# Basic Object Types¶

**Note**:

There are reading-comprehension exercises included throughout the text. These are meant to help you put your reading to practice. Solutions for the exercises are included at the bottom of this page.

You will see the term “object” be used frequently throughout this text.
In Python, the term “object” is quite the catch-all; including numbers,
strings of characters, lists, functions - a Python object is essentially
anything that you can assign to a variable. That being said, there are
different *types* of objects: Python treats integers as a different
*type* of object than a string, for instance.

The different object types have manifestly different purposes and thus have different built-in functions available to them. Here, we will review some of the basic types that are built into Python, as a natural entry point to writing code. We will cover:

- numbers (integers, floating-point numbers, and complex numbers)
- booleans
- the “null” type
- strings
- lists

The built-in function `isinstance`

will allow us to check if an object
is of a given type. You can also use the built-in `type`

function to
check an object’s type. For example, the following code checks if an
object is an integer:

```
# assign the variable `x` to the integer 1
>>> x = 1
# checking the type of `x`
>>> type(x)
int
# verifying that `x` is an integer-type object
>>> isinstance(x, int)
True
```

In a later module, you will learn “object-oriented” programming, which will allow you to create your own, customized objects!

## Number Types¶

Python has three basic types of numbers: integers, “floating-point” numbers, and complex numbers. Familiar mathematical symbols can be used to perform arithmetic on all of these numbers (comparison operators like “greater than” are not defined for complex numbers):

Operation | Description |
---|---|

`x + y` |
Sum of two numbers |

`x - y` |
Difference of two numbers |

`x * y` |
Product of two numbers |

`x / y` |
Quotient of two numbers |

`x // y` |
Quotient of two numbers, returned as an integer |

`x % y` |
`x` “modulo”: `y` : The remainder of `x / y` |

`x ** y` |
`x` raised to the power `y` |

`-x` |
A negated number |

`abs(x)` |
The absolute value of a number |

`x == y` |
Check if two numbers have the same value |

`x != y` |
Check if two numbers have different values |

`x > y` |
Check if `x` is greater than `y` |

`x >= y` |
Check if `x` is greater than or equal to `y` |

`x < y` |
Check if `x` is less than `y` |

`x <= y` |
Check if `x` is less than or equal to `y` |

These operations obey the familiar order of operations from your mathematics class, with parentheses available for association:

```
# multiplication takes precedence over addition
>>> 1 + 2 * 3
7
# grouping operations with parentheses
>>> (1 + 2) * 3
9
# finding the remainder of division
>>> 11 % 5
1
# checking an inequality
>>> (2 ** 3) < (2 ** 4)
True
```

It should be noted that in many other programming languages, including
the out-dated Python 2, dividing two integers would always return an
integer - even if mathematically the result should be a fraction. In
Python 3, *the quotient of two integers will always return a
floating-point number* (i.e. a number that includes a decimal point):

```
# In many other languages, 3 / 2 returns the integer 1.
# In Python 3, division always returns a floating-point number:
>>> 3 / 2
1.5
>>> 4 / 2
2.0
```

The `//`

operator is known as the “floor-divide” operator: it performs
division between two numbers and returns the result as an integer by
discarding any decimal-places for that number (thus returning the
“floor” of that number). This can be used to perform the
integer-division traditionally used in other programming languages:

```
# floor-division
>>> 1 // 3 # 0.3333.. -> 0
0
>>> 3 // 2 # 1.5 -> 1
1
```

**Reading Comprehension: Understanding the modulo operator**

The modulo operator, `%`

, is not commonly seen in mathematics
textbooks. It is, however, a very useful operation to have at our
disposal. `x % y`

(said as x “mod” y in programmer’s jargon) returns
the *remainder* of `x / y`

. For example:

- \(\frac{3}{2} = 1 + \frac{1}{2}\). 2 “goes into” 3 one time,
leaving a remainder of 1. Thus
`3 % 2`

returns`1`

- \(\frac{9}{3} = 3\). 3 “goes into” 9 three times, and leaves no
remainder. Thus
`9 % 3`

returns`0`

Given this description of the “mod” operator, simplify the following by hand, and then use the IPython console to check your work:

`1 % 5`

`2 % 5`

`22 % 1`

`22 % 2`

`22 % 3`

`22 % 4`

`22 % 5`

`22 % 6`

Now, given any integer, `n`

, what are the possible values that
`n % 2`

can return? See if you can come up with a simple rule for
explaining the behavior of `n % 2`

.

