5. Python Essentials

5.1. Overview

We have covered a lot of material quite quickly, with a focus on examples.

Now let’s cover some core features of Python in a more systematic way.

This approach is less exciting but helps clear up some details.

5.2. Data Types

Computer programs typically keep track of a range of data types.

For example, 1.5 is a floating point number, while 1 is an integer.

Programs need to distinguish between these two types for various reasons.

One is that they are stored in memory differently.

Another is that arithmetic operations are different

  • For example, floating point arithmetic is implemented on most machines by a specialized Floating Point Unit (FPU).

In general, floats are more informative but arithmetic operations on integers are faster and more accurate.

Python provides numerous other built-in Python data types, some of which we’ve already met

  • strings, lists, etc.

Let’s learn a bit more about them.

5.2.1. Primitive Data Types

One simple data type is Boolean values, which can be either True or False

x = True

We can check the type of any object in memory using the type() function.


In the next line of code, the interpreter evaluates the expression on the right of = and binds y to this value

y = 100 < 10

In arithmetic expressions, True is converted to 1 and False is converted 0.

This is called Boolean arithmetic and is often useful in programming.

Here are some examples

x + y
x * y
True + True
bools = [True, True, False, True]  # List of Boolean values


Complex numbers are another primitive data type in Python

x = complex(1, 2)
y = complex(2, 1)
print(x * y)


5.2.2. Containers

Python has several basic types for storing collections of (possibly heterogeneous) data.

We’ve already discussed lists.

A related data type is tuples, which are “immutable” lists

x = ('a', 'b')  # Parentheses instead of the square brackets
x = 'a', 'b'    # Or no brackets --- the meaning is identical
('a', 'b')

In Python, an object is called immutable if, once created, the object cannot be changed.

Conversely, an object is mutable if it can still be altered after creation.

Python lists are mutable

x = [1, 2]
x[0] = 10
[10, 2]

But tuples are not

x = (1, 2)
x[0] = 10
TypeError                                 Traceback (most recent call last)
<ipython-input-13-d1b2647f6c81> in <module>
      1 x = (1, 2)
----> 2 x[0] = 10

TypeError: 'tuple' object does not support item assignment

We’ll say more about the role of mutable and immutable data a bit later.

Tuples (and lists) can be “unpacked” as follows

integers = (10, 20, 30)
x, y, z = integers

You’ve actually seen an example of this already.

Tuple unpacking is convenient and we’ll use it often. Slice Notation

To access multiple elements of a list or tuple, you can use Python’s slice notation.

For example,

a = [2, 4, 6, 8]
[4, 6, 8]
[4, 6]

The general rule is that a[m:n] returns n - m elements, starting at a[m].

Negative numbers are also permissible

a[-2:]  # Last two elements of the list
[6, 8]

The same slice notation works on tuples and strings

s = 'foobar'
s[-3:]  # Select the last three elements
'bar' Sets and Dictionaries

Two other container types we should mention before moving on are sets and dictionaries.

Dictionaries are much like lists, except that the items are named instead of numbered

d = {'name': 'Frodo', 'age': 33}

The names 'name' and 'age' are called the keys.

The objects that the keys are mapped to ('Frodo' and 33) are called the values.

Sets are unordered collections without duplicates, and set methods provide the usual set-theoretic operations

s1 = {'a', 'b'}
s2 = {'b', 'c'}

The set() function creates sets from sequences

s3 = set(('foo', 'bar', 'foo'))
{'bar', 'foo'}

5.3. Input and Output

Let’s briefly review reading and writing to text files, starting with writing

f = open('newfile.txt', 'w')   # Open 'newfile.txt' for writing
f.write('Testing\n')           # Here '\n' means new line
f.write('Testing again')


  • The built-in function open() creates a file object for writing to.

  • Both write() and close() are methods of file objects.

Where is this file that we’ve created?

Recall that Python maintains a concept of the present working directory (pwd) that can be located from with Jupyter or IPython via


If a path is not specified, then this is where Python writes to.

We can also use Python to read the contents of newline.txt as follows

f = open('newfile.txt', 'r')
out = f.read()
'Testing\nTesting again'
Testing again

5.3.1. Paths

Note that if newfile.txt is not in the present working directory then this call to open() fails.

In this case, you can shift the file to the pwd or specify the full path to the file

f = open('insert_full_path_to_file/newfile.txt', 'r')

5.4. Iterating

One of the most important tasks in computing is stepping through a sequence of data and performing a given action.

