Tutorial Release 10.4 The Sage Development Team https://doc.sagemath.org/pdf/en/tutorial/sage_tutorial.pdf

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.

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tutorials/CoCalc's Official Sage Tutorial / Chapter One / Chapter One (Introduction).ipynb

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Kernel: SageMath 10.4

CHAPTER ONE

INTRODUCTION

This tutorial should take at most 3-4 hours to fully work through. 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.

Though much of Sage is implemented using Python, no Python background is needed to read this tutorial. You will want to learn Python (a very fun language!) at some point, and there are many excellent free resources for doing so: the Python Beginner’s Guide lists many options. If you just want to quickly try out Sage, this tutorial is the place to start. For example:

In [26]:

2 + 2

Out[26]:

4

In [27]:

factor(-2007)

Out[27]:

-1 * 3^2 * 223

In [28]:

A = matrix(4,4, range(16)); A

Out[28]:

[ 0  1  2  3]
[ 4  5  6  7]
[ 8  9 10 11]
[12 13 14 15]

In [29]:

factor(A.charpoly())

Out[29]:

x^2 * (x^2 - 30*x - 80)

In [30]:

m = matrix(ZZ,2, range(4))
m[0,0] = m[0,0] - 3
m

Out[30]:

[-3  1]
[ 2  3]

In [31]:

E = EllipticCurve([1,2,3,4,5]);
E

Out[31]:

Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field

In [32]:

E.anlist(10)

Out[32]:

[0, 1, 1, 0, -1, -3, 0, -1, -3, -3, -3]

In [33]:

E.rank()

Out[33]:

1

In [34]:

k = 1/(sqrt(3)*I + 3/4 + sqrt(73)*5/9); k

Out[34]:

36/(20*sqrt(73) + 36*I*sqrt(3) + 27)

In [35]:

N(k)

Out[35]:

0.165495678130644 - 0.0521492082074256*I

In [36]:

N(k,30)

Out[36]:

0.16549568 - 0.052149208*I

In [37]:

latex(k)

Out[37]:

\frac{36}{20 \, \sqrt{73} + 36 i \, \sqrt{3} + 27}

In [38]:

from sage.all import *
Integer(2) + Integer(2)

Out[38]:

4

In [39]:

factor(-Integer(2007))

Out[39]:

-1 * 3^2 * 223

In [40]:

A = matrix(Integer(4),Integer(4), range(Integer(16))); A

Out[40]:

[ 0  1  2  3]
[ 4  5  6  7]
[ 8  9 10 11]
[12 13 14 15]

In [41]:

factor(A.charpoly())

Out[41]:

x^2 * (x^2 - 30*x - 80)

In [42]:

m = matrix(ZZ,Integer(2), range(Integer(4)))
m[Integer(0),Integer(0)] = m[Integer(0),Integer(0)] - Integer(3)
m

Out[42]:

[-3  1]
[ 2  3]

In [43]:

E = EllipticCurve([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]);
E

Out[43]:

Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field

In [44]:

E.anlist(Integer(10))

Out[44]:

[0, 1, 1, 0, -1, -3, 0, -1, -3, -3, -3]

In [45]:

E.rank()

Out[45]:

1

In [46]:

k = Integer(1)/(sqrt(Integer(3))*I + Integer(3)/Integer(4) + sqrt(Integer(73))*Integer(5)/Integer(9)); k

Out[46]:

36/(20*sqrt(73) + 36*I*sqrt(3) + 27)

In [47]:

N(k)

Out[47]:

0.165495678130644 - 0.0521492082074256*I

In [48]:

N(k,Integer(30))

Out[48]:

0.16549568 - 0.052149208*I

In [49]:

latex(k)

Out[49]:

\frac{36}{20 \, \sqrt{73} + 36 i \, \sqrt{3} + 27}

CHAPTER ONE

INTRODUCTION

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