### Python’s math module¶

The standard library’s math module provides us with many more mathematical functions, like logarithms and trigonometric functions. A complete listing of them can be found in the official Python documentation. This module must be imported into your code in order to use its functions:

```
# using the `math` module to use
# additional mathematical functions
>>> import math
>>> math.sqrt(4.)
2.0
# base-10 log
>>> math.log10(10.)
1.0
# 4! = 4*3*2*1
>>> math.factorial(4)
24
```

### Integers¶

As in traditional mathematics, an integer is any “whole” number: \(\dots, -3, -2, -1, 0, 1, 2, 3, \dots\).

Integers belong to the built-in type `int`

, which can be used convert
objects to integers:

```
>>> type(-3)
int
# `1.3` is not an integer-type object
>>> isinstance(1.3, int)
False
# converting a string to an integer
>>> int("10")
10
# converting a floating-point number to an integer
>>> int(1.3)
1
```

You can create as large an integer as you’d like; Python will allocate as much memory as needed (and ultimately, as is available) to store an integer’s exact value:

```
# you can make an integer as large as you'd like
>>> large_int = 281938481039848500192847576920
```

Integers have some built-in functions available to them, which are detailed in the official documentation. The utility of these will likely be quite obscure to new programmers. Just note that they are here for now.

### Floating-Point Numbers¶

A “floating-point” number is a number with a decimal point. Referred to as a “float” for short, this can be used to represent any number, up to a limited number of digits.

These objects belong to the built-in type `float`

, which can be used
to convert objects to floats:

```
# examples of "floating-point" numbers
>>> 100. ** 0.5
10.0
>>> 1 / 3
0.3333333333333333
>>> 1 / 2
0.5
>>> type(-2.1)
float
# the integer 10 is not a float-type object
>>> isinstance(10, float)
False
# including a decimal makes the number a float
>>> isinstance(10., float)
True
# converting a string to a floating-point number
>>> float("10.456")
10.456
# converting an integer to a floating-point number
>>> float(-22)
-22.0
```

Floats have a couple of built-in functions available to them, as detailed in the official documentation.

#### Scientific Notation¶

A float can also be created using familiar scientific notation. The
character `e`

is used to represent \(\times 10\), and the
proceeding number is the exponent. Here are some examples of traditional
scientific notation, and their corresponding representation in Python:

\(1.38 \times 10^{-4} \rightarrow\) `1.38e-04`

\(-4.2 \times 10^{10} \rightarrow\) `-4.2e10`

Python will automatically display a float that possesses many digits in scientific notation:

```
# python will display many-digit numbers using
# scientific notation
>>> 0.0000001 # seven leading-zeros
1e-07
```

#### Understanding Numerical Precision¶

Whereas a Python integer can be made to be as large as you’d like, a
floating-point number is *limited in the number of digits it can store*.
That is, your computer will only use a set amount of memory, 8 bytes (32
bits) on most machines, to store the value of a floating-point number.

In effect, this means that a float can only be represented with a
*numerical precision* of approximately 16 decimal places, when that
number is written in scientific notation. The computer will not be able
to reliably represent a number’s digits beyond those accounted for by
the allotted 8 bytes. For instance, the following Python integer is
defined with 23 digits, but when this number is converted to a float, it
only retains 16 decimal places in scientific notation:

```
# Demonstrating the finite-precision of a float.
# An integer with 23 digits - Python will use as
# much memory as needed to store an integer
>>> 18181818181818181818181
18181818181818181818181
# Converted to a float, it retains only
# 16 decimal places, when written in scientific
# notation. This is the precision permitted by
# 8 bytes of memory.
>>> float(18181818181818181818181)
1.8181818181818182e+22
```

The computer cannot keep track of those last 6 decimal places, because doing so would require more than 8 bytes of memory to store the entire value of that float. As such, attempting to modify a number in decimal places beyond its numerical precision does not have any effect:

```
# changing a float beyond its precision has no effect
>>> 1. + 1e-16
1.0
```

Even in light of this discussion on float precision, you may be shocked and dismayed to see the following outcome of float arithmetic:

```
# the finite-precision of floats
# result in non-obvious behavior
>>> 0.1 + 0.1 + 0.1 - 0.3 == 0.
False
# the effects of having finite numerical precision
>>> 0.1 + 0.1 + 0.1 - 0.3
5.551115123125783e-17
```

This is not a quirk of Python; this is a well-understood aspect of dealing with floating-point numbers with limited numerical precision. To accommodate this, don’t check if two floats are “equal”. Rather, you should check if they are “close enough in value”. Let me emphasize this:

**You should never check to see if two floats are exactly equal in
value. Instead, you should only check that two floats are approximately
equal to one another**.

