The Sharat's

The Python Dictionary

The Python Dictionary is a key–value style data structure that is tightly integrated with the language syntax and semantics. Understanding them well can help us use them better and investigate subtle problems more efficiently.

This is my attempt to document this topic in more depth. Though I included a small section about the syntax and basic usage of dictionaries, it’ll be helpful if you have some beginner–intermediate level experience with Python.

This article is written for Python 3.6 installed via Anaconda on Xubuntu. Here’s the platform details:

$ python -V
Python 3.6.1 :: Anaconda custom (64-bit)
$ uname -isro
Linux 4.10.0-33-generic x86_64 GNU/Linux

Note: This is not intended as a substitute for official documentation. The official documentation is a reference and there will be some overlap. This document is intended as a supplement that covers more depth and practical nuances.

Introduction

Dictionaries (type dict) are a very powerful data structure, not just in Python. They are present in almost every modern high level language, sometimes called maps, hashes or associative arrays. Python’s syntax for dictionaries inspired the syntax of the JSON serialization format.

Dictionaries are a fundamental part of Python language and integrate tightly with the semantics and APIs of the standard library. This can be seen in the fact that we have a special syntax just to create these data structures.

Usage

Syntax

As a quick primer, here’s the syntax for defining a dictionary:

country_currencies = {
    'India': 'Rupee',
    'Russia': 'Ruble',
    'USA': 'Dollar',
    'Japan': 'Yen',
}

API

Again, we quickly run down the common operations on dictionaries.

# Get the value of a key.
indian_currency = country_currencies['India']

# Set the value of a key.
country_currencies['France'] = 'Euro'

# Delete a key.
del country_currencies['USA']

# Check for presence of a key.
'Russia' in country_currencies

# Get if key present, otherwise return `None`.
# (Takes a second parameter which is returned when key is missing).
country_currencies.get('USA')

# Set only if the key is not already present.
country_currencies.setdefault('France', 'Franc')

Contents

The contents of dictionaries are made up two components. The keys and the values. The keys form the index using which we can retrieve the values. Each key uniquely identifies a value within the dictionary.

Key Types

The keys form the index of the dictionary. In most practical cases, keys tend to be strings. Tuples are often used as well. In fact, values of any immutable, hashable types can be used as keys.

So, what is a hashable type? The official documentation of the __hash__ method gives the full detail of what it is and what are considered hashable. Simply put, if passing an object to the hash builtin function doesn’t raise an exception, the object is hashable and can be used as a key in a dictionary.

However, in practice, we should avoid using mutable objects as keys (even if they are hashable). Especially, if mutation changes the hash of the object.

For example, consider the following User class.

class User:
    def __init__(self, first_name, last_name):
        self.first_name = first_name
        self.last_name = last_name

Let’s inspect the hash values of User objects.

>>> ned = User('Ned', 'Stark')
>>> hash(ned)
8784834659087
>>> ned.first_name = 'Robb'
>>> hash(ned)
8784834659087

If you try the above code, you might see a different number. That’s because Python default hashing algorithm includes a random salt.

As seen above, the hash value did not change even though the object was modified. These User objects can be used as keys for a dictionary since they meet the requirement, but it should be kept in mind that they are mutable.

>>> ned = User('Ned', 'Stark')
>>> d = {ned: 123}
>>> d[ned]
123
>>> ned.first_name = 'Robb'
>>> d[ned]
123

If that doesn’t seem confusing, try this:

>>> robb = ned
>>> ned = User('Ned', 'Start')
>>> robb.first_name
'Robb'
>>> robb in d  # Robb isn't in our dictionary!
True
>>> ned.first_name
'Ned'
>>> ned in d  # We gave Ned Stark a value right?
False

This can quickly cause headaches and hard-to-find problems.

To fix this, if someone later decides to customize the hashing of this class by adding the following method:

    def __hash__(self):
        return hash((self.first_name, self.last_name))

Now, the hash of the object changes when we change the first_name.

>>> ned = User('Ned', 'Stark')
>>> hash(ned)
4091961891370636651
>>> ned.first_name = 'Robb'
>>> hash(ned)
-7890115541605828979

Using these objects as keys can be confusing as well:

>>> ned = User('Ned', 'Stark')
>>> d = {ned: 123}
>>> d[ned]
123
>>> ned.first_name = 'Robb'
>>> d[ned]
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
KeyError: <__main__.User object at 0x7fd60e7c5828>
>>> ned.first_name = 'Ned'
>>> d[ned]
123

In essence, using mutable types as keys in a dictionary can lead to confusing results in a fairly large codebase.

