User:Revolver/Writings

Math notation suggestions

Here are some suggestions for wikipedia math notation:

When writing in HTML (not math) mode, don't use '-', use '& minus;', (without the space). This produces a longer minus sign, not a hyphen.
I've been ignorant of this one for a long time, apparently: when writing out function abbreviations in math mode (i.e. sin, cos, exp, ln, etc.) use '\sin', '\cos', etc. If the function isn't standard, use '\operatorname{xyz}' to write xyz. If you have some text to write in math mode that isn't a function abbreviation, you can use '\mbox{...}'.
Write the cyclic group with n elements as Z/nZ (or even Z/n if you get tired of seeing too many Z's..., although I prefer Z/nZ), not as Z_n. I know the latter is used by a lot of people and textbooks, but I have more than one good reason for asking for this. First of all, and most importantly, when n = p is prime, Z_p is the standard notation for the ring of p-adic integers, and the cyclic group is not isomorphic to this (indeed, one notation stands for a group, the other for a ring, so they can hardly even be compared to each other. We could think of the quotient ring, but this isn't ring-isomorphic, either.) To make matters even worse, some people use Z_p to denote the localization of the integers at the prime ideal (p). The correct relationship between these is that the ring of p-adic integers is the completion of the localization. So, the best way to notate this is really to let the localization be Z_(p) and reserve Z_p for the p-adic integers. I hope this amount of potential confusion is enough to persuade people that we should use Z/nZ or Z/n for cyclic groups. Another reason to do this is that the Z/nZ notation is unambiguous -- it can mean only one thing. Lastly, this notation explicitly realises the nature of the finite cyclic group as a quotient group of the integers. For my comments on the notation C_n (often used by group theorists and geometers, as opposed to number theorists), see Talk:Cyclic group
If you're going to use 'log' to mean 'log base 10', write out the base explicitly, or at least mention at the start of the article what the base will be throughout the article. I lost the war on this one ('ln' vs 'log' for natural logarithm) but maybe I can win this battle. Lots of people use 'log' for both base 10 and base 2 (and some of us, even for base e!, although I'll surrender and use 'ln'), so it's important to know. Don't just assume everyone knows that 'log' means 'log base 10'.

20 favorite math books

The following are 20 of my favorite math books. These are not necessarily the books I've found most useful, but they are the ones I've found most fun and the ones that had the most influence on me going into mathematics. This list will probably give you a good idea of my mathematical tastes. Unfortunately, many of these books are no longer in print.

Asimov on Numbers, Isaac Asimov, ASIN 067149404X
Counterexamples in Topology, Steen and Seebach, ISBN 048668735X
Disquisitiones Arithmeticae, Gauss, ISBN 0300094736
Fourier Analysis, Korner, ISBN 0521389917
The Higher Arithmetic, H. Davenport, ISBN 0521634466
The Historical Development of the Calculus, C. H. Edwards, ISBN 0387943137
How to Solve It, Polya, ISBN 0691023565
Introduction to General Topology, K. D. Joshi, ASIN 0470275561
Introduction to Set Theory, J. D. Monk, ASIN 0898740061
Introduction to the Theory of Numbers, Hardy and Wright, ISBN 0198531710
Introduction to the Theory of Numbers, Niven, ISBN 0471625469
Irrational Numbers, Niven, ASIN 0883850117
The Mathematical Experience, Davis and Hersh, ISBN 0395929687
Mathematics and the Unexpected, Ivar Ekeland, ISBN 0226199908
Measure and Category, John Oxtoby, ISBN 0387905081
Primes of the Form x² + ny², David Cox, ISBN 0471190799
Proofs From the Book, ed. Aigner and Ziegler, ISBN 3540678654
Theory of Algebraic Integers, Dedekind (Stillwell preface), ISBN 0521565189
Visual Complex Analysis, Tristan Needham, ISBN 0198534469
The World of Mathematics, J. R. Newman, ISBN 0671829505