In this short article I propose well known programs that will give you a lot of flexibility and math crunching fun. All of them have advantages and drawbacks and none of them can be considered perfect or infallible, but I consider them some of the best available today in their respective categories. They are rather general purose softwares, but there are plenty of other specialized open source programs if you have specific needs.
I’ve chosen one program for each of the 3 macro categories: symbolic, numeric and statistical computing, but you can expect quite a bit of overlapping and shared functionalities. Try the three of them, try the suggested alternatives and settle with the ones that you like and that meet your needs the best.
Follow the link to see his selections. I hope to try these out now that I have my field exam out of the way.