This document is one of More SageMath Tutorials. You may edit it on github. \(\def\NN{\mathbb{N}}\) \(\def\ZZ{\mathbb{Z}}\) \(\def\QQ{\mathbb{Q}}\) \(\def\RR{\mathbb{R}}\) \(\def\CC{\mathbb{C}}\)
Finding help¶
Here are several ways of getting help from within Sage.
Tab completion¶
Does Sage have a command for defining a permutation? (Hint: Start typing
Perm and then hit the tab key.)
sage: Perm
?: documentation and examples¶
To see documentation and examples for the Permutation command, type
Permutation? or Permutation( and hit tab (or enter).
sage: Permutation
Exercises:
Create the permutation \(51324\) and assign it to the variable
p.sage: # edit here
Find the inverse and the length of
p. (Hint: to see the methods available top, you can type ‘p.‘ and hit tab.)sage: p.
Does
phave the pattern \(123\)? What about \(1234\)? And \(312\)?sage: p.
??: get source code¶
To see the how the inverse of p is computed, type p.inverse?? and
hit tab (or enter).
sage: p.inverse()
Searching documentation¶
There are other ways to get help.
Click on Help on the top right of this page.
Use the command ‘search_doc’:
sage: search_doc()
Use the command ‘search_src’:
sage: search_src()
Use the command ‘search_def’:
sage: search_def()
Exercises:
Use ‘search_doc’ to find information about Taylor series, then define the function \(f(t) = sin(t)\) and find its Taylor series expanded about \(t=0\) up to degree \(14\).
sage: search_doc()
Can you guess an expression for the \(n\)-th term of the Taylor series of \(f\)? (Hint: you might find the command
sloane_finduseful in finding an expression for the denominators.)sage: sloane_find()
Project Euler¶
Several of your exercises will from from the Project Euler website:
Project Euler is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems.
Each problem has been designed according to a “one-minute rule”, which means that although it may take several hours to design a successful algorithm with more difficult problems, an efficient implementation will allow a solution to be obtained on a modestly powered computer in less than one minute.
Exercise: Go to the Project Euler website ( www.projecteuler.net ) and create an account.
Project Euler Problem 3¶
The prime factors of \(13195\) are \(5\), \(7\), \(13\) and \(29\).
What is the largest prime factor of the number \(600851475143\)?
(After you solve this problem, visit the Project Euler website and enter your answer. Visit the forums and read some of the other solutions. Pick one that you like best.)
Project Euler Problem 5¶
\(2520\) is the smallest number that can be divided by each of the numbers from \(1\) to \(10\) without any remainder. What is the smallest number that is evenly divisible by all of the numbers from \(1\) to \(20\)?
(After you solve this problem, visit the Project Euler website and enter your answer. Visit the forums and read some of the other solutions. Pick one that you like best.)