So, to avoid these potential problems, it’s best to use numbers, strings or tuples (containing numbers or strings) as keys for dictionaries. If you have to use other types, keep the hashing semantics in mind and document the reasons well.

Retrieving Keys

Dictionaries have a .keys method that returns an object of type dict_keys which is an iterable (technically, a view) of the keys of the dictionary. Note that this method used to return an ordinary list in Python 2.

>>> countries = country_currencies.keys()
>>> countries
dict_keys(['India', 'Russia', 'USA', 'Japan'])
>>> import collections
>>> isinstance(countries, collections.Iterable)
True

Note that the order of the keys is not retained/defined. Don’t rely on the order even if they seem predictable. It might vary across Python implementations and versions even. Use an OrderedDict when ordering is needed. More on this in a later section.

It should be noted that starting in Python 3.6, order of keys is preserved. This is an unintended side affect of using a more efficient dict implementation. As such, the Python documentation explicitly states that this is an implementation detail and should not be relied upon. Read more.

So, what’s special about dict_keys, as opposed to a list? Look look!

>>> countries
dict_keys(['India', 'Russia', 'USA', 'Japan'])
>>> country_currencies['France'] = 'Euro'
>>> countries
dict_keys(['India', 'Russia', 'USA', 'Japan', 'France'])

See? The dict_keys object is a view of the keys of the original dictionary object. When the dictionary’s keys change, so does the keys view. Of course, we can make a set of currently available keys by passing it to set builtin. This set would be independent of the dictionary.

>>> set(countries)
{'Japan', 'USA', 'Russia', 'India', 'France'}

Most people would suggest and use list here, instead of set. I personally feel a set is semantically more correct since a list indicates the contents have a specific ordering and does not convey that the contents are hashable, immutable, and more importantly, unique. A set shares these features of dictionary keys.

Additionally, the dict_keys objects are themselves set-like. They implement the Set abstraction. So, we don’t need to convert them to a set in order to do set operations on them. For example, here’s an intersection operation:

>>> isinstance(countries, collections.abc.Set)
True
>>> countries & {'India', 'China'}
{'India'}

Using Tuples for Keys

Here’s a quick example of using tuples as keys in a dictionary:

>>> data = {
...     ('a', 1): 'a1',
...     ('a', 2): 'a2',
...     ('b', 1): 'b1',
...     ('b', 2): 'b2',
... }
>>> data['a', 2]
'a2'

Note that only tuples that contain hashable types (or further such tuples) can be used as keys. Lists or dictionaries, on the other hand, cannot be used since they are not hashable.

Retrieving Values

Values are what the keys index. Naturally, values don’t have to be unique, unlike keys. There’s no restrictions on what types can be used as values in a dictionary.

We can get a sequence of values in a dict with the .values method. This returns a dict_values object.

>>> currencies = country_currencies.values()
>>> currencies
dict_values(['Rupee', 'Ruble', 'Dollar', 'Euro', 'Yen'])
>>> type(currencies)
<class 'dict_values'>
>>> isinstance(currencies, collections.abc.Set)
False

This is live as well!

>>> del country_currencies['France']
>>> currencies
dict_values(['Rupee', 'Ruble', 'Dollar', 'Yen'])

This can be passed to list to get a list of values. Using set here is probably not a good idea since unlike the keys, values don’t have to be unique or hashable.

Items Collection

Dictionaries also provide a .items method that returns all the key–value pairs as a sequence of 2-tuples.

>>> pairs = country_currencies.items()
>>> pairs
dict_items([('India', 'Rupee'), ('Russia', 'Ruble'), ('USA', 'Dollar'), ('Japan', 'Yen')])

Again, just like with .keys or .values, the sequence is live and the order of items is not defined.

The .items method is probably mostly used with the for statement to loop over the key–value pairs.

for country, currency in country_currencies:
    print(f"{country}'s currency is {currency}.")

The above code uses f-strings introduced in Python 3.6. In older versions of Python, the .format method or the modulo (%) operator should be used.