One of Python’s strengths is its simple, flexible interface to this kind of iteration via the for loop.

5.4.1. Looping over Different Objects

Many Python objects are “iterable”, in the sense that they can be looped over.

To give an example, let’s write the file us_cities.txt, which lists US cities and their population, to the present working directory.

%%file us_cities.txt
new york: 8244910
los angeles: 3819702
chicago: 2707120
houston: 2145146
philadelphia: 1536471
phoenix: 1469471
san antonio: 1359758
san diego: 1326179
dallas: 1223229
Writing us_cities.txt

Here %%file is an IPython cell magic.

Suppose that we want to make the information more readable, by capitalizing names and adding commas to mark thousands.

The program below reads the data in and makes the conversion:

data_file = open('us_cities.txt', 'r')
for line in data_file:
    city, population = line.split(':')         # Tuple unpacking
    city = city.title()                        # Capitalize city names
    population = f'{int(population):,}'        # Add commas to numbers
    print(city.ljust(15) + population)
New York       8,244,910
Los Angeles    3,819,702
Chicago        2,707,120
Houston        2,145,146
Philadelphia   1,536,471
Phoenix        1,469,471
San Antonio    1,359,758
San Diego      1,326,179
Dallas         1,223,229

Here format() is a string method used for inserting variables into strings.

The reformatting of each line is the result of three different string methods, the details of which can be left till later.

The interesting part of this program for us is line 2, which shows that

  1. The file object data_file is iterable, in the sense that it can be placed to the right of in within a for loop.

  2. Iteration steps through each line in the file.

This leads to the clean, convenient syntax shown in our program.

Many other kinds of objects are iterable, and we’ll discuss some of them later on.

5.4.2. Looping without Indices

One thing you might have noticed is that Python tends to favor looping without explicit indexing.

For example,

x_values = [1, 2, 3]  # Some iterable x
for x in x_values:
    print(x * x)

is preferred to

for i in range(len(x_values)):
    print(x_values[i] * x_values[i])

When you compare these two alternatives, you can see why the first one is preferred.

Python provides some facilities to simplify looping without indices.

One is zip(), which is used for stepping through pairs from two sequences.

For example, try running the following code

countries = ('Japan', 'Korea', 'China')
cities = ('Tokyo', 'Seoul', 'Beijing')
for country, city in zip(countries, cities):
    print(f'The capital of {country} is {city}')
The capital of Japan is Tokyo
The capital of Korea is Seoul
The capital of China is Beijing

The zip() function is also useful for creating dictionaries — for example

names = ['Tom', 'John']
marks = ['E', 'F']
dict(zip(names, marks))
{'Tom': 'E', 'John': 'F'}

If we actually need the index from a list, one option is to use enumerate().

To understand what enumerate() does, consider the following example

letter_list = ['a', 'b', 'c']
for index, letter in enumerate(letter_list):
    print(f"letter_list[{index}] = '{letter}'")
letter_list[0] = 'a'
letter_list[1] = 'b'
letter_list[2] = 'c'

5.4.3. List Comprehensions

We can also simplify the code for generating the list of random draws considerably by using something called a list comprehension.

List comprehensions are an elegant Python tool for creating lists.

Consider the following example, where the list comprehension is on the right-hand side of the second line

animals = ['dog', 'cat', 'bird']
plurals = [animal + 's' for animal in animals]
['dogs', 'cats', 'birds']

Here’s another example

range(0, 8)
doubles = [2 * x for x in range(8)]
[0, 2, 4, 6, 8, 10, 12, 14]

5.5. Comparisons and Logical Operators

5.5.1. Comparisons

Many different kinds of expressions evaluate to one of the Boolean values (i.e., True or False).

A common type is comparisons, such as

x, y = 1, 2
x < y
x > y

One of the nice features of Python is that we can chain inequalities

1 < 2 < 3
1 <= 2 <= 3

As we saw earlier, when testing for equality we use ==

x = 1    # Assignment
x == 2   # Comparison

For “not equal” use !=

1 != 2

Note that when testing conditions, we can use any valid Python expression

x = 'yes' if 42 else 'no'
x = 'yes' if [] else 'no'

What’s going on here?

The rule is:

  • Expressions that evaluate to zero, empty sequences or containers (strings, lists, etc.) and None are all equivalent to False.

