#### Overview of this book

Python is the preferred choice of developers, engineers, data scientists, and hobbyists everywhere. It is a great language that can power your applications and provide great speed, safety, and scalability. It can be used for simple scripting or sophisticated web applications. By exposing Python as a series of simple recipes, this book gives you insight into specific language features in a particular context. Having a tangible context helps make the language or a given standard library feature easier to understand. This book comes with 133 recipes on the latest version of Python 3.8. The recipes will benefit everyone, from beginners just starting out with Python to experts. You'll not only learn Python programming concepts but also how to build complex applications. The recipes will touch upon all necessary Python concepts related to data structures, object oriented programming, functional programming, and statistical programming. You will get acquainted with the nuances of Python syntax and how to effectively take advantage of it. By the end of this Python book, you will be equipped with knowledge of testing, web services, configuration, and application integration tips and tricks. You will be armed with the knowledge of how to create applications with flexible logging, powerful configuration, command-line options, automated unit tests, and good documentation.

# Choosing between float, decimal, and fraction

Python offers several ways to work with rational numbers and approximations of irrational numbers. We have three basic choices:

• Float
• Decimal
• Fraction

With so many choices, when do we use each?

It's important to be sure about our core mathematical expectations. If we're not sure what kind of data we have, or what kinds of results we want to get, we really shouldn't be coding yet. We need to take a step back and review things with a pencil and paper.

There are three general cases for math that involve numbers beyond integers, which are:

1. Currency: Dollars, cents, euros, and so on. Currency generally has a fixed number of decimal places. Rounding rules are used to determine what 7.25% of \$2.95 is, rounded to the nearest penny.
2. Rational Numbers or Fractions: When we're working with American units like feet and inches, or cooking measurements in cups and fluid ounces, we often need to work in fractions. When we scale a recipe that serves eight, for example, down to five people, we're doing fractional math using a scaling factor of `5/8`. How do we apply this scaling to `2/3` cup of rice and still get a measurement that fits an American kitchen gadget?
3. Irrational Numbers: This includes all other kinds of calculations. It's important to note that digital computers can only approximate these numbers, and we'll occasionally see odd little artifacts of this approximation. Float approximations are very fast, but sometimes suffer from truncation issues.

When we have one of the first two cases, we should avoid floating-point numbers.

## How to do it...

We'll look at each of the three cases separately. First, we'll look at computing with currency. Then, we'll look at rational numbers, and after that, irrational or floating-point numbers. Finally, we'll look at making explicit conversions among these various types.

### Doing currency calculations

When working with currency, we should always use the `decimal` module. If we try to use the values of Python's built-in `float` type, we can run into problems with the rounding and truncation of numbers:

1. To work with currency, we'll do this. Import the `Decimal` class from the `decimal` module:
``````>>> from decimal import Decimal
``````
2. Create `Decimal` objects from strings or integers. In this case, we want 7.25%, which is 7.25/100. We can compute the value using `Decimal` objects. We could have used `Decimal('0.0725')` instead of doing the division explicitly. The result is a hair over \$0.21. It's computed correctly to the full number of decimal places:
``````>>> tax_rate = Decimal('7.25')/Decimal(100)
>>> purchase_amount = Decimal('2.95')
>>> tax_rate * purchase_amount
Decimal('0.213875')
``````
3. To round to the nearest penny, create a `penny` object:
``````>>> penny = Decimal('0.01')
``````
4. Quantize your data using this `penny` object:
``````>>> total_amount = purchase_amount + tax_rate * purchase_amount
>>> total_amount.quantize(penny)
Decimal('3.16')
``````

This shows how we can use the default rounding rule of `ROUND_HALF_EVEN`.

Every financial wizard (and many world currencies) have different rules for rounding. The `Decimal` module offers every variation. We might, for example, do something like this:

``````>>> import decimal
>>> total_amount.quantize(penny, decimal.ROUND_UP)
Decimal('3.17')
``````

This shows the consequences of using a different rounding rule.

### Fraction calculations

When we're doing calculations that have exact fraction values, we can use the `fractions` module. This provides us with handy rational numbers that we can use. In this example, we want to scale a recipe for eight down to five people, using `5/8` of each ingredient. When we need 2 cups of sugar, what does that turn out to be?

To work with fractions, we'll do this:

1. Import the `Fraction` class from the `fractions` module:
``````>>> from fractions import Fraction
``````
2. Create `Fraction` objects from strings, integers, or pairs of integers. If you create fraction objects from floating-point values, you may see unpleasant artifacts of float approximations. When the denominator is a power of 2, – , , and so on, converting from float to fraction can work out exactly. We created one fraction from a string, `'2.5'`. We created the second fraction from a floating-point calculation, `5/8`. Because the denominator is a power of 2, this works out exactly:
``````>>> sugar_cups = Fraction('2.5')
>>> scale_factor = Fraction(5/8)
>>> sugar_cups * scale_factor
Fraction(25, 16)
``````
3. The result, , is a complex-looking fraction. What's a nearby fraction that might be simpler?
``````>>> Fraction(24,16)
Fraction(3, 2)
``````

We can see that we'll use almost a cup and a half of sugar to scale the recipe for five people instead of eight.

