (area of circle) * (circumference) * (1/2). It's less a question of why it doesn't work, than why your brother thought it would work. Maybe if you posted the derivation, we could point out the problem with it.
99 = 3 * 33, so 1 and 11 aren't the only values x or y can take.
There's 1 solution to your problem.
Don't be surprised that you're getting "ad hoc" methods, you've thrown together a few random arbitrary conditions, especially the "twin number" bit.
I think that's equivalent to \sum_{p=2}^{n} \frac{\left \lfloor n/p \right \rfloor}{p} , where the square brackets represent the floor function, and p runs through the primes less than or equal to n.
I don't know if that helps at all, and no doubt it can be simplified more so.
Now I hate this kind of discussion, so I won't really get involved, but mathematics does not attempt to prove anything "in terms of the physical world", it has no concern for silly things like atoms and quantum systems.
The generally accepted order of operations are BIDMAS (or BODMAS, same thing), which is:
Brackets, Indicies, Division and Multiplication, Addition and Subtraction.
So, brackets are computed first, then indices (powers), then division and multiplication have equal priority, and are done left...
There aren't any proofs if you keep on asking "why?", at some point or another you're going to have to accept things as true, without proof: these are axioms. Some common axioms for arithmetic are:
1) a + (b + c) = (a + b) + c
2) a + b = b + a
3) a + 0 = a
4) a + (-a) = 0
5) a*b = b*a
6)...
It's proven, after hundreds of pages, in this book:
http://en.wikipedia.org/wiki/Principia_mathematica
I definitely can't explain how it's done, it's all in symbolic logic, and I guess you start with a degree level mathematical education.
For the first equation, your first step is wrong. Where you've attempted to subtract x from both sides, you've actually subtracted x from the left side, but only \frac{x}{4} from the left side. Adding, for example, 2 to a fraction, is not the same as adding 2 to the numerator of the fraction. In...
Your lack of brackets or LaTeX is both annoying and confusing.
For your first question, the answer x < -3 only comes about if the equation is 2x + 5 < \frac{x-1}{4}, so I'm assuming that's what it is. From your working, it's a bit confusing as to what equation you're trying to solve...
Euler's formula says that e^{ix} = \cos x + i \sin x.
Let x = 2 \pi n, where n is any integer. Then e^{ix} = \cos x + i \sin x = 1 + 0i = 1.
Therefore, e^{ix} = 1 has an infinite number of solutions, all of the form x = 2 \pi n. The two you brought up, x=0 and x = 2 \pi are just the ones...
It's certainly easy, from a practical point of view, to put in the values first, so you know if you're multiplying/dividing by a negative number, which would change the inequality sign. Mathematically, if you consider separate cases for each variable being positive or negative, it makes no...