Wikipedia:Reference desk/Archives/Mathematics/2008 July 4

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July 4

Looking for an algebra book

I'm looking for an algebra book that will help prepare me for a basic undergraduate text in algebra (i.e. abstract algebra) and / or real analysis. I currently own a copy of Michael Spivak's "Calculus" but the exercises and jumps in logic are way too hard; I want to re-cement my understanding of numbers, operations, rational expressions, inequalities, polynomials, rearranging etc. before going back to it.

More specifically, I would like this book to have a general tone (I don't need examples like "four apples + ten apples" or real world examples of problems) and present a lot of problems with solutions, but focus heavily on the basics. Most books I have found so far rush in to advanced topics too quickly or cover too many subjects; I just want to focus on the basics with a lot of exercises. Thankyous! 81.187.252.174 (talk) 08:03, 4 July 2008 (UTC)

You might try Geoff Smith's Introductory Mathematics: Algebra and Analysis in the Springer SUMS series, although it could be too basic for you. Worth a look though. 163.1.148.158 (talk) 09:12, 4 July 2008 (UTC)

I actually checked that one out, it's a nice book, but it is a very brief overview of several subjects such as group theory, set theory, vectors and complex numbers, all of which I'm not particularily interested in at the moment. Furthermore the exercises were sparse. I'm looking for a book that focuses more on algebraic basics in R, such as a chapter devoted to manipulation of rational expressions, or rearranging equations or solving polynomial equations. 81.187.252.174 (talk) 09:48, 4 July 2008 (UTC)

At my school, we use Spivak (I have a copy) for graduate analysis. So that is definitely not the book you want for basic analysis. For real analysis, my most favorite book (which is very well written for beginners by the way) is James Kirkwood's "An Introduction to Real Analysis". But this is only real analysis. There is no algebra here. For, Schaum's outline for both group theory and abstract algebra will be tremendously helpful. All three of these books will be available (very cheaply) at half.com or some such website.A Real Kaiser (talk) 01:32, 5 July 2008 (UTC)

Long line

Long line (topology) says, "And if we tried to glue together more than ω₁ copies of [0,1), the resulting space would no longer be locally homeomorphic to R." Why is that? Why can't you add another copy of [0,1) onto the end? --Tango (talk) 14:09, 4 July 2008 (UTC)

You wouldn't have a countable local basis at the newly added 0, so it can't be locally homeo to R there. Algebraist 14:16, 4 July 2008 (UTC)

I think that there is some miscommunication here with regards to the meaning of ω₁. This is not the first ordinal following the first uncountable ordinal, but rather the first cardinal following the first uncountable cardinal. Therefore, it is not ω₀+1. Algebraist's answer therefore applies to ω₁. Oded (talk) 02:10, 5 July 2008 (UTC)

Crazy manifolds

What would happen if you replaced Rⁿ in the definition of a manifold with some other topological space? Are there any other spaces which have non-trivial manifolds? --Tango (talk) 14:11, 4 July 2008 (UTC)

Apparently, people have studied slight generalizations, at least: see manifold#Broad definition. On a related note, fibre bundles are spaces that are in some sense 'locally trivial'. Algebraist 14:18, 4 July 2008 (UTC)

There's one particularly useful generalization. If you replace $\mathbb {R} ^{n}$ by the closed upper half space in $\mathbb {R} ^{n}$, then you get a manifold with boundary. (That is, every point has a neighborhood homeomorphic to $\mathbb {R} ^{n}$ or homeomorphic to a closed half space.) Oded (talk) 02:31, 5 July 2008 (UTC)

Replace R with the p--adic rationals Q_p and restrict to analytic charts, and you get the definition of a p-adic analytic manifold---a totally disconnected space equipped with an analytic structure based on Q_pⁿ. Tesseran (talk) 00:05, 6 July 2008 (UTC)

That's interesting - thanks! I'm just studying p-adic numbers for my final year project, I'll have to look into p-adic manifolds. --Tango (talk) 01:16, 8 July 2008 (UTC)

Remember to write it up into a WP article. :) Algebraist 14:22, 8 July 2008 (UTC)

Calculating your mark

hey there! In our school, the regular school work is worth 80% of our mark and our exam is worth 20%. Our meany-weeny teachers won't tell us our exam mark, but I want to figure it out. Say I had a 100% before the exam and that dropped to a 90% after the exam, how do I calculate how much I scored on the exam? Keep in mine that the 100% is worth 80% of my mark and the x% that i got on my exam is worth 20%. THank you all!! 99.240.177.206 (talk) 21:24, 4 July 2008 (UTC)

Your final mark F is given by $F=0.8S+0.2E$ where S is your schoolwork mark and E is your exam. So now put in the information you have and you get: $0.9=0.8\times 1+0.2E$ and you can work out what E is in terms of percentages fairly easily. x42bn6 Talk Mess 21:49, 4 July 2008 (UTC)

Try NOT to work in percentages. People brain kinda switch off when they have to work in percentages. Instead work in 'unity' units.

100% = 1

70% = 0.7

30% = 0.3

2% = 0.02

122.107.135.140 (talk) 22:31, 4 July 2008 (UTC)

Also, see this recent/similar question. -hydnjo talk 01:02, 5 July 2008 (UTC)