First, I'll say a bit about how I use Sage: in the first place, I use it for my own research, and also use it for my teaching -- Mostly I use it in my lectures to do demonstrations, hopefully will incorporate Sage into the homework.

I can install it anywhere I want -- my office computer, my laptop, the computer in the lecture hall, anywhere. I spend exactly zero time wondering if what I'm doing is allowed by the site license, wondering when we'll get the next version, and so on. Free means my students can use it. Everyone can download and use Sage, so I can take a demo from class, publish it, and *everyone* they can look at it after class and interact with it themselves, instead of just watching me fiddle with it. Sure, Mathematica has their "Player" application, but with Sage, my students can easily alter the demo for themselves, which isn't possible with the Mathematica Player.

Sage is built out of nearly 100 open-source packages and features a unified interface. Sage can be used to study elementary and advanced, pure and applied mathematics. This includes a huge range of mathematics, including 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. It combines various software packages and seamlessly integrates their functionality into a common experience. It is well-suited for education and research.

The user interface is a notebook in a web browser or the command line. Using the notebook, Sage connects either locally to your own Sage installation or to a Sage server on the network. Inside the Sage notebook you can create embedded graphics, beautifully typeset mathematical expressions, add and delete input, and share your work across the network.

The following showcase presents some of Sage's capabilities, screenshots and gives you an overall impression of what Sage is. The examples show the lines of code in Sage on the left side, accompanied by an explanation on the right. They only show the very basic concepts of how Sage works. Please refer to the documentation material for more detailed explanations or visit the library to see Sage in action.

Video Introduction to Sage by William Stein — first video of several explaining Sage, better than a bunch of screenshots...

• Quickstart with Sage — explains basic concepts and gives you an impression of what Sage is.

• Sage Graphics — shows examples of useful plots and visualizations.

• Sage in Research — presents features useful for higher mathematics.

• Sage Benchmarks — compares speed or memory usage of Sage with other systems.

• Testimonials, Success Stories and News — read what others write about Sage.

• Screenshots — they give an impression how a Sage notebook looks like.

• Pictures — collection of images drawn by Sage.

• Interact — examples of Sage's interactive drawing capabilities.