# Why SageMath?¶

## Choosing the best ICT tool for a math or physics lesson¶

For many years, attempts have been made to use ICT tools and programming in science classes. Often the software solution has been is selected by professionals in one specialized field. Sometimes the choice is lobbied by the manufacturer of the given system. As a result, this leads to uncoordinated activities limited to individual subjects. A student learns in lessons informatics tools and languages not useful during other subjects. The lessons of physics and math are enriched with dedicated software that is not used in computer science. This is in inherently wrong procedure - we use the specialized tools to individual tasks. But, what if there is a general purpose tool and language which can be practically without a compromise used in a range of tasks in school education?

Let’s consider what features a computer system should have to break the stereotype above? Let’s find a solution at once having the following characteristics:

1. WIDE: The system should be based on popular and open language wide-ranging programming.

Wide-ranging programming languages can be used make computer games as well as scientific applications or education. On the other hand, there are many so-called domain-specific languages created for a single application. Such languages do a great job, but they are not good at all other tasks. An example is the Matlab language, which despite its own popularity is not the solution adopted in teaching computer science. Languages like Python allow practically perform all tasks which are specific to Matlab, but their specificity allows admit them to teach computer science (see https://docs.scipy.org/doc/numpy-dev/user/numpy-for-matlab-users.html). Important feature of the system is avoiding vendor lock-in, which is often the case when using domain specyfic languages.

2. INTERACT: Programming language should allow for interactive work.

Such a claim virtually eliminates compiled languages (C/C++). In order to use the computer system interactively, the most adequate seem to be languages with dynamic typing and introspection. This requirement is met by the most of domain specific languages provided by Computer Algebra Systems, but also many general-purpose languages such as Python.

3. FREE - The system should be widely available.

Unrestricted access to the system is best exploited by the open software. In addition, the open software give the possibility of the insight into every algorithm used. It is important both in science and education. Availability is also related to the technical aspects of the installation software. Opportunity to work in “cloud” with only internet browser is very desirable feature of such a system.

4. POWER - The capabilities of the system should allow to deploy it uncompromisingly for all scientific subjects.

Such a requirement eliminates languages that are not enough widespread, and do not have large enought set of implemetned tools and libraries. Python is an interesting example, because it is known for its ease of creating interfaces various libraries written in other languages. This feature is heavily exploited by the SageMath system which contains hundreds of scientific libraries linked by common interface in Python.

5. PROF - The system should allow a smooth transition from school computer lab to professional use in scientific research and industrial.

There is no reason to teach at a “small” system school at school, and then study or work using the “big”-one. The best way to go would be to the even in primary school the same language and system, which is used by scientists, but of course limit this usage a small part of it. It saves a lot of time and effort halpe to make good habits right from the earliest period of study. It should be emphasized that often the cost of software licenses for proprietary systems are urging to use simpler and smaller one in schools. This problem does not exist when we relay on open software.

 System WIDE INTERACT FREE POWER PROF SageMath YES YES YES YES YES Python/Scipy YES YES YES YES YES Mathematica YES YES NO YES YES C/C++ YES NO YES YES/NO YES Geogebra NO YES YES NO NO Java YES NO YES YES/NO YES

The above analysis shows that solutions based on Python meet all requirements. Moreover, Python is a language of increasing importance in computer science. Both the standard Python interpreter and the SageMath algebra system can provide similar capabilities. Definitely in the Maths or Physics class, SageMath will offer a shorter path to solution as a computer algebra system. But before we discuss these systems let’s answer the question of what is a computer algebra system?

### What is a Computer Algebra System?¶

Computer Algebra System (CAS) is a computer program that supports symbolic calculations. Consider, for example, the following code in Python:

This program will print an approximate value for the expression after Substituting the variables: math:a = 23, b = 3: 1.0197. If we do not perform the first two substitutions the interpreter will signal an error.

The situation is different in case of CAS system. Here only we inform the system that the variables: math: a, b will be symbols and we can Expand the algebraic expression containing these symbols. For example, executing:

We will get an algebraic expression.

Modern computer algebra systems are not limited to the manipulation of mathematical formulas. As a rule, they are equipped with a numerical computing system and a rich set of visualization tools. As of today most of the possibilities of CAS systems are similar and the main differences are the programming language and the license for the software.

The proposed approach is based on SageMath, which is a free and open source software. This eliminates the cost of licenses. In addition, SageMath uses the popular Python language, which students can learn during IT lessons.

## What is SageMath? (from Python to SageMath)¶

### Python¶

Python has been developing since the nineties in the last century. However, its ubiquitous popularity has started in last decade. In the United States most programming projects is writen in this language. Python has extensive standard libraries and is characterized by the clear and concise syntax. Importantly Python supports different programming methods: procedural, object oriented and functional. Thanks to these advantages, Norway is the first country systematically introduced that programming language into schools.

### Ecosystem Scipy¶

Python is a language widely used for scientific research and education. The most known set of tools is called scipy ecosystem. It contains:

• NumPy, the basic package for numerical calculation similar in the philosophy and functionality to well known Matlab software
• the SciPy library of numerical methods
• Matplotlib, a graphing package
• SymPy, symbolic computation library (CAS)

### SageMath¶

SageMath is a complete Computer Algebra System. First version of SageMath was released on February 24, 2005 as free and open source software in accordance with the terms of the GNU General Public License. One can say that Sage is an “overlay” on Python, which integrates many specialized mathematical packages and hundreds of thousands unique line of code to add new features. Capabilities and the flexibility of SageMath is immeasurable, so it is worthwhile to implement the above programming language also in school. It is not without significance that this is an open source software and as such free. Teachers and students can access the platform at any time and place, if they only have access to the internet.

### Ecosystem Scipy vs SageMath¶

The SageMath computer algebra system is a huge collection of tools and it includes, among other things, tools from the Scipy ecosystem. The essential difference is, however, a common interface for all tools. Way using SageMath is optimized for interactive work and convenience of mathematicians. Running SageMath one has a Python 2.7 interpreter available with two key differences:

1. Each command is processed by the so-called preparser before
will be sent to Python interpreter. Preparser changes input in the following way:
• replaces the power of 2^3 to the Python syntax 2**3
• literals like e.g.: 1 and 1.0 are transformed to constructors: Integer(1) and RealNumber(1.0) respectively
2. About 2000 useful functions are automatically loaded like plot,simplify, etc., and a symbolic variable x is predefined.

Therefore, for example, to solve a square equation in SageMath, it is enough write solve(x^2 + 2*x + 1 == 0, x) and we will get the answer. The same can be accomplished in “pure” Python but one needed to load the appropriate modules and define the variable and only then proceed to the proper computations.

These advantages of SageMath have prompted us to apply this system in physics, mathematics and chemistry classes. However, it should be noted that usingf SageMath is actually Programming in Python and if the students get this skill during IT lessons then there will be only a small threshold to be overcomed for effective application of SageMath system for e.g. mathematics or physics. As a result, the solution is based on the SageMath system will provide a very efficient tool with very small overhead and will reuse potential students skills in Python programming.