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First steps towards programming

In this worksheet, you will learn to define functions, write for loops, conditional statements and and continue to learn about lists.

Functions

To define a function in Sage, use the def command and a colon after the variable names. For example:

sage: def is_even(n):
....:     return n % 2 == 0

Notice that body of the function (the line: return n % 2 == 0 ) is indented. The indentation defines the body of the function.

Exercises:

  1. Define a function called square that returns the square of a number:

    sage: # edit here

If statement

Below is an example of an if statement:

sage: if n % 2 == 0:
....:     return True
....: else:
....:     return False

Notice again how the commands in the first block and the second block of the statement are indented.

The following example defines the factorial_function , which takes a number \(n\) and returns the product \(n(n-1)(n-2)\cdots1\):

sage: def factorial_function(n):
....:     if n == 0:
....:         return 1
....:     elif n == 1:
....:         return 1
....:     else:
....:         return n*factorial_function(n-1)

Exercises:

  1. Define a function sign that returns the sign of a number:

    sage: # edit here

For loops

The following example uses a for loop to create a list of all the numbers between 1 and 1000 that are multiples of 3 and 5:

sage: nums = []
sage: for i in range(1,1001):
....:     if i % 3 == 0 and i % 5 == 0:
....:         nums.append(i)
sage: print nums
[15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 210, 225, 240, 255, 270, 285, 300, 315, 330, 345, 360, 375, 390, 405, 420, 435, 450, 465, 480, 495, 510, 525, 540, 555, 570, 585, 600, 615, 630, 645, 660, 675, 690, 705, 720, 735, 750, 765, 780, 795, 810, 825, 840, 855, 870, 885, 900, 915, 930, 945, 960, 975, 990]

Exercises:

  1. Recall that the Fibonacci sequence is the sequence of numbers that begins with \(F_0 = 0\), \(F_1=1\), and that satisfies the equation \(F_n = F_{n-1} + F_{n-2}\) for all \(n\geq2\). Define a function that returns a list of the first m terms in the Fibonacci sequence:

    sage: # edit here

Project Euler Problem 2

Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be:

\[1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \ldots\]

Find the sum of all the even-valued terms in the sequence which do not exceed four million.

sage: # edit here

Slicing lists

You can slice a list to obtain only part of it. The syntax is L[start:stop:step].

  1. Let L = range(100), and try the following L[0:3], L[:3], L[1:], L[1:-1], L[::2].

    sage: # edit here

  2. Create a list L.

    sage: # edit here

  3. Use a slice to obtain the reversal of L .

    sage: # edit here

  4. Revese the list L using L.reverse() .

    sage: # edit here

  5. What is the difference between these methods of reversing a list?

    sage: # edit here

Project Euler Problem 4

A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is \(9009 = 91 \times 99\).

Find the largest palindrome made from the product of two 3-digit numbers.

Hints:

  • 7%3 returns the remainder of 7 divided by 3.
  • 7//3 returns the integer quotient of 7 by 3.
sage: # edit here