This page does not attempt to cover all the different options for running LaTeX on your PC. I discuss a couple of configurations that I'm familiar with. They're easy and simple, and they'll probably meet your needs. But chances are that if you look elsewhere, you'll find another configuration you like.
TeX/LaTeX can run on any hardware and OS in common use today (and quite a few exotic ones), and is free to use and distribute. Numerous "distributions" of latex are available over the internet or on software distribution CDs (for example, as part of Linux distributions). There are also commercial add-ons for various purposes (which I won't talk about here).
To use LaTeX, you basically need to take care of two things:
The TeX/LaTeX system: The style and font files (which are mostly system-independent) and the formatting and viewing software (which is system-dependent, but generally ported to multiple OSs).
How best to take care of these two things depends on your system, but also on your background and purposes.
Let's start with the easy cases:
Linux distributions come with LaTeX, so if you're running (or thinking of running) Linux on your computer, you're on easy street. (Just look through the software packages in the distribution disk and add the TeX/LaTeX bundle to the installed packages, because it will probably not be installed by default).
I don't use a Macintosh so I know very little about this, but I believe that latex is very well supported. Take a look at one of the following sites for more information.
The TeXShop homepage (A TeX/LaTeX editor/front end for the Macintosh)
As explained above, you need a TeX distribution and a suitable editor for your documents. Let's take these up one at a time.
I use the MikTeX distribution, which is popular and works well. Other distributions are probably just as good, I just haven't tried them under Windows. From the MikTeX Home Page you'll download an installer program, which you then run to configure and fetch the TeX system. Go through all the steps before you install a front end.
Since latex files are plain-text, you can edit them using any editor you want, including Microsoft Word or Notepad. (I've often done that when editing a file on a computer that does not have TeX installed). However, it is far more convenient to use one of the many "latex-aware" editors, or front ends, which facilitate editing and can call the latex formatting and viewing programs for you. (The editor will show you the source document; it will help you open an external viewer, usually Acrobat or Yap, to view the formatted version of your document). Here are just three choices that I know about. I recommend TeXnicCenter, unless you know that one of the others fits your needs better.
TeXnicCenter This GUI-oriented front end is simple and straightforward. Install MikTeX before you install it; TeXnicCenter will detect an existing MikTeX installation and configure itself to call the right programs when you press the "Build" or "View" buttons. It gives you a simple Word-like editor with buttons that insert formatting tags (e.g., for boldface or for math symbols), so it's nice if you're new to LaTeX and don't know the latex names for various things.
WinEdt Is another GUI-oriented editor/front end. This one is extensively (and laboriously) configurable. I haven't used it but have heard good things about it. If you outgrow TeXnicCenter and don't fancy typing control-key combinations in emacs, you could give it a try.
emacs The ubiquitous editor for unix also runs on Windows. Unix enthusiasts sometimes talk their friends into trying out latex, and then advise them to edit their latex file with emacs. Unless you're already familiar with emacs, I advise against it: Learning LaTeX is work enough, why complicate it with learning emacs? Use one of the many GUI-oriented front ends available. (Besides, let's admit it: emacs is great, but it's a funky old beast :-).
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