The `math`

module has a very nice function for this; `math.isclose`

will check if the relative difference between two numbers is less than
\(1 \times 10^{-9}\). You can change this tolerance value along with
the type of tolerance-checking used by the function; see its
documentation
here.
Because in the previous example we comparing values that are close to 0,
we will check if their absolute difference is sufficiently small:

```
# checking if two float values are "almost equal"
>>> import math
# check:
# | (0.1 + 0.1 + 0.1 - 0.3) - 0 | < 1x10^{-9}
>>> math.isclose((0.1 + 0.1 + 0.1 - 0.3), 0., abs_tol=1e-9)
True
```

If you do not heed this lesson, it is inevitable that you will end up with serious, hard-to-find bugs in your code. Lastly, when doing numerical work in Python (and any other programming language), you must understand that the finite numerical precision of floating-point numbers is a source of error, akin to error associated with imprecision with a measuring device, and should be accounted for in your analysis (if error analysis is warranted).

Python’s decimal
module can be used
to define higher (or lower) precision numbers than permitted by the
standard 8-byte floats. Furthermore, all arithmetic involving decimal
numbers from this module is guaranteed to be *exact*, meaning that
`0.1 + 0.1 + 0.1 - 0.3`

would be exactly `0.`

. There is also a
built-in fractions
module,
which provides tools for working with exact representations of rational
numbers. Although we will not be using them here, it is very important
to keep in mind that these modules exist and that floating point numbers
are not the only way around the number line in Python.

### Complex Numbers¶

In mathematics, a “complex number” is a number with the form \(a + bi\), where \(a\) and \(b\) are real-valued numbers, and \(i\) is defined to be the number that satisfies the relationship \(i^2 = -1\). Because no real-valued number satisfies this relationship, \(i\) is called the “imaginary number”.

Weirdo electrical engineers use the symbol \(j\) in place of
\(i\), which is why Python displays the complex number
\(2 + 3i\) as `2+3j`

(this is actually because \(i\) typically
denotes current; we like electrical engineers too).

Along with the `a + bj`

syntax, built-in type `complex`

can be used
to create complex-type numbers:

```
# creating complex numbers
>>> 2 + 3j
(2+3j)
>>> complex(2, 3)
(2+3j)
>>> type(2+3j)
complex
>>> isinstance(2-4j, complex)
True
```

You can access `a`

and `b`

from `a + bj`

, the real and imaginary
parts of the complex number, respectively.

```
# Accessing the real and imaginary parts of
# a complex number.
>>> x = complex(1.2, -3.4)
>>> x.real
1.2
>>> x.imag
-3.4
```

The `cmath`

(“complex math”) module provides a collection of
mathematical functions defined for complex numbers. For a complete
listing of these functions, refer to the official
documentation.

**Reading Comprehension: Working with numbers in Python**

1. In Python, performing an arithmetic operation, such as addition or multiplication, on two integers will return an integer, and performing an operation on two floats will return a float:

```
>>> 2 * 3
6
>>> 2.0 * 3.0
6.0
```

For which operation, among `+ - * / **`

, does this *not* hold?

2. What type of number will be returned if you perform a mathematical operation using an integer and a floating-point number? Does this hold for all the arithmetic operations? Determine this by trial and error.

3. Given the function \(f(x) = e^{|x - 2|}\), make use of the
`math`

module to compute \(f(-0.2)\).

