How To Use Sagemath

Leo Migdal

-Nov 17, 2025, 4:12 AM

Sage is free, open-source math software that supports research and teaching in algebra, geometry, number theory, cryptography, numerical computation, and related areas. Both the Sage development model and the technology in Sage itself are distinguished by an extremely strong emphasis on openness, community, cooperation, and collaboration: we are building the car, not reinventing the wheel. The overall goal of Sage is to create a viable, free, open-source alternative to Maple, Mathematica, Magma, and MATLAB. This tutorial is the best way to become familiar with Sage in only a few hours. You can read it in HTML or PDF versions, or from the Sage notebook (click Help, then click Tutorial to interactively work through the tutorial from within Sage). This work is licensed under a Creative Commons Attribution-Share Alike 3.0 License.

This tutorial is divided into four parts. This part, About this tutorial, discusses some basic properties of SageMath, introduces you to the structure of the tutorial, and explains how to contribute to the project if you so desire. The second part, SageMath as a Calculator, covers topics such as how to do arithmetic, evaluate functions, create simple graphs, solve equations and do basic calculus. We call this section SageMath as a Calculator because most of the topics covered are those that are commonly done with a standard graphing calculator. The target audience for this section is any motivated pre-calculus or calculus student. Programming in SageMath introduces the reader to some more advanced topics such as how SageMath handles numbers; how to define and use variables and functions; how to manipulate lists, strings, and sets; and SageMath...

The final part, Mathematical Structures, introduces the reader to topics that one finds in a college-level curriculum: linear algebra, number theory, groups, rings, fields, etc. Since this tutorial is an introduction to SageMath, we will be using examples to demonstrate ideas and the reader is encouraged to follow along as we progress by entering the commands into their own... We have included exercises for practice and problems for more extensive exploration of a given topic. The reader is also encouraged to do many of these. This Sage document is the first in a series of tutorials developed for the MAA PREP Workshop “Sage: Using Open-Source Mathematics Software with Undergraduates” (funding provided by NSF DUE 0817071). It is licensed under the Creative Commons Attribution-ShareAlike 3.0 license (CC BY-SA).

If you are unsure how to log on to a Sage server, start using a local installation, or to create a new worksheet, you might find the prelude on logging in helpful. Otherwise, you can continue with this tutorial, which has the following sections: See Evaluating in the Jupyter notebook for the Jupyter notebook See Jupyter Annotation for the Jupyter notebook SageMath is mainly a computer algebra system based on Python, which is very popular in discrete mathematics. SageMath is a free open-source software.

As a "consequence", not all of the documentations are written for generic computer users (see this meme). The main aim of this article is an attempt of making up the gaps. You can have a quick try of SageMath online via SageCell. It also provides a way to embed Sage computations into any webpage like this. You can also use CoCalc in the cloud (online). When using it, it is recommended to start with creating a Jupyter Notebook with kernel the lastest SageMath.

If you do hope to download and install SageMath, it will be complicated. Here are some useful links providing some hints: Windows, Binaries, Sources. After installation, it is strongly suggested to use the shortcut "SageMath n.m Notebook" which links to a Jupyter Notebook. There are two official introductory documentations: This part of the tutorial examines commands that allow you to use SageMath much like a graphing calculator. The chapter on arithmetic and functions and the chapter on solving equations and inequalities serve as a foundation for the rest of the material.

The chapters on plotting, statistics and calculus are independent of each other, although plotting may be useful to read next since plotting graphs is so is useful in calculus and in statistics. The basic arithmetic operators are +, -, *, and / for addition, subtraction, multiplication and division, while ^ is used for exponents. The - symbol in front of a number indicates that it is negative. As we would expect, SageMath adheres to the standard order of operations, PEMDAS (parenthesis, exponents, multiplication, division, addition, subtraction). When dividing two integers, there is a subtlety: SageMath will return a either a fraction or its decimal approximation. Unlike most graphing calculators, SageMath will attempt to be as precise as possible and will return the fraction unless told otherwise.

One way to tell SageMath that we want the decimal approximation is to include a decimal in the expression itself. Sage can perform various computations related to basic algebra and calculus: for example, finding solutions to equations, differentiation, integration, and Laplace transforms. See the Sage Constructions documentation for more examples. In all these examples, it is important to note that the variables in the functions are defined to be var(...). As an example: If you get a NameError, check to see if you misspelled something, or forgot to define a variable with var(...).

The solve function solves equations. To use it, first specify some variables; then the arguments to solve are an equation (or a system of equations), together with the variables for which to solve: You can solve equations for one variable in terms of others: Sage is free, open-source math software that supports research and teaching in algebra, geometry, number theory, cryptography, numerical computation, and related areas. Both the Sage development model and the technology in Sage itself are distinguished by an extremely strong emphasis on openness, community, cooperation, and collaboration: we are building the car, not reinventing the wheel. The overall goal of Sage is to create a viable, free, open-source alternative to Maple, Mathematica, Magma, and MATLAB.

This tutorial is the best way to become familiar with Sage in only a few hours. You can read it in HTML or PDF versions, or from the Sage notebook (click Help, then click Tutorial to interactively work through the tutorial from within Sage). This work is licensed under a Creative Commons Attribution-Share Alike 3.0 License.

How To Use Sagemath

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