The dict_items object also implements the Set abstraction.

>>> isinstance(pairs, collections.abc.Set)
True

However, the abstraction’s methods only work if the dictionary’s values are hashable, not just the keys. So, for the dictionary we are working with, the pairs object can be used as a set.

>>> pairs & {('India', 'Rupee'), ('UK', 'Pound')}
{('India', 'Rupee')}

But if we try this on a dictionary whose values are not hashable, say, lists, then it fails.

>>> number_types = {
...     'even': [2, 4, 6, 8],
...     'odd': [1, 3, 5, 7, 9],
... }
>>> pairs = number_types.items()
>>> pairs
dict_items([('even', [2, 4, 6, 8]), ('odd', [1, 3, 5, 7, 9])])
>>> isinstance(pairs, collections.abc.Set)
True

Let’s try intersecting this with an empty set.

>>> pairs & set()
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
TypeError: unhashable type: 'list'

As the error says, list is not hashable. So, although the isinstance tells us that this is a Set, whether it can actually be used as such, depends on it’s contents. This is not incorrect, actually, I feel it’s just a consequence of Python’s dynamic nature.

Typing

Dictionaries in Python are what I call a homogeneous data structure. What that means is that they are best used by having all the keys be of the same type and similarly for values. This is enforced in comparable data structures in statically typed languages like Java’s Map or Haskell’s HashMap. But since Python is a dynamic language, such restrictions are not placed. We can have keys / values of several different types within the same dictionary.

data = {
    'a': 1,
    42: 'yay!',
    ('a', 'b', 2): True,
}

This is still a valid dictionary, although an extremely sad and ugly one (totally my opinion :D).

If using homogeneous dictionaries, the type annotations syntax can be used to declare the type signatures. We use typing.Dict for this purpose as illustrated below.

from typing import Dict, Tuple

number_map: Dict[int, int] = {1: 10, 2: 20, 3: 30}
data_map: Dict[Tuple[str, int], str] = {('a', 1): 'a1', ('a', 2): 'a2'}

This is new in Python 3.6. Before 3.6, annotations are only supported for function arguments. Read more.

Additionally, the typing module itself is new in Python 3.5. Read more.

The general structure is Dict[<key-type>, <value-type>]. So, Dict[str, int] denotes a dictionary that maps string keys to integer values.

Note that these type annotations are not checked at runtime. They’re mere help to IDEs, static checkers and human readers. Python’s dynamic nature is not affected by these annotations.

However, if such type annotations are declared, you could use a static analyzer like mypy to perform type checks. I won’t be discussing that here.

Creating Dictionaries

There are a few other ways to create dictionaries besides the {} syntax. Here’s a few of them.

Calling dict

The dict callable can be used to create dictionaries from a list of tuples or bypassing the keys and values as keyword arguments.

>>> dict([('Chromium', 24), ('Phosphorus', 15), ('Silver', 47)])
{'Chromium': 24, 'Phosphorus': 15, 'Silver': 47}

This is obviously more convenient than the dictionary syntax only if we already have such a list. If we have the keys and corresponding values in different lists, we can zip them up and pass the result to dict.

>>> dict(zip(
...     ['Sulfer', 'Calcium', 'Gold'],  # Keys
...     [16, 20, 79],  # Values
... ))
{'Sulfer': 16, 'Calcium': 20, 'Gold': 79}

Of course, we can pass keyword arguments directly to dict, in addition to the above even.

>>> dict(dict([('Chromium', 24), ('Phosphorus', 15)]), Sodium=11, Nitrogen=7)
{'Chromium': 24, 'Phosphorus': 15, 'Sodium': 11, 'Nitrogen': 7}
>>> dict(Sodium=11, Nitrogen=7)
{'Sodium': 11, 'Nitrogen': 7}

The second form is better written using the Python syntax. That is more natural to a potential future reader, and, slightly faster1 as well.

Comprehensions

Python 3 (and 2.7) added support for dict comprehensions which are very similar to list comprehensions, but with a small variation in syntax.

>>> dict((i, i**2) for i in range(5))  # Using the `dict` builtin.
{0: 0, 1: 1, 2: 4, 3: 9, 4: 16}
>>> {i: i**2 for i in range(5)}  # Using a dict comprehension.
{0: 0, 1: 1, 2: 4, 3: 9, 4: 16}

The above two examples create the same dictionary. However, as pointed out in PEP 274, the dict comprehension is more succinct and makes the intent clearer.

