Wikipedia:Reference desk/Archives/Mathematics/2015 May 4

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May 4

Math book recommendation needed

When I was in school, I had a math book entitled "mathematics for science and engineering", or something like that. It wasn't a textbook used in my school--I bought it because I liked math. It wasn't a "tome"--about 1-1/3" thick as I recall, and not overly verbose. The topics were more advanced than what was taught at my level in school. It was a good book for my younger self. It showed me what was there to learn beyond my level, and the presentation was clear, not too wordy, and seemed to have an emphasis on techniques useful in science and engineering.

I'd like to get a similar book that's available in the US, for a high school student. Would be good if it's not too pricey. Can someone recommend one? Thanks. --98.114.98.58 (talk) 01:53, 4 May 2015 (UTC)

What kind of math do you want to study? Even in science and engineering there are different mathematical concepts for different disciplines. For example, electricity and magnetism requires a lot of vector calculus while most simple rigid-body mechanics problems that I know of can be solved with ordinary single-variable calculus.--Jasper Deng (talk) 05:07, 4 May 2015 (UTC)

The book is not for me. For the person the book is for, it's way too early to talk about specialization. What I'm looking for is a book that can serve as a source of materials for "serendipitous" learning--a book with a variety of topics that can be browsed and studied more or less in random order--whatever catches the eye while flipping through the pages. The book Mathematics 1001: Absolutely Everything That Matters in Mathematics in 1001 Bite-Sized Explanations is kind of like that, but the treatment of each topic is too superficial. In contrast, the book I had when I was young actually covered useful math techniques, not just giving you concepts and interesting facts. --98.114.98.58 (talk) 12:36, 5 May 2015 (UTC)

I was more on the pure side, and these days work mostly in economics, econometrics, and the philosophy of mathematical economics. When I was at high school the books that most interested me were Cundy and Rollett's "Mathematical Models" (a magnificent book) and Martin Gardner's various works. (I also attempted Principia Mathematica way too early!)

On the science and engineering side, the book that I found most useful is now out of print - it was old when I got it - but is still available second-hand on Amazon: "Advanced Level Applied Mathematics" by F.G.J. Norton. It's pretty much the benchmark for a good first textbook for those planning to study applied mathematics and engineering at college. I also recommend "A Student's Guide to Vectors and Tensors" by Daniel Fleisch, which is a little more advanced. Also good is "Div, Grad, Curl and All That" by H.M. Schey.

On the purer side, an enthusiastic high school student with an interest in maths will enjoy "Five Golden Rules" by John Casti, and, a bit more advanced but really good, "Field Theory and Its Classical Problems" by Charles Robert Hadlock. All are on Amazon; all are also on my own bookshelf, and have been well used. RomanSpa (talk) 10:25, 4 May 2015 (UTC)

Thanks for the suggestions. I did look them up but not all of the was on Amazon. Principia Mathematica was a tour de force in a way, but I found the unusual notations difficult to get used to. Once I learned that it was ultimately an unsuccessful attempt, the interest in actually working through it was just lost. --98.114.98.58 (talk) 12:36, 5 May 2015 (UTC)

It's only unsuccessful in a rather uninteresting way. (Lest I be accused of being silly, I'll note that the proof that it's unsuccessful is interesting, but the fact that it's unsuccessful is merely a foundation on which subsequent ideas can be built.) RomanSpa (talk) 15:36, 5 May 2015 (UTC)

+1 for Cundy & Rollett, which I'm glad (and slightly surprised) to see is still in print; and anything by Martin Gardner is worth reading. As a teenager I was inspired by the books of W W Sawyer, from whom I learned the basics of differential calculus (Mathematician's Delight) before doing it at school, and non-euclidean geometry (Prelude to Mathematics). AndrewWTaylor (talk) 13:01, 5 May 2015 (UTC)

My father loved "Mathematician's Delight", but I found it less interesting. If you're looking for a book suitable for browsing, perhaps David Wells' "The Penguin Dictionary of Curious and Interesting Numbers" might do the trick... RomanSpa (talk) 15:40, 5 May 2015 (UTC)

Periods of binomial coefficients in finite groups

Hello! Where could I find some properties of the function ${\binom {n}{k}}$ with recursive definition in the group $\mathbb {Z} _{m}$, with $m$ not prime? In particular, I was wondering about what could be the minimum $\tau (k,m)$ such that ${\binom {n}{k}}{\bmod {m}}={\binom {n+\tau (k,m)}{k}}{\bmod {m}}$ for all $n\in \mathbb {N}$, but I had not been able to find anything on the internet.--Nickanc (talk) 18:37, 4 May 2015 (UTC)

The section "Variations and generalizations" in the article Lucas theorem mentions an article by Granvilles which talks about the binomial coeffs mod m in the case where m is a prime power. This seems to be the only case you need to consider because you can apply the Chinese remainder theorem to get the answer for m once yous have it for relatively prime factors for m. For m=2 the period of c_k(n)=${\binom {n}{k}}{\bmod {m}}$ is the next power of two after k, which is A062383 though it's not mentioned there. This should follow from Lucas theorem though I haven't worked it out in detail. Presumably you could use Granville's results in the same way to get the period for any prime power and from there get the period for any m. --RDBury (talk) 00:51, 6 May 2015 (UTC)

Conic sections and curvy cutting

I have been thinking of conic sections lately, and I notice that conic sections are usually cut with a flat plane. Making conic sections seems to assume that the slices are cut straight across. But what happens if you don't cut straight across? Instead of a normal straight knife, you make your own "knife" that is shaped into a semi-circle with the outline of a square at the convex side of the semi-circle. Now, you attempt to make a conic section of the cone-shaped pile of clay. But instead of cutting straight across, you stop midway, turn the handle so that the knife gets some clay outside the usual trajectory, and then pass through the clay out the other end. I am not sure if my visualization of this unusual cutting makes sense to anybody without a picture, but if it does, is there a way to calculate the volume of both pieces of the cone-shaped clay-mound after the cut? 140.254.136.157 (talk) 19:14, 4 May 2015 (UTC)

There are really three cuts going on here, assuming I understand the description correctly, and it is useful to think of them individually. The first cut (before you rotate the "knife") forms a part of the intersection of the cone with a cylinder. The second cut, as you rotate the knife, forms part of the intersection of the cone with a sphere, and the last cut forms part of the intersection of the cone with the same cylinder (I am assuming that the rotation is through 180 degrees, otherwise this would be the intersection with some different elliptic cylinder, but otherwise not much would be changed). Each of these boundary curves are (parts of) quartic curves (here I am assuming that these cuts are in "general position"). The volume would be given by elliptic integrals. Sławomir Biały (talk) 21:46, 4 May 2015 (UTC)