### Augmented Assignment Statements¶

Python provides a nice “shortcut” for updating a variable via an
arithmetic operation. For example, suppose you want to increase the
value of `x`

by 1. Currently, we would update `x`

as follows:

```
# incrementing `x` by 1
>>> x = 5
>>> x = x + 1
>>> x
6
```

We can make use of a special assignment operation `+=`

to perform this
update in an abbreviated way.

```
# using `+=` to increment `x` by 1
>>> x = 5
>>> x += 1 # equivalent to: `x = x + 1`
>>> x
6
```

`+=`

is a type of *augmented assignment statement*. In general, an
augmented assignment performs a mathematical operation on a variable,
and then updates that variable using the result. Augmented assignment
statements are available for all of the arithmetic operations. Assuming
`x`

and `n`

are both types numbers, the following summarizes the
available arithmetic augmented assignment statements that we can perform
on `x`

, using `n`

:

`x += n`

\(\rightarrow\)`x = x + n`

`x -= n`

\(\rightarrow\)`x = x - n`

`x *= n`

\(\rightarrow\)`x = x * n`

`x /= n`

\(\rightarrow\)`x = x / n`

`x //= n`

\(\rightarrow\)`x = x // n`

`x **= n`

\(\rightarrow\)`x = x ** n`

## The Boolean Type¶

There are two boolean-type objects: `True`

and `False`

; they belong
to the built-in type `bool`

. We have already seen that the
`isinstance`

function either returns `True`

or `False`

, as a given
object either is or isn’t and instance of a specific type.

```
# the two boolean-objects: `True` and `False`
>>> type(True)
bool
# `False` is a boolean-type object
>>> isinstance(False, bool)
True
```

`True`

and `False`

must be specified with capital letters in Python.
These should not be confused with strings; note that there are no
quotation marks used here.

Python provides familiar operators for performing basic boolean logic:

`and`

`or`

`not`

```
# demonstrating boolean-logic operators
>>> True or False
True
>>> True and False
False
>>> not False
True
```

Grouping with parentheses also works, and multiple operators can be used in a single line:

```
>>> (True or False) and True
True
```

Comparison statements used in basic mathematics naturally return boolean objects.

```
>>> 2 < 3
True
>>> 10.5 < 0
False
>>> (2 < 4) and not (4 != -1)
False
```

The `bool`

type has additional utilities, which will be discussed in
the “Conditional Statements” section.

**Reading Comprehension: Boolean expressions**

1. Assuming `x`

is a an integer-type, write a comparison statement
that will return `True`

if `x`

is an even number, and `False`

otherwise. (Hint: recall the purpose of the `%`

operator)

2. Assuming `x`

and `y`

are both real-valued numbers (i.e. not
complex numbers), write a line of code that will return `False`

if:
`x`

and `y`

are within 0.9 of one another, and `x`

is a positive
number. (Hint: try writing the expression that will return `True`

for
this condition, and then negate it)

3. Write an expression that returns `True`

if `x`

is a boolean-type
object or a float-type object.

## The None-Type¶

There is a simple type, `NoneType`

that has exactly one object:
`None`

. `None`

is used to represent “null”… nothing.

```
# `None` is the *only* object belonging to NoneType
>>> type(None)
NoneType
```

As such, instead of checking if an object belongs to NoneType, you
should simply check if the object is `None`

. Python reserves `is`

as
an operation that checks if two objects are identical. This is different
than `==`

, which checks if two objects are associated with the same
value or state:

```
# Check if an object "is" None, instead
# of checking if it is of NoneType
>>> x = 22
>>> x is None
False
>>> x is not None
True
>>> y = None
>>> y is None
True
```

`None`

appears frequently, and is often used as a placeholder in code.
Here is a simple example where `None`

could be useful; don’t worry
that this code may not make perfect sense to you yet:

```
# Demonstrating the use of `None` as a placeholder
# In this code, we want to get the first
# item in a list that is greater than 10, and notify
# the user if there is no such number
large_num = None
for number in [1, 2, 3, 4]:
if number > 10:
large_num = number
break
if large_num is None:
print("The list did not contain any number larger than 10")
```

## Strings¶

### Introducing the string type¶

The string type is used to store written characters. A string can be formed using:

- single quotes:
`'Hello world'`

- double quotes:
`"Hello world"`

- triple quotes:
`"""Hello world"""`

or`'''Hello world'''`

```
# Strings contain written characters, even those
# not found in the english alphabet!
>>> "hello, 你好, Olá, 123"
'hello, 你好, Olá, 123'
```

By default, Python 3 uses UTF-8 unicode to represent this wide variety of characters. Don’t worry about this detail beyond making note of it, for now.

