Variables, conditionals and loops

Variables, conditionals and loops

After this meeting students should:

• Understand the coursework exercise and the group aspect
• Understand the how to use a for loop and a while loop
• Know what they need to do to prepare for their seventh tutorial and how to make groups.

Coursework

Explain goal of coursework:

To build and present a Python library to solve a mathematical problem of your choice.

How it will be assessed (15 minute presentation and 2 page (max) paper).

Groups of 4. Self select and see instructions in email.

I recommend you meet regularly.

Have an agenda for every meeting.

Single point of contact.

Problem

Explain to students that we will be solving the following problem:

A perfect number is a positive integer whose divisors (excluding itself) sum to
itself. For example, :math:`6` is a perfect number because :math:`6 = 1 + 2 +
3`.

1. How many perfect numbers are there less than 20?
2. What is the next perfect number?

Solution

Group exercise (breakout rooms of 3): ask students to spend 5 minutes writing a plan to tackle that problem (not necessarily carrying out each step).

We start by writing a function that checks if a number is perfect::

>>> def is_perfect(n):
...     """
...     Checks if a number n is perfect.
...     """
...     return sum(i for i in range(1, n) if (n % i == 0)) == n
>>> is_perfect(5)
False
>>> is_perfect(6)
True

Now we can check all numbers up until 20::

>>> checks = [is_perfect(n) for n in range(1, 21)]

Note that we can combine all booleans using all::

>>> all(checks)
False

Or any::

>>> any(checks)
True

We cam also count them (as before) using sum::

>>> sum(checks)
1

We are there using a list comprehension (as before) but we can also use a for loop::

>>> count = 0
>>> for n in range(1, 21):
...     if is_perfect(n):
...         count += 1
>>> count
1

Discuss how the for loop works.

Now to answer the other question, we do not know how many times we need to iterate the code so we use a while loop::

>>> n = 7
>>> while not is_perfect(n):
...     n += 1
>>> n
28

Indeed the prime divisors of 28 are::

>>> import sympy as sym
>>> sym.primefactors(28)
[2, 7]

And::

>>> 1 + 2 + 4 + 7 + 14
28

We can repeat the above to find the 3rd perfect number::

>>> n = 29
>>> while not is_perfect(n):
...     n += 1
>>> n
496

Indeed the divisors of 496 are::

>>> import sympy as sym
>>> sym.primefactors(496)
[2, 31]

And::

>>> 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
496

After class communication

Send the following email after class::

Hi all,

A recording of today's class is available at <>.

In this class I went over 2 separate things:

1. The group coursework (please read the end of this email where action is
   required).
2. Variables, loops and conditionals.

I recommend working through the following chapter of the class text:
https://vknight.org/pfm/building-tools/01-variables-conditionals-loops/introduction/main.html

IMPORTANT ACTION REQUIRED

For your group coursework you have until <DEADLINE> to form groups with 4
people. I am letting you self select groups. If you do not have a group by
<DEADLINE> I will create a group for you.

---

Please get in touch if I can assist with anything,
Vince

Some common difficulties

Create a set::

>>> set_1 = set((-1, 1, 1, 2, 3, 5))
>>> set_1
{1, 2, 3, 5, -1}
>>> set_2 = set(range(10))
>>> set_2
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

Union::

>>> set_1 | set_2
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, -1}

Intersection::

>>> set_1 & set_2
{1, 2, 3, 5}

Set subtraction::

>>> set_1 - set_2
{-1}

Set inclusion::

>>> set_1 <= set_2
False

Reasons for using sets:

• The ability to to the above operations.
• Immediate removal of duplicates.
• Checking inclusion.

Run the following timing examples::

N = 10 ** 4
numbers_as_range = range(N)
numbers_as_list = list(numbers_as_range)
numbers_as_tuple = tuple(numbers_as_range)
numbers_as_set = set(numbers_as_range)

Testing the list::

%%timeit
5000 in numbers_as_list

Testing the tuple::

%%timeit
5000 in numbers_as_tuple

Testing the set::

%%timeit
5000 in numbers_as_set

Testing the range itself::

%%timeit
5000 in numbers_as_range

Modulo ++++++

Give some examples of modula arithmetic::

>>> 5 % 2
1
>>> 17 % 9
8
>>> for i in range(24 * 5):
...     print(i % 24)
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