Public Appearance

Unsurprisingly, dictionaries pop up in a lot of places in Python. Here’s a few ones.

Keyword Arguments

When defining a function that takes arbitrary keyword arguments, they are passed to the function as a dictionary.

>>> def construct(**counts):
...     print(counts)
...     print(len(counts), type(counts))
...
>>> construct(a=1, b=2, c=3)
{'a': 1, 'b': 2, 'c': 3}
3 <class 'dict'>

Of course, we can pass a dictionary’s data as keyword arguments to a function using similar syntax.

>>> kw_args = {'a': 1, 'b': 2, 'c': 3}
>>> construct(**kw_args)
{'a': 1, 'b': 2, 'c': 3}
3 <class 'dict'>

Namespaces

The globals builtin function gives a dictionary of all names and their values in the current global namespace. We can modify this dictionary to define new names or delete existing ones, although that’s probably a bad idea.

>>> len(globals())
25
>>> globals()['x'] = 123
>>> x
123
>>> del globals()['x']
>>> x
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
NameError: name 'x' is not defined

The locals builtin returns a dictionary of names and values from the local scope, for e.g., the private local scope inside of a function or method.

The vars builtin takes an object as an argument and returns the names available as properties on this objects. Specifically, it returns the __dict__ property’s value of the given object. When called without any arguments, then it returns names and values from the local scope. In other words, vars() is locals() return True.

Serialization

Dictionaries, being key–value data structures, extend naturally to be stored into key–value databases and other NoSQL data stores. However, here we’ll look at forms of serializing them into text and binary forms for transmission or for saving to disk.

JSON

Nowadays, the thought of serializing a python dictionary is usually followed by using the json module to dump and load using the JSON format. No surprise since it’s extremely convenient and there’s quality parsers and writers for almost every programming language today. The syntax as well, although not too convenient to write by hand, is still very simple, lightweight and easy to read. It helps that the syntax is quite close to Python’s own syntax for dictionaries.

Here’s a quick example:

>>> import json
>>> json.dumps(country_currencies)
'{"India": "Rupee", "Russia": "Ruble", "USA": "Dollar", "Japan": "Yen"}'
>>> json.loads(json.dumps(country_currencies)) == country_currencies
True

In short, these four functions from json module are enough to know the basic usage.

Method Description
.dump(obj, fp) Turn obj into JSON and write it to the fp file-like object.
.dumps(obj) Turn obj into JSON and return the resulting string.
.load(fp) Read valid JSON from fp file-like object and return the resulting object.
.loads(data) Parse data as a valid JSON string and return the resulting object.

As convenient as this is, it is important to know the changes to data types that will result because of this. JSON only supports numbers, strings and booleans as primary data types and arrays & maps as analogues to lists and dicts. As a result of this, if there are tuples somewhere in the dictionary, then they will be turned into lists when the dict is serialized and deserialized with JSON. A similar situation occurs for dates and any other data type not directly supported by the JSON spec.

Pickling

Unlike the above, pickling (using the pickle module) serializes objects into binary data and can handle a much wider range of data types. For this reason, pickled data can only be loaded by Python, not other languages (well, not yet at least).

The pickle module has similar dump, dumps, load and loads methods just like for the above discussed json module.

The Item Syntax

The syntax used to get an item from a dictionary, given it’s index, is data[key]. This is mostly equivalent to calling the __getitem__ method, like the following:

data.__getitem__(key)

But obviously, we’d prefer the square bracket syntax. But understanding that underneath the syntax, it’s just a method call, lets us implement the __getitem__ method in our own classes and get the item syntax on our objects.

Here’s a simple example:

>>> class Store:
...     def __getitem__(self, name):
...             return name.upper()
...
>>> store = Store()
>>> store['Hello there!']
'HELLO THERE!'

Similar to this is the __setitem__ which is used to set the value using the item syntax.

# The following two are equivalent.
data[key] = value
data.__setitem__(key, value)

Note that this should be used responsibly. This feature gets into borderline operator overloading category. In almost all cases (including the above example), using a normal named method on your classes should be a better option than overriding the item syntax. Since a normal method would have a name which makes the intent clearer.