### Floating-point approximations

Python's built-in `float` type can represent a wide variety of values. The trade-off here is that `float` often involves an approximation. In a few cases—specifically when doing division that involves powers of 2—it can be as exact as `fraction`. In all other cases, there may be small discrepancies that reveal the differences between the implementation of `float` and the mathematical ideal of an irrational number:

1. To work with `float`, we often need to round values to make them look sensible. It's important to recognize that all `float` calculations are an approximation:
``````>>> (19/155)*(155/19)
0.9999999999999999
``````
2. Mathematically, the value should be `1`. Because of the approximations used for `float`, the answer isn't exact. It's not wrong by much, but it's wrong. In this example, we'll use `round(answer, 3)` to round to three digits, creating a value that's more useful:
``````>>> answer = (19/155)*(155/19)
1.0
``````
3. Know the error term. In this case, we know what the exact answer is supposed to be, so we can compare our calculation with the known correct answer. This gives us the general error value that can creep into floating-point numbers:
``````>>> 1-answer
1.1102230246251565e-16
``````

For most floating-point errors, this is the typical value—about 10-16. Python has clever rules that hide this error some of the time by doing some automatic rounding. For this calculation, however, the error wasn't hidden.

This is a very important consequence.

Don't compare floating-point values for exact equality.

When we see code that uses an exact `==` test between floating-point numbers, there are going to be problems when the approximations differ by a single bit.

### Converting numbers from one type into another

We can use the `float()` function to create a `float` value from another value. It looks like this:

``````>>> float(total_amount)
3.163875
>>> float(sugar_cups * scale_factor)
1.5625
``````

In the first example, we converted a `Decimal` value into `float`. In the second example, we converted a `Fraction` value into `float`.

It rarely works out well to try to convert `float` into `Decimal` or `Fraction`:

``````>>> Fraction(19/155)
Fraction(8832866365939553, 72057594037927936)
>>> Decimal(19/155)
Decimal('0.12258064516129031640279123394066118635237216949462890625')
``````

In the first example, we did a calculation among integers to create a `float` value that has a known truncation problem. When we created a `Fraction` from that truncated `float` value, we got some terrible - looking numbers that exposed the details of the truncation.

Similarly, the second example tries to create a `Decimal` value from a `float` value that has a truncation problem, resulting in a complicated value.

## How it works...

For these numeric types, Python offers a variety of operators: `+`, `-`, `*`, `/`, `//`, `%`, and `**`. These are for addition, subtraction, multiplication, true division, truncated division, modulo, and raising to a power, respectively. We'll look at the two division operators in the Choosing between true division and floor division recipe.

Python is adept at converting numbers between the various types. We can mix `int` and `float` values; the integers will be promoted to floating-point to provide the most accurate answer possible. Similarly, we can mix `int` and `Fraction` and the results will be a `Fraction` object. We can also mix `int` and `Decimal`. We cannot casually mix `Decimal` with `float` or `Fraction`; we need to provide explicit conversions in that case.

It's important to note that `float` values are really approximations. The Python syntax allows us to write numbers as decimal values; however, that's not how they're processed internally.

We can write a value like this in Python, using ordinary base-10 values:

``````>>> 8.066e+67
8.066e+67
``````

The actual value used internally will involve a binary approximation of the decimal value we wrote. The internal value for this example, `8.066e+67`, is this:

``````>>> (6737037547376141/(2**53))*(2**226)
8.066e+67
``````

The numerator is a big number, `6737037547376141`. The denominator is always 253. Since the denominator is fixed, the resulting fraction can only have 53 meaningful bits of data. This is why values can get truncated. This leads to tiny discrepancies between our idealized abstraction and actual numbers. The exponent (2226) is required to scale the fraction up to the proper range.

Mathematically, .

We can use `math.frexp()` to see these internal details of a number:

``````>>> import math
>>> math.frexp(8.066E+67)
(0.7479614202861186, 226)
``````

The two parts are called the mantissa (or significand) and the exponent. If we multiply the mantissa by 253, we always get a whole number, which is the numerator of the binary fraction.

The error we noticed earlier matches this quite nicely: 10-16 ≈ 2-53.

Unlike the built-in `float`, a `Fraction` is an exact ratio of two integer values. As we saw in the Working with large and small integers recipe, integers in Python can be very large. We can create ratios that involve integers with a large number of digits. We're not limited by a fixed denominator.

A `Decimal` value, similarly, is based on a very large integer value, as well as a scaling factor to determine where the decimal place goes. These numbers can be huge and won't suffer from peculiar representation issues.

Why use floating-point? Two reasons: Not all computable numbers can be represented as fractions. That's why mathematicians introduced (or perhaps discovered) irrational numbers. The built-in float type is as close as we can get to the mathematical abstraction of irrational numbers. A value like , for example, can't be represented as a fraction. Also, float values are very fast on modern processors.

## There's more...

The Python `math` module contains several specialized functions for working with floating-point values. This module includes common elementary functions such as square root, logarithms, and various trigonometry functions. It also has some other functions such as gamma, factorial, and the Gaussian error function.

The `math` module includes several functions that can help us do more accurate floating-point calculations. For example, the `math.fsum()` function will compute a floating-point sum more carefully than the built-in `sum()` function. It's less susceptible to approximation issues.

We can also make use of the `math.isclose()` function to compare two floating-point values to see if they're nearly equal:

``````>>> (19/155)*(155/19) == 1.0
False
>>> math.isclose((19/155)*(155/19), 1)
True
``````

This function provides us with a way to compare floating-point numbers meaningfully for near-equality.

Python also offers complex numbers. A complex number has a real and an imaginary part. In Python, we write `3.14+2.78j` to represent the complex number . Python will comfortably convert between float and complex. We have the usual group of operators available for complex numbers.

To support complex numbers, there's the `cmath` package. The `cmath.sqrt()` function, for example, will return a complex value rather than raise an exception when extracting the square root of a negative number. Here's an example:

``````>>> math.sqrt(-2)
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
ValueError: math domain error
>>> cmath.sqrt(-2)
1.4142135623730951j
``````

This is essential when working with complex numbers.