Strings belong to the built-in `str`

type, which can be used to
convert non-string objects into strings.

```
# the type `str`
>>> type("hello")
str
>>> isinstance("83", str)
True
# Using the type `str` to convert non-string objects
# into strings.
>>> str(10.34)
'10.34'
>>> str(True)
'True'
```

Once a string is formed, it cannot be changed (without creating an
entirely new string). Thus a given string object cannot be “mutated” - a
string is an *immutable* object.

As the string stores a *sequence* of characters, Python provides a means
for accessing individual characters and subsequences of characters from
a string:

```
>>> sentence = "The cat in the hat."
>>> sentence[0]
'T'
>>> sentence[0:3]
'The'
```

Strings are not the only sequence-type in Python; lists and tuples are examples of sequences as well. We will reserve a separate section to learn about the common interface that Python has for all of its types that are sequential in nature, including the “indexing” and “slicing” demonstrated here.

### String essentials¶

We will only scratch the surface with strings, touching on some essentials. Please refer to the official Python tutorial for a more extensive, but still informal, overview of strings.

In a string, `\n`

is treated as a single character. It denotes a
new-line in a string, and will be rendered thusly when the string is
printed. Similarly, `\t`

will render as a tab-character.

```
# using `\n` to create a newline
>>> print("hi...\n...bye")
hi...
...bye
```

Using triple quotes allows you to write a block-string, meaning that you can include text on multiple lines, and it is all still treated as one string:

```
# using triple-quotes to write a multi-line string
>>> x = """ I am a string.
I am part of the same string.
me... too!"""
>>> x
' I am a string.\nI am part of the same string.\n me... too!'
```

Python’s strings have a large number of fantastic, built-in functions
available to them. It is *very important* that you familiarize yourself
with these functions by looking over the official
documentation.
To demonstrate a few of these:

```
# demonstrating a few of the built-in functions for strings
>>> "hello".capitalize()
'Hello'
# join a list of strings, using "..."
>>> "...".join(["item1", "item2", "item3"])
'item1...item2...item3'
# split a string wherever ", " occurs
>>> 'item1, item2, item3'.split(", ")
['item1', 'item2', 'item3']
# does this string end with ".py"?
>>> "script.py".endswith(".py")
True
# does this string start with "sc"?
>>> "script.py".startswith("sc")
True
# insert objects into a string, in its
# "formatting" fields {}
>>> "x: {}, y: {}, z: {}".format(3.2, 8.4, -1.0)
'x: 3.2, y: 8.4, z: -1.0'
# Are the characters in the string
# numerical digits?
>>> "7".isdigit()
True
```

### Official documentation for strings¶

It is highly recommended that you take time to read over all of the functions that are built-in to a string.

**Reading Comprehension: Strings**

To answer some of the following questions, you will need to peruse the documentation for the built-in functions of strings. It may take a bit of experimentation to understand the documentation’s use of square-brackets to indicate optional inputs for a function.

1. Use a function that will take the string `"cat"`

, and returns the
string `" cat "`

(which has a length of 11, including the letters
c, a, t). Now, change the way you call the function so that it returns
`"----cat----"`

instead.

2. Replace the first three periods of this string with a
space-character: `"I.am.aware.that.spaces.are.a.thing"`

- Remove the whitespace from both ends of:
`" basket "`

4. Create a string that will print as (the second line begins with a tab-character):

```
Hello
over there
```

- Convert the integer
`12`

to the string`"12"`

.

## Lists¶

A `list`

is a type of Python object that allows us to store a sequence
of other objects. One of its major utilities is that it provides us with
means for updating the contents of a list later on.

