#### Overview of this book

If you’re a Python developer who wants to discover how to take the power of functional programming (FP) and bring it into your own programs, then this book is essential for you, even if you know next to nothing about the paradigm. Starting with a general overview of functional concepts, you’ll explore common functional features such as first-class and higher-order functions, pure functions, and more. You’ll see how these are accomplished in Python 3.6 to give you the core foundations you’ll build upon. After that, you’ll discover common functional optimizations for Python to help your apps reach even higher speeds. You’ll learn FP concepts such as lazy evaluation using Python’s generator functions and expressions. Moving forward, you’ll learn to design and implement decorators to create composite functions. You'll also explore data preparation techniques and data exploration in depth, and see how the Python standard library fits the functional programming model. Finally, to top off your journey into the world of functional Python, you’ll at look at the PyMonad project and some larger examples to put everything into perspective.
Title Page
Packt Upsell
Contributors
Preface
Free Chapter
Understanding Functional Programming
Introducing Essential Functional Concepts
Functions, Iterators, and Generators
Working with Collections
Recursions and Reductions
The Itertools Module
More Itertools Techniques
The Functools Module
Decorator Design Techniques
Conditional Expressions and the Operator Module
A Functional Approach to Web Services
Optimizations and Improvements
Other Books You May Enjoy
Index

## Simple numerical recursions

We can consider all numeric operations to be defined by recursions. For more details, read about the Peano axioms that define the essential features of numbers at: http://en.wikipedia.org/wiki/Peano_axioms.

From these axioms, we can see that addition is defined recursively using more primitive notions of the next number, or successor of a number, n

.

To simplify the presentation, we'll assume that we can define a predecessor function,

, such that

, as long as

. This formalizes the idea that a number is the successor of the number's predecessor.

Addition between two natural numbers could be defined recursively as follows:

If we use the more common

and

and

, we can see that

.

This translates neatly into Python, as shown in the following command snippet:

```def add(a: int, b: int) -> int:
if a == 0:
return b
else: