Free, mathematical syntax that includes visualization tools and plotting
GNU Octave is an advanced programming language capable of meeting the demands of a wide range of scientific and mathematical disciplines. The math that the typical person learns in high school is far more complicated than simple arithmetic and even advanced formats like trigonometry and calculus. The development of computers have allowed mathematicians the means to work with numbers on a scale they never could, and that's only grown as technology has developed. Scientific programming languages allow you to take high concept mathematics and see how they work in practice through the use of machine learning, and GNU Octave is one of the best of these frameworks available. Since its conception in 1988, it's grown exponentially, and it currently serves as one of the preeminent scientific programming languages around. GNU Octave is capable of high end numerical computations, but its actual application is far more versatile. It's found a significant degree of adoption in both academic and industrial circles as a result.
GNU Octave is built off of the fundamentals of C++, and that's a huge boon. Anyone familiar with the C languages should be able to jump in to using GNU Octave without a huge amount of fuss, and the wide adoption of C++ based languages means that it's more accessible than it might otherwise be. Even more important, it's largely compatible with Matlab. Since Matlab is one of the most popular mathematical programming languages around, this largely extends its value to experienced users. Scripts written in Matlab can largely be dropped right into a GNU Octave program without concerns that the results will bring up errors or contradictions.
The command line interface that GNU Octave uses supports both linear and nonlinear mathematics, and it comes with a range of very useful tools that can accommodate amateur mathematicians while also providing all the utilities you'd need for most advanced mathematical computations. These include support for differentials and differential-algebraic equations as well as the manipulation of polynomials. To make things easier and more adaptable to the needs of the user, GNU Octave can be run via the console, as a standalone application, or as the component of a shell script.
One of the main aspects that's allowed GNU Octave to have such a long shelf life is the fact that it's both free and open source. The genesis of the language was humble, intended as a complementary component of a chemistry class, but 30 years of tinkering and a developing community of developers have transformed it into far much more, and the language has grown in some important and fundamental ways since its early days. But make no bones about it. This is a difficult language to learn, and anyone without experience in associated language can expect to spend a long time learning its ins and out.