    • for example, [] and () are equivalent to False in an if clause

  • All other values are equivalent to True.

    • for example, 42 is equivalent to True in an if clause

5.5.2. Combining Expressions

We can combine expressions using and, or and not.

These are the standard logical connectives (conjunction, disjunction and denial)

1 < 2 and 'f' in 'foo'
1 < 2 and 'g' in 'foo'
1 < 2 or 'g' in 'foo'
not True
not not True


  • P and Q is True if both are True, else False

  • P or Q is False if both are False, else True

5.6. More Functions

Let’s talk a bit more about functions, which are all important for good programming style.

5.6.1. The Flexibility of Python Functions

As we discussed in the previous lecture, Python functions are very flexible.

In particular

  • Any number of functions can be defined in a given file.

  • Functions can be (and often are) defined inside other functions.

  • Any object can be passed to a function as an argument, including other functions.

  • A function can return any kind of object, including functions.

We already gave an example of how straightforward it is to pass a function to a function.

Note that a function can have arbitrarily many return statements (including zero).

Execution of the function terminates when the first return is hit, allowing code like the following example

def f(x):
    if x < 0:
        return 'negative'
    return 'nonnegative'

Functions without a return statement automatically return the special Python object None.

5.6.2. Docstrings

Python has a system for adding comments to functions, modules, etc. called docstrings.

The nice thing about docstrings is that they are available at run-time.

Try running this

def f(x):
    This function squares its argument
    return x**2

After running this code, the docstring is available

Type:       function
String Form:<function f at 0x2223320>
File:       /home/john/temp/temp.py
Definition: f(x)
Docstring:  This function squares its argument
Type:       function
String Form:<function f at 0x2223320>
File:       /home/john/temp/temp.py
Definition: f(x)
def f(x):
    This function squares its argument
    return x**2

With one question mark we bring up the docstring, and with two we get the source code as well.

5.6.3. One-Line Functions: lambda

The lambda keyword is used to create simple functions on one line.

For example, the definitions

def f(x):
    return x**3


f = lambda x: x**3

are entirely equivalent.

To see why lambda is useful, suppose that we want to calculate \(\int_0^2 x^3 dx\) (and have forgotten our high-school calculus).

The SciPy library has a function called quad that will do this calculation for us.

The syntax of the quad function is quad(f, a, b) where f is a function and a and b are numbers.

To create the function \(f(x) = x^3\) we can use lambda as follows

from scipy.integrate import quad

quad(lambda x: x**3, 0, 2)
(4.0, 4.440892098500626e-14)

Here the function created by lambda is said to be anonymous because it was never given a name.

5.6.4. Keyword Arguments

In a previous lecture, you came across the statement

plt.plot(x, 'b-', label="white noise")

In this call to Matplotlib’s plot function, notice that the last argument is passed in name=argument syntax.

This is called a keyword argument, with label being the keyword.

Non-keyword arguments are called positional arguments, since their meaning is determined by order

  • plot(x, 'b-', label="white noise") is different from plot('b-', x, label="white noise")

Keyword arguments are particularly useful when a function has a lot of arguments, in which case it’s hard to remember the right order.

You can adopt keyword arguments in user-defined functions with no difficulty.

The next example illustrates the syntax

def f(x, a=1, b=1):
    return a + b * x

The keyword argument values we supplied in the definition of f become the default values


They can be modified as follows

f(2, a=4, b=5)

5.7. Coding Style and PEP8

To learn more about the Python programming philosophy type import this at the prompt.

Among other things, Python strongly favors consistency in programming style.

We’ve all heard the saying about consistency and little minds.

In programming, as in mathematics, the opposite is true

  • A mathematical paper where the symbols \(\cup\) and \(\cap\) were reversed would be very hard to read, even if the author told you so on the first page.

In Python, the standard style is set out in PEP8.

(Occasionally we’ll deviate from PEP8 in these lectures to better match mathematical notation)

5.8. Exercises

Solve the following exercises.

(For some, the built-in function sum() comes in handy).

5.8.1. Exercise 1

Part 1: Given two numeric lists or tuples x_vals and y_vals of equal length, compute their inner product using zip().

Part 2: In one line, count the number of even numbers in 0,…,99.

  • Hint: x % 2 returns 0 if x is even, 1 otherwise.

Part 3: Given pairs = ((2, 5), (4, 2), (9, 8), (12, 10)), count the number of pairs (a, b) such that both a and b are even.