A list object is created using square-brackets, and its contents are
separated by commas: `[item1, item2, ..., itemN]`

. Its contents need
not be of the same type of object.

```
# a list-type object stores a sequence of other objects
>>> [3.5, None, 3.5, True, "hello"]
[3.5, None, 3.5, True, 'hello']
>>> type([1, 2, 3])
list
>>> isinstance([1, 2], list)
True
# constructing an empty list
>>> []
[]
# constructing a list with only one member
>>> ["hello"]
["hello"]
```

You can also include variables, equations, and other Python expressions in the list constructor; Python will simplify these expressions and construct the list with the resulting objects.

```
# the list constructor will simplify expressions
# and store their resulting objects
>>> x = "hello"
>>> [2 < 3, x.capitalize(), 5**2, [1, 2]]
[True, 'Hello', 25, [1, 2]]
```

The built-in `list`

type can be used to convert other types of
sequences (and more generally, any *iterable* object, which we will
discuss later) into a list:

```
# `list` forms a list out of the contents of other sequences
>>> list("apple")
['a', 'p', 'p', 'l', 'e']
```

### Lists are sequences¶

Like a string, the ordering of a list’s contents matters, meaning that a list is sequential in nature.

```
# A list's ordering matters
>>> [1, "a", True] == [1, True, "a"]
False
```

Thus a list supports the same mechanism for accessing its contents, via indexing and slicing, as does a string. Indexing and slicing will be covered in detail in the next section.

```
# Accessing the contents of a list with indexing and slicing
>>> x = [2, 4, 6, 8, 10]
# `x` contains five items
>>> len(x)
5
# access the 0th item in the list via "indexing"
>>> x[0]
2
# access a subsequence of the list via "slicing"
>>> x[1:3]
[4, 6]
```

### Lists can be “mutated”¶

We will encounter other types of containers in Python, what makes the
list stand out is that the *contents of a list can be changed after the
list has already been constructed*. Thus a list is an example of a
*mutable* object.

```
# changing a list after it has been constructed
>>> x = [2, 4, 6, 8, 10]
>>> y = [2, 4, 6, 8, 10]
# "set" the string 'apple' into position-1 of `x`
>>> x[1] = "apple"
>>> x
[2, 'apple', 6, 8, 10]
# replace a subsequence of `y`
>>> y[1:4] = [-3, -4, -5]
>>> y
[2, -3, -4, -5, 10]
```

The built-in list-functions “append” and “extend” allow use to add one item and multiple items to the end of a list, respectively:

```
>>> x = [2, 4, 6, 8, 10]
# use `append` to add a single object to the end of a list
>>> x.append("moo")
>>> x
[2, 4, 6, 8, 10, 'moo']
# use `extend` to add a sequence of items to the end of a list
>>> x.extend([True, False, None])
>>> x
[2, 4, 6, 8, 10, 'moo', True, False, None]
```

The “pop” and “remove” functions allow us to remove an item from a list based on its position in the list, or by specifying the item itself, respectively.

```
>>> x = ["a", "b", "c", "d"]
# pop the position-1 item out from a list
# `pop` will return the item that gets removed.
>>> x.pop(1)
'b'
>>> x
['a', 'c', 'd']
# remove the object "d" from the list
>>> x.remove("d")
>>> x
['a', 'c']
```

### Official documentation for lists¶

It is highly recommended that you take time to read over all of the functions that are built-in to a list. These are all designed to allow us to either inspect or mutate the contents of a list.

**Reading Comprehension: Lists**

To answer some of the following questions, you will need to peruse the documentation for the built-in functions of lists.

- Create a list whose sole entry is the
`None`

object.

2. Assign to the variable `k`

a list that contains an integer, a
boolean, and a string, in that order. Then, add two more entries to the
end of the list: a float and a complex number.

3. Alphabetize the list of names:
`["Jane", "Adam", "Ryan", "Bob", "Zordon", "Jack", "Jackenzie"]`

.

## Summary¶

The term “object” is a catch-all in Python, meaning anything that we can assign to a variable. Objects behave differently from one another according to what “type” a given object is.

We reviewed several fundamental object types in Python:

`int`

,`float`

,`complex`

: the numerical types`bool`

: the boolean type.`True`

and`False`

are the only boolean-type objects`NoneType`

: the “null” type;`None`

is*the only object that belongs to this type*`str`

: the string type`list`

: the list type

The built-in function `type`

permits us to check the type of any
object:

```
>>> type(3.2)
float
>>> type(True)
bool
```

The built-in function `isinstance`

should be used to check if an
object is of a specific type:

```
>>> x = 2 + 3
>>> isinstance(x, int)
True
```

The only exception to this is if you want to check if an object is of
the type `NoneType`

, since this is only possible if the object is
`None`

; thus it is “cleaner” to directly check this:

```
>>> x = None
>>> x is None
True
```

Objects of different types have different built-in functions available to them:

```
>>> x = "I am a farmer.. moo"
>>> x.upper()
'I AM A FARMER.. MOO'
>>> y = 0.5
>>> y.as_integer_ratio()
(1, 2)
>>> z = [1, 2]
>>> z.append(3)
>>> z
[1, 2, 3]
```

You should leverage the official documentation, for which links were provided throughout this section, whenever you are wondering how to best do a specific task with a given type of object.

