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updated 3/6/13

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Lyx 
Lyx
is software designed for typing beautiful and professional looking
math. Internally, it uses
a system called LaTeX, a math typesetting program. LaTeX gives
you the ability to type integral signs, square roots,
summations, etc. It can handle just about any math notation that
you need. It is also designed to handle the formatting for
you. You type the text, and LaTex formats it automagically.
However, LaTeX requires the user to know a bunch of programming code in
order to use those symbols. Lyx provides a graphical user
interface so that you don't need to learn the codes. In other
words, it provides
nice looking menus for you to use. It's like the equation editor
from Word on steroids. Note that you can do much more with Lyx
than just pretty math type. It's great for organizing a document
with titles, subtitles, theorems, proofs, corollaries, equations,
tables, figures, etc. That may not sound like much until you have
to start numbering everything. As much as I like Word's numbering
system, it's pretty limited. Lyx can keep track of multiple
number systems in the same document. For example, you may want to
number exercises, theorems, and figures separately. Lyx does it
for you! This software comes highly recommended. 
Equation Editor 
The
equation editor in MSOffice is often overlooked as a piece of math
software, but the version Microsoft introduced in 2007 and 2010 is
quite robust. For
the casual mathematician, the equation editor is often all you will
need to type high quality mathematics. It provides a user interface for
most math symbols an amateur
mathematician would want. It supports keyboard shortcuts for many
more of them (most are the LaTex codes which is nice feature).
The link to the
left is my list of keyboard shortcuts for the symbols that I use the
most. It's not exhaustive, but it's a good start. The biggest
feature it lacks is a number system to number your equations. 
wxMaxima Maxima 
Maxima
is a computer algebra system that is designed to manipulate symbolic or
numeric expressions. For instance, that means it can
differentiate and
integrate functions. It can solve systems of equations It
has built in graphing capability for 2 and 3 dimensions. If
you've ever used Maple or Mathematica, it is a poor man's version of
that. It doesn't have near the feature set of the two big
players, but what it does, it does well. And of course, it does
it
free! As it is a CAS, it does have a steep learning curve (albeit
with a poor help menu).
It tries to alleviate that with some panels on the left to help the
user. Essentially, it's a privative GUI so that you don't have to
know the commands. Note that on
a Windows machine, you will actually run wxMaxima which is a shell for
running Maxima (which is cross platform). The program wxMaxima
will install Maxima automatically. From Maxima's website, you can
download a version for just about any system. 
Sage 
Sage
is a multipurpose math environment. It combines many existing
math software packages into one easy to use interface. To see the
full list of almost 100 packages, click here.
I have just started using it as I wanted a 3D plotter for Vector
Calculus. Being a complete math environment, the learning curve
is fairly steep. Google seems to be the best way to find examples
of how
to get Sage to do what you want. Once you learn how to do it,
it's
not hard. It just takes time, but I think it is worth the effort. From their website, it supports "basic algebra, calculus, elementary to very advanced number theory, cryptography, numerical computation, commutative algebra, group theory, combinatorics, graph theory, exact linear algebra and much more". Note that it runs through the web so it is cross platform with no need to install. However, you will need to create an account with them to use it. However, if you have a google account or a yahoo account, you can also use that. If you want to, you do have the ability to set it up on a server and run a local copy on your own network. 