5.8.2. Exercise 2

Consider the polynomial

(5.1)\[p(x) = a_0 + a_1 x + a_2 x^2 + \cdots a_n x^n = \sum_{i=0}^n a_i x^i\]

Write a function p such that p(x, coeff) that computes the value in (5.1) given a point x and a list of coefficients coeff.

Try to use enumerate() in your loop.

5.8.3. Exercise 3

Write a function that takes a string as an argument and returns the number of capital letters in the string.

Hint: 'foo'.upper() returns 'FOO'.

5.8.4. Exercise 4

Write a function that takes two sequences seq_a and seq_b as arguments and returns True if every element in seq_a is also an element of seq_b, else False.

  • By “sequence” we mean a list, a tuple or a string.

  • Do the exercise without using sets and set methods.

5.8.5. Exercise 5

When we cover the numerical libraries, we will see they include many alternatives for interpolation and function approximation.

Nevertheless, let’s write our own function approximation routine as an exercise.

In particular, without using any imports, write a function linapprox that takes as arguments

  • A function f mapping some interval \([a, b]\) into \(\mathbb R\).

  • Two scalars a and b providing the limits of this interval.

  • An integer n determining the number of grid points.

  • A number x satisfying a <= x <= b.

and returns the piecewise linear interpolation of f at x, based on n evenly spaced grid points a = point[0] < point[1] < ... < point[n-1] = b.

Aim for clarity, not efficiency.

5.8.6. Exercise 6

Using list comprehension syntax, we can simplify the loop in the following code.

import numpy as np

n = 100
ϵ_values = []
for i in range(n):
    e = np.random.randn()

5.9. Solutions

5.9.1. Exercise 1 Part 1 Solution:

Here’s one possible solution

x_vals = [1, 2, 3]
y_vals = [1, 1, 1]
sum([x * y for x, y in zip(x_vals, y_vals)])

This also works

sum(x * y for x, y in zip(x_vals, y_vals))
6 Part 2 Solution:

One solution is

sum([x % 2 == 0 for x in range(100)])

This also works:

sum(x % 2 == 0 for x in range(100))

Some less natural alternatives that nonetheless help to illustrate the flexibility of list comprehensions are

len([x for x in range(100) if x % 2 == 0])


sum([1 for x in range(100) if x % 2 == 0])
50 Part 3 Solution

Here’s one possibility

pairs = ((2, 5), (4, 2), (9, 8), (12, 10))
sum([x % 2 == 0 and y % 2 == 0 for x, y in pairs])

5.9.2. Exercise 2

def p(x, coeff):
    return sum(a * x**i for i, a in enumerate(coeff))
p(1, (2, 4))

5.9.3. Exercise 3

Here’s one solution:

def f(string):
    count = 0
    for letter in string:
        if letter == letter.upper() and letter.isalpha():
            count += 1
    return count

f('The Rain in Spain')

An alternative, more pythonic solution:

def count_uppercase_chars(s):
    return sum([c.isupper() for c in s])

count_uppercase_chars('The Rain in Spain')

5.9.4. Exercise 4

Here’s a solution:

def f(seq_a, seq_b):
    is_subset = True
    for a in seq_a:
        if a not in seq_b:
            is_subset = False
    return is_subset

# == test == #

print(f([1, 2], [1, 2, 3]))
print(f([1, 2, 3], [1, 2]))

Of course, if we use the sets data type then the solution is easier

def f(seq_a, seq_b):
    return set(seq_a).issubset(set(seq_b))

5.9.5. Exercise 5

def linapprox(f, a, b, n, x):
    Evaluates the piecewise linear interpolant of f at x on the interval
    [a, b], with n evenly spaced grid points.

        f : function
            The function to approximate

        x, a, b : scalars (floats or integers)
            Evaluation point and endpoints, with a <= x <= b

        n : integer
            Number of grid points

        A float. The interpolant evaluated at x

    length_of_interval = b - a
    num_subintervals = n - 1
    step = length_of_interval / num_subintervals

    # === find first grid point larger than x === #
    point = a
    while point <= x:
        point += step

    # === x must lie between the gridpoints (point - step) and point === #
    u, v = point - step, point

    return f(u) + (x - u) * (f(v) - f(u)) / (v - u)

5.9.6. Exercise 6

Here’s one solution.

n = 100
ϵ_values = [np.random.randn() for i in range(n)]