Flavors

Python’s standard library comes with a few flavors of dictionaries that provide some nice additional functionality. These data structures are all available in the collections module.

The following are subclasses of dict and have all the features of Python’s dictionaries.

The OrderedDict

The collections.OrderedDict is a dictionary that remembers the order in which keys are inserted. The order remembered is the insertion order. So, if we add a new key to the dict, it will be at the end of the key sequence. But if we change the value of an existing key, it’s position in the ordering is unchanged.

Create a new OrderedDict:

>>> from collections import OrderedDict
>>> planet_satellites = OrderedDict(
...     Mercury=0,
...     Venus=0,
...     Earth=1,
...     Mars=2,
...     Jupiter=69,
...     Saturn=62,
...     Uranus=27,
...     Neptune=14,
... )
>>> from pprint import pprint
>>> pprint(planet_satellites)
OrderedDict([('Mercury', 0),
             ('Venus', 0),
             ('Earth', 1),
             ('Mars', 2),
             ('Jupiter', 69),
             ('Saturn', 62),
             ('Uranus', 27),
             ('Neptune', 14)])

Note that we use the pprint function to show the OrderedDict objects in a convenient way.

They are just dictionaries under the hood.

>>> isinstance(planet_satellites, dict)
True
>>> planet_satellites['Mars']
2

These objects support being reversed as well:

>>> rev_planets = OrderedDict(reversed(planet_satellites.items()))
>>> pprint(rev_planets)
OrderedDict([('Neptune', 14),
             ('Uranus', 27),
             ('Saturn', 62),
             ('Jupiter', 69),
             ('Mars', 2),
             ('Earth', 1),
             ('Venus', 0),
             ('Mercury', 0)])

The results of .keys and .values methods also retain the ordering. Refer to the official documentation linked above for full details.

The defaultdict

The name defaultdict is unfortunate as it doesn’t adhere to any naming conventions. I’d love to see it renamed to default_dict or even DefaultDict, but it’s probably easier to just live with it.

A defaultdict can understand how to initialize new keys. Consider the following code. Here, we have a piece of text and we want a dictionary mapping each letter in the text to it’s count of occurrences.

text = 'lorem ipsum dolor sit amet'
counts = {}
for letter in text:
    if letter not in counts:
        counts[letter] = 0
    counts[letter] += 1
print(counts)

Of course, there’s better ways to do this, but for the sake of example, let’s bear with this implementation.

Notice how we check if the letter is not already present in the dict and if so, we initialize it to zero. A defaultdict can learn this method of initialization. It takes a function as its first argument which returns the value of a new key when accessed. So, we can replace the above code to use defaultdict like:

from collections import defaultdict
text = 'lorem ipsum dolor sit amet'
counts = defaultdict(int)
for letter in text:
    counts[letter] += 1
print(counts)

When we try to get the value of a letter from counts, and that letter doesn’t already exist in counts, defaultdict will call int, with no arguments, and puts the return value into counts[letter]. Precisely what we were doing in our previous example. So, what does int return when called with no arguments? You guessed it, zero!

>>> int()
0
>>> float()
0.0
>>> str()
''
>>> bool()
False
>>> list()
[]
>>> dict()
{}
>>> set()
set()

As illustrated above, calling the data type builtins with no arguments return the falsy value of that data type. We can use this fact and pass these builtins to defaultdict constructor depending on the need. If we wanted a different initial value, say 42, we could use a lambda function like lambda: 42 instead.

The ChainMap

The ChainMap is an abstraction over a chain of dictionaries in order of precedence. Essentially, it holds a list of dictionaries and when a key is indexed, each of these dictionaries are searched for this key and the value of the first match is returned.

This is better illustrated with an example. Let’s create a ChainMap with dummy data:

>>> from collections import ChainMap
>>> data = ChainMap({'a': 1, 'b': 2, 'c': 3}, {'c': 30, 'd': 40, 'e': 50})
>>> data
ChainMap({'a': 1, 'b': 2, 'c': 3}, {'c': 30, 'd': 40, 'e': 50})
>>> data.maps  # A list of maps in the chain.
[{'a': 1, 'b': 2, 'c': 3}, {'c': 30, 'd': 40, 'e': 50}]

Let’s try indexing:

>>> data['a']
1
>>> data['e']
50
>>> data['c']
3

Here, the 'a' is indexed from the first dictionary, 'e' is indexed from the second dictionary and 'c' is indexed from the first dictionary.

