Working with Lists

Author: Franco Saliola <saliola at gmail.com>

To create a list of objects, use square brackets.

1. Create the list [63, 12, -10, 'a', 12] , assign it to the variable L , and print the list. (Hint : Variable assignment in Sage/Python is done with = . For example, a = 3 defines the a to be 3.)

   {{{id=0| # edit here /// }}}

2. Use the len command to find the length of the list L.

   {{{id=1| # edit here /// }}}

3. To access an element of the list, use the syntax L[i] , where i is the index of the item. What is L[3]?

   {{{id=2| # edit here /// }}}

4. What is L[1]?

   {{{id=3| # edit here /// }}}

5. What is the index of the first item of L?

   {{{id=4| # edit here /// }}}

6. What is L[-1], L[-2]?

   {{{id=5| # edit here /// }}}

7. Access the last item in L.

   {{{id=6| # edit here /// }}}

An important concept about lists is that (like dictionaries, but unlike many objects in Sage), they can be directly changed. This property is known as mutability.

1. Change L[3] to 17.

   {{{id=7| # edit here /// }}}

This concept can lead to some confusion at first. See if you can guess what the output of the following commands will be.:

{{{id=8| a = [1,2,3] b = a b[0] = 7 print a, b /// }}}

This result makes sense when you understand that a and b are both labels attached to the same list. Compare that result with the following.:

{{{id=9| a = 2 b = a b = b + 1 print a, b /// }}}

In this case, we changed the object that b is attached to (to the object 2 plus the object 1, which is the object 3), while a continues to be attached to the object 2. This concept will be useful to keep in mind, as we discuss some methods which can be used to modify lists.

1. By typing L.<tab key>, you get a list of methods for L. Use one of these methods to append 17 to the end of L.

   {{{id=10| # edit here /// }}}

2. Insert the letter ‘b’ at index position 2 (do not change the element in position 2, but add a new element).

   {{{id=11| # edit here /// }}}

3. Remove the second occurrence of from L.

   {{{id=12| # edit here /// }}}

4. Redefine L to be the list [3, 1, 4, 1, 5, -1, 0].

   {{{id=13| # edit here /// }}}

5. Reverse the list L.

   {{{id=14| # edit here /// }}}

6. Sort the list L.

   {{{id=15| # edit here /// }}}

1. Guess the result of the following commands.

   {{{id=16| L = [3, 1, 2] M = L.sort() print L, M /// }}}

2. Now try the following.

   {{{id=17| L = [3, 1, 2] M = sorted(L) print L, M /// }}}

The range command

The range command provides an easy way to construct a list of integers.

1. Read the documentation (type: range? and hit enter or tab). Use it to create the list .

   {{{id=18| # edit here /// }}}

2. Create the list of even numbers between 1 and 100 (including 100).

   {{{id=19| # edit here /// }}}

3. The step argument in the range command can be negative. Use range to construct the list .

   {{{id=20| # edit here /// }}}

4. Sage (but not Python!) includes syntax to simplify creating lists like the above easier. What is the output of the command [2, 4, .., 100] ?

   {{{id=21| # edit here /// }}}

5. Create the list using Sage’s special syntax. Compare this with the output of range(1,5,0.5) .

   {{{id=22| # edit here /// }}}

List Comprehensions

We already know how to create the list :

{{{id=23| range(1,11) /// [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] }}}

Using a list comprehension, we can now create the list

{{{id=24| [i^2 for i in range(1,11)] /// [1, 4, 9, 16, 25, 36, 49, 64, 81, 100] }}}

Exercises:

1. Create two lists:

   

   {{{id=25| # edit here /// }}}

2. Use a list comprehension to construct the list

   

   {{{id=26| # edit here /// }}}

3. Using a list comprehension and the command sum, compute

   

   {{{id=27| # edit here /// }}}

Project Euler Problem 6

The sum of the squares of the first ten natural numbers is:

The square of the sum of the first ten natural numbers is:

Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is

Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.

{{{id=28| # edit here /// }}}

Filtering lists with a list comprehension

A list can be filtered using a list comprehension. For example, to create a list of the squares of the prime numbers between 1 and 100, we use a list comprehension as follows:

{{{id=29| [p^2 for p in [1,2,..,100] if is_prime(p)] /// }}}

Exercise: Use a list comprehension to list all the natural numbers below 20 that are multiples of 3 or 5. Hints:

• To get the remainder of 7 divided by 3 use 7 % 3.
• To test for equality use two equal signs (==); for example, 3 == 7.

{{{id=30| # edit here /// }}}

Project Euler Problem 1

If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.

Find the sum of all the multiples of 3 or 5 below 1000.

{{{id=31| # edit here /// }}}