5. More Mathematics with Python

Overview

By the end of this chapter, you will be able to grasp the basic concepts of sequences and series and write Python functions that implement these concepts. You will understand the relationships between basic trigonometric functions and their applications, such as the famous Pythagorean theorem. You will practice vector calculus and know where it is applicable by performing vector algebra in Python. Finally, you will feel happy knowing that complex numbers are not any less a type of number; they are intimately connected to trigonometry and are useful for real-world applications.

Introduction

In the previous chapter, we covered functions and algebra with Python, starting with basic functions before working through transformations and solving equations. In this chapter, we'll introduce sequences and series, which have many applications in the real world, such as finance, and also form the basis for an understanding of calculus. Additionally, we will explore trigonometry, vectors, and complex numbers to give us a better understanding of the mathematical world.

The core skills of any exceptional Python programmer include a solid understanding of the background mathematics and an effective application of them. Think of vectors and complex numbers as valuable extensions to our mathematical toolbox that, later on, will contribute to efficiently describing, quantifying, and tackling real-world problems from the finance, science, or business and social domains.

Sequences and series, among others, appear in situations where profits, losses, dividends, or...

Sequences and Series

If you were to participate in a TV show where the $10,000 question was "Given the numbers 2, 4, 8, 16, and 32, what comes next in the sequence?", what would your best guess be? If your response is 64, then congratulations—you just came closer to understanding one of the key concepts in mathematical abstraction: that of a sequence. A sequence is, pretty much like in the ordinary sense of the word, a particular order in which things follow each other. Here, things are (in most cases) integers or real numbers that are related. The order of the elements matters. The elements are also called the members or terms of the sequence.

For example, in the preceding sequence of the TV show you participated in, every term stems from the number prior being multiplied by 2; there is no end in this sequence as there is no end in the number of terms (integer numbers) you can come up with. In other instances, elements in a sequence can appear more than once. Think...

Trigonometry

Trigonometry is about studying triangles and, in particular, the relation of their angles to their edges. The ratio of two of the three edges (sides) of a triangle gives information about a particular angle, and to such a pair of sides, we give it a certain name and call it a function. The beauty of trigonometry and mathematics in general is that these functions, which are born inside a triangle, make (abstract) sense in any other situation where triangles are not present and operate as independent mathematical objects. Hence, functions such as the tangent, cosine, and sine are found across most fields of mathematics, physics, and engineering without any reference to the triangle.

Let's look at the most fundamental trigonometric functions and their usage.

Basic Trigonometric Functions

We will start by defining a right-angled triangle (or simply a right triangle), triangle ABC. One of its angles (the angle BCA in the following diagram) is a right angle, that...

Vectors

Vectors are abstract mathematical objects with a magnitude (size) and direction (orientation). A vector is represented by an arrow that has a base (tail) and a head. The head shows the direction of the vector, while the length of the arrow's body shows its magnitude.

A scalar, in contrast to a vector, is a sole number. It's a non-vector, that is, a pure integer, real or complex (as we shall see later), that has no elements and hence no direction.

Vectors are symbolized by either a bold-faced letter A, a letter with an arrow on top, or simply by a regular letter, if there is no ambiguity regarding the notation in the discussion. The magnitude of the vector, A, is stylized as |A| or simply A. Now, let's have a look at the various vector operations.

Vector Operations

Simply put, a vector is a collection (think of a list or array) of two, three, or more numbers that form a mathematical object. This object lives in a particular geometrical space called...

Complex Numbers

Mathematical ideas have been evolving regarding numbers and their relationships since ancient numerical systems. Historically, mathematical ideas have evolved from concrete to abstract ones. For instance, a set of natural numbers was created so that all physical objects in the world around us directly correspond to some number within this set. Since arithmetic and algebra have developed, it has become clear that numbers beyond the naturals or integers are necessary, so decimal and rational numbers were introduced. Similarly, around the times of Pythagoras, it was found that rational numbers cannot solve all numerical problems that we could construct with the geometry that was known at that time. This happened when irrational numbers—numbers that result from taking the square root of other numbers and that have no representation as ratios—were introduced.

Complex numbers are an extension of real numbers and include some special numbers that can provide...

Summary

In this chapter, you have been provided with a general and helpful exposition of the most central mathematical concepts in sequences, series, trigonometry, vectors, and complex numbers and, more importantly, their implementation in Python using concrete and short examples. As a real-life example, we examined a retirement plan and the progression of our savings in time. However, numerous other situations can be modeled after sequences or series and be studied by applying vectors or complex numbers. These concepts and methods are widely used in physics, engineering, data science, and more. Linear algebra, that is, the study of vectors, matrices, and tensors, heavily relies on understanding the concept of geometry and vectors and appears almost everywhere in data science and when studying neural networks. Geometry and trigonometry, on the other hand, are explicitly used to model physical motion (in video games, for instance) and more advanced concepts in geospatial applications...