As a final reminder, we saw that that an integer in Python is able to hold a value with arbitrarily-many digits. A floating-point number, on the other hand, is restricted in the number of “significant” digits it can hold. Thus, where it is perfectly fine to check if two integers are exactly equal:

```
# checking if two integers are equal is great!
>>> 2 + 2 == 4
True
```

you should never rely on two floats being exactly equal. Instead, check if they are close in value:

```
# checking if two floats are equal is lame!
>>> 0.1 + 0.1 + 0.1 - 0.3 == 0.
False
>>> abs((0.1 + 0.1 + 0.1 - 0.3) - 0.) < 1e-12
True
```

It is very important to remember this issue of the limited numerical precision of a floating point number.

## Links to Official Documentation¶

## Reading Comprehension Exercise Solutions:¶

**Understanding the modulo operator: Solution**

If `n`

is an integer, then 2 will divide into it evenly, and thus
there is no remainder. If `n`

is odd, then `n / 2`

must have a
remainder of 1. Thus:

`n % 2`

= 0 if`n`

is an even number`n % 2`

= 1 if`n`

is an odd number

**Working with numbers in Python: Solution**

1. The division operator, `/`

, will return a floating-point number
even if it is operating on two integer-type numbers.

2. For all of the arithmetic operations, `+ - * / **`

, operating on a
floating-point number and an integer will return a floating-point
number:

```
>>> 2 * 3.0
6.0
>>> -1.0 + 2
1.0
```

```
# 3. Given the function f(x)=e^|x−2|f(x), make use
# of the math module to compute f(−0.2).
>>> from math import exp
>>> x = -0.2
>>> exp(abs(x - 2))
9.025013499434122
```

**Boolean expressions: Solutions**

```
# 1. Write an a comparison-statement that will return
# True if x is an even number
x%2 == 0
```

```
# 2. Write a line of code that will return False if:
# x and y are within 0.9 of one another, and x is a positive number.
not (abs(x - y) < 0.9 and 0 < x)
# alternatively,
abs(x - y) > 0.9 or 0 > x
```

```
# 3. Write an expression that returns True if
# x is a boolean-type object or a float-type object
isinstance(x, bool) or isinstance(x, float)
```

**Strings: Solutions**

```
# 1. Use a function that will take the string "cat"
# and returns the string " cat "
>>> "cat".center(11)
' cat '
>>> "cat".center(11, "-")
'----cat----'
```

```
# 2. Replace the first three periods of this string with
# a space-character:
>>> "I.am.aware.that.spaces.are.a.thing".replace(".", " ", 3)
'I am aware that.spaces.are.a.thing'
```

```
# 3. Remove the whitespace from both ends
# of: " basket "
>>> " basket ".strip()
'basket'
```

```
# 4.
>>> print("Hello\n\tover there")
Hello
over there
```

```
# 5. Convert the integer 12 to the string "12"
>>> str(12)
'12'
```

**Lists: Solutions**

```
# Create a list whose sole entry is the None object.
>>> [None]
[None]
```

```
# 2. Assign the variable k to a list that contains an
# integer, a boolean, and a string, in that order.
# Then, add two more entries to the end of the
# list - a float and a complex number.
>>> k = [4, False, "moo"]
>>> k.extend([3.14, complex(9, -2)])
>>> k
[4, False, 'moo', 3.14, (9-2j)]
```

```
# 3. Alphabetize the list of names:
>>> names = ["Jane", "Adam", "Ryan", "Bob", "Zordon", "Jack", "Jackenzie"]
# The documentation for `sort` says that the sorting happens "in-place". This
# means that the original list is replaced by the sorted list. The alternative
# would be that `sort` returns a new, sorted list, without changing the list
# assigned to `names`.
>>> names.sort()
>>> names
['Adam', 'Bob', 'Jack', 'Jackenzie', 'Jane', 'Ryan', 'Zordon']
```