As mentioned in the documentation, writes, updates and deletes, however, operate on the first dictionary alone.

>>> data['a'] = 91
>>> data
ChainMap({'a': 91, 'b': 2, 'c': 3}, {'c': 30, 'd': 40, 'e': 50})
>>> data['e'] = 951
>>> data
ChainMap({'a': 91, 'b': 2, 'c': 3, 'e': 951}, {'c': 30, 'd': 40, 'e': 50})
>>> data['c'] = 93
>>> data
ChainMap({'a': 91, 'b': 2, 'c': 93, 'e': 951}, {'c': 30, 'd': 40, 'e': 50})

Of course, if we explicitly want to modify the last dictionary, it can be indexed directly:

>>> data.maps[-1]['c'] = 999
>>> data
ChainMap({'a': 91, 'b': 2, 'c': 93, 'e': 951}, {'c': 999, 'd': 40, 'e': 50})

The ChainMap is useful to hold tiers of configuration parameters for an application, in a form similar to the following:

ChainMap(user_settings, default_settings)

We can have multiple tiers depending the situation. The user can modify the dictionary as they fit and all writes and updates will be made only on the first dictionary, user_settings. Whereas, when one tries to get the value of a configuration parameter, it automatically falls back to default_settings if it isn’t present in user_settings.

The Counter

Counter dictionaries can be used to keep counts of any (hashable) objects. The keys are these hashable objects and the values are their counts. The official docs on this gives some clever examples and uses so I recommend you go read this up there, instead of redoing it here.

Custom Flavor

Although rarely needed in practice, we can create our own flavors of dictionary types. One way to achieve this would be to extend the dict type directly, but usually the easier way to deal with this is to use the UserDict class.

Here’s an example dictionary type that works with string keys and is case-insensitive. A good use for something like this is for HTTP headers. (The requests library does something similar.)

from collections import UserDict


class CaselessDict(UserDict):

    def __getitem__(self, name):
        return self.data[name.lower()]

    def __setitem__(self, name, value):
        self.data[name.lower()] = value

As seen above, the UserDict class provides a .data attribute that can be used as the underlying store dictionary.

Let’s try it out.

>>> data = CaselessDict(accept='application/json')
>>> data['accept']
'application/json'
>>> data['Accept']
'application/json'
>>> data['ACCEPT']
'application/json'

Disassembling

Now, let’s disassemble a few common operations on dictionaries. I won’t be going into the details of how to interpret the disassembled instructions in this article. We use the dis function (from the aptly named dis module) for this.

Let’s try this a very simple function.

>>> dis.dis(lambda: {'a': 1})
  1           0 LOAD_CONST               1 ('a')
              2 LOAD_CONST               2 (1)
              4 BUILD_MAP                1
              6 RETURN_VALUE

Here, we see the BUILD_MAP opcode that takes a count which is the length of the dictionary to build. From the official docs,

Pushes a new dictionary object onto the stack. Pops 2 * count items so that the dictionary holds count entries: {..., TOS3: TOS2, TOS1: TOS}.

Now let’s do this with two elements in the dict.

>>> dis.dis(lambda: {'a': 1, 'b': 2})
  1           0 LOAD_CONST               1 (1)
              2 LOAD_CONST               2 (2)
              4 LOAD_CONST               3 (('a', 'b'))
              6 BUILD_CONST_KEY_MAP      2
              8 RETURN_VALUE

Here, we see a different opcode, BUILD_CONST_KEY_MAP which also takes the length of the dict as an argument. This is also explained best from the docs,

The version of BUILD_MAP specialized for constant keys. count values are consumed from the stack. The top element on the stack contains a tuple of keys.

Conclusion

Dictionaries in Python (or any other language for that matter) are a very powerful multi-purpose data structure and are extremely handy and easy to use in Python. I hoped to put the things I learned about them in this article. If you see any inaccuracies or if there’s something that makes for a good addition to this article, let me know in the comments below.

Thank you for reading. Please let me know what you think. If you have any topics you’d like me to cover in a future article, put in a comment.

References

The official documentation, mostly. Wikipedia for data used in examples.


  1. I read the proof for this a long time ago, but I don’t remember where :). 

Discuss on: Reddit.