Sam Nelson -- Mathematical Writing Tools

Mathematical writing involves a lot of special symbols, equations, matrices, pictures and such things which are not easy to create in usual word-processing software. Moreover, many mathematicians (myself included) feel that the equation editors included with e.g. Microsoft Word and such are inadequate and produce unsatisfactory results.

Happily, you can produce professional-quality mathematical documents yourself with the industry-standard mathematical typesetting software for free! Real mathematics is written in a typesetting language known as LaTeX, pronounced "lay-tek" or "lah-tek" (but never "lay-tex"). LaTeX is not a word processor but rather a typesetting language similar in some ways to HTML; documents have a header section and body section, for instance, and most formatting uses "begin" and "end" tags. You use your favorite text editor to write the LaTeX source code in a file ending with a ".tex". Next, you compile the source code into a .pdf file using a program called "pdflatex." Any picture files you want included will need to be in the same directory as your .tex file as either .png files or .pdf files.

The tools you will need to produce professional-looking mathematical writing are all free software, though if you're determined to pay there are commercial packages available, e.g. Scientific Workplace and others. To get LaTeXing, you'll need:

A text editor such as Notepad, TextEdit, or emacs, and
The LaTeX compiler software. If you're using Microsoft Windows, you'll want to download the latest version of MiKTeX; for MacOS, the simplest option is to install MacTex; for Linux, you'll want to install the "texlive" package using your favorite package installer (e.g. apt-get, synaptic, rpm etc.)
Alternatively, there are free integrated LaTeX environments which include both a LaTeX compiler and a custom text editor, such as Kile, Lyx and TeXshop.
To make diagrams, I recommend the GIMP and/or Inkscape, both of which are free and available for Windows, MacOS and Linux.
You can also draw images in LaTeX itself using macro packages like PGF and TikZ or LaTeX's native picture environment
If you want to make presentations for giving talks, the Beamer package is included with most LaTeX distributions, but can be downloaded separately if needed.

To learn the LaTeX typesetting language, the usual thing is to start with a sample document like this one and play with it to see how it works. There are also excellent LaTeX manuals which will walk you thorough the features of the language from the basic to the obscure. A freely downloadable reference guide is here, while the canonical guide is here. Many more examples are available at texample.net.