6. Loops

6.1. Reusing variables

Consider this code fragment:

a = 5
print(a)
b = 6
print(a)
a = a + b
print(a)
a = 3
a = a + 1
print(a)

Very Quick Activity

What is the value of the variable a at the various print function in the above code?

A very common pattern we’ll use is incrementing a variable used as a counter:
```
a = a + 1
```
This reminds me of the +1 thing we used to do on calculators in elementary school.
Try this a few times:
```
a += 1
```
What does this do?

Activity

So far, if we want Python to do the same thing over and over, we have to tell it explicitly by repeating those instructions over and over.
- There has to be a better way!
We want to automate the process of repeating things.
If I can put a block of instructions into a function and call that function…
… why can’t I put a block of instructions somewhere and say “Hey, do that block of instructions until I tell you to stop”?
The while statement allows us to do exactly this.
While some condition is true, keep doing the code in the indented block:
```
a = 1
while a < 11:
    print(a)
    a = a + 1
```
That code will print the numbers from 1 to 10. Take a minute to note three things:
- Before the while statement, we initialize the loop variable a
- The while statement is followed by a condition (which can be any boolean function/statement/expression!). If the condition is True, the body of the loop gets executed, otherwise it gets skipped. (don’t forget the : !)
- What would happen if we didn’t have a = a + 1?

Activity — Featuring LOOPS

Consider this code:

def do_stuff(n):
    answer = 1
    while n > 1:
        answer = answer * n
        n = n - 1
    return answer

Activity

What does the code above do? Trace through it, using pen and paper, for a few example values of n!

The pattern a = a + 1 shows up so often that Python permits a shorthand for it: a += 1. If you like the shorthand, use it. If you don’t: don’t. It’s not mandatory; just saves some typing.
while loops can get complicated quickly. Much of the time, it is by no means obvious what they do (if only the coder wrote comments).
If you’re faced with such a loop, trace through the execution of the loop by building a table of values.
Let’s trace do_stuff(4). We’ll look at the values of n and answer right after the while statement.

Activity

Write a function int_sum(n) that takes a single integer n as a parameter and returns the sum of all of the numbers between 1 and n.

Trace through your function for the call int_sum(5)

Activity

Modify int_sum(n) so that it prints out a Trace table, like the one you did by hand, every time it runs.

Don’t worry about formatting the table, just print out the values.

Activity

Find the solution to the equation (for what value of x is this statement true?):

No need to worry about degrees/radians here. Just use cos and sin.

Okay, that’s a tough one, so you get some help. How do we go about it?
Let’s use something called Newton’s Method .
Here’s what you do:
- Pick a value x between 0 and 1. Any will do. Seriously.
- Compute:
- The answer to that equation is an approximation of the solution
- It’s not a very good approximation yet. What to do?
- Set x equal to the new approximation and plug in to the formula again.
- New approximation.
- Still not good enough? Guess what?
- Set x equal to the new approximation and plug in to the formula again.
What you want to do is:
- write a function approx_x that, given an approximation for x, computes the formula I gave you
- write another function, that calls this function while x != approx_x

What you just saw, Newton’s method, is an example of an algorithm.
An algorithm is a description of a series of steps to solve a problem.
Algorithms can be presented in natural language, but are easier to turn into a program when presented in a formal language.
Finding an algorithm to solve most problems is very hard. You can make a career, get tenure, make millions of dollars in patent licensing, etc., “just” by developing algorithms.
As programmers though, we usually leverage existing algorithms and other things to make our lives easier.
The two most important concepts that will be introduced in this course (or really, what a computer scientist spends years learning) are:
- ALGORITHM
- DATA STRUCTURE

Activity

Write down (in English) an algorithm for printing out the sum of all the even numbers between 1 and n.

Now convert the algorithm into a Python function.

Test it.