As this chapter solely recaps some of the mathematical notation we will use in later chapters, there are no code examples given and hence no technical requirements for this particular chapter.

For later chapters, you will be able to find code examples at the GitHub repository: https://github.com/PacktPublishing/15-Math-Concepts-Every-Data-Scientist-Should-Know

In this section, we introduce notation for describing sets of numbers. We will focus on the real numbers and the complex numbers.

Notation for numbers and fields

As this is a book about data science, we will be dealing with numbers. So, it will be worthwhile recapping the notation we use to refer to the most common sets of numbers.

Most of the numbers we will deal with in this book will be real numbers, such as 4.6, 1, or -2.3. We can think of them as “living” on the real number line shown in Figure 1.1. The real number line is a one-dimensional continuous structure. There are an infinite number of real numbers. We denote the set of all real numbers by the symbol ℝ.

Figure 1.1: The real number line

Obviously, there will be situations where we want to restrict our datasets to, say, just integer-valued numbers. This would be the case if we were analyzing count data, such as the number of items of a particular...

In this section, we introduce notation to describe vectors and matrices, which are key mathematical objects that we will encounter again and again throughout this book.

Vectors

In many circumstances, we will want to represent a set of numbers together. For example, the numbers 7.3 and 1.2 might represent the values of two features that correspond to a data point in a training set. We often group these numbers together in brackets and write them as (7.3, 1.2) or [7.3, 1.2]. Because of the similarity to the way we write spatial coordinates, we tend to call a collection of numbers that are held together a vector. A vector can be two-dimensional, as in the example just given, or d-dimensional, meaning it contains d components, and so might look like .

We can write a vector in two ways. We can write it as a row vector, going across the page, such as the following vector:

d-dimensional row vector

Eq. 8

Alternatively, we can write it as a column vector going...

In this section, we introduce notation for doing the most basic operations we can do with numbers, namely adding them together or multiplying them together. We’ll then introduce notation for working with logarithms.

Sums and the 𝚺 notation

When we want to add several numbers together, we can use the summation, or , notation. For example, if we want to represent the addition of the numbers , we use the notation to write this as follows:

Eq. 13

This notation is shorthand for writing . This essentially defines what the notation represents – that is, the following:

Eq. 14

In the left-hand side (LHS) of Eq. 14, the integer indexing variable, , takes the values between 1 (indicated beneath the symbol) and 5 (indicated above the symbol) and we interpret the LHS as “take all the numbers for the values of indicated by the symbol and add them together.”

You may wonder whether the shorthand...

In this section, we won’t go into the fundamentals of differential calculus, but instead just recap some basic results and notation. Therefore, we are assuming you already have some basic familiarity with differentiation and integration.

Differentiation

Let’s start with what the derivative of a function or curve intuitively represents. An example curve is shown in Figure 1.5. The derivative of this function is denoted by the following symbol:

Eq. 29

The derivative of is itself a function of . The numerical value of the derivative evaluated at a particular value of , let’s say at , is the gradient (or slope) of the tangent to the curve at . As such, we can think of the derivative as defining the local gradient value of the curve. This is illustrated in Figure 1.5 as well:

Figure 1.5: The derivative as the gradient of the tangent to the curve

Sometimes, when we want to be explicit...

In this section, we recap the notation that is used in analyzing and describing the behavior of functions. This includes notation for describing limits, notation for describing the relative ordering of functions, and notation for describing standard approximations of functions.

Limits

When we are talking about limits, we are talking about the mathematical behavior of some quantity, often a function, as some other quantity approaches a particular value, often infinity. Let’s make that more concrete. Consider the function . As gets bigger and bigger, then clearly, the value of gets smaller and smaller, until eventually, as becomes infinitely large, becomes zero. We say that approaches its limit of 0 as approaches . Mathematically, we write this as follows:

Eq. 46

The word lim denotes the fact that we are talking about a limit, while the symbols describe what limit we are talking about. The RHS of Eq. 46 gives the actual limiting value.

Sometimes...

Our final section regards binomial coefficients. They are part of the mathematical field of combinatorics, but we will introduce them in the context of the binomial distribution, which we will meet multiple times in the book.

Binomial coefficients

Along with the normal or Gaussian distribution, the binomial distribution is one of the most common distributions we will encounter as data scientists. It is the distribution of the number of times, , we observe a particular outcome in a set of observations, where in each observation there are only two possibilities that can occur. Given we are interested only in the total number, , of successful outcomes of a particular type, a large part of calculating the associated probability comes down to calculating how many ways we can distribute or arrange the successes between the observations. The answer is given by the binomial coefficient . This is defined mathematically as follows:

Eq. 62

Here, means factorial...

We have completed our brief recap of the main notation and terminology that we will need for our tour of math concepts in data science. We are now ready for that tour, so let’s begin.

In the next chapter, we will meet our first key math concept when we learn about random variables and probability distributions.

1. An example of one of the more well-known “big book of integrals” is Table of Integrals, Series, and Products by I.S. Gradshteyn and I.M. Ryzhik, 8th Edition, Daniel Zwillinger (Editor), Academic Press (Cambridge, Massachusetts, USA), 2014. This is the book of integrals that I use.