Set of numbers (Real, integer, rational, natural and irrational numbers)

In this unit, we shall give a brief, yet more meaningful introduction to the concepts of sets of numbers, the set of real numbers being the most important, and being denoted by $\mathbb{R}$.

But first, to get to the real numbers we start at the set of natural numbers.

Natural numbers $\mathbb{N}$

Natural numbers are those who from the beginning of time have been used to count. In most countries they have adopted the Arabic numerals, so called because it was the Arabs who introduced them in Europe, but it was in India where they were invented.

The set of natural numbers is denoted as $\mathbb{N}$; so:

$$\mathbb{N}=\{1,2,3,4,5,6\dots \}$$

Natural numbers are characterized by two properties:

• The number 1 is the first natural number and each natural number is formed by adding 1 to the previous one.
• When we subtract or divide two natural numbers the result is not necessarily a natural number, so we say that natural numbers are not closed under these two operations. Natural numbers are only closed under addition and multiplication, ie, the addition or multiplication of two natural numbers always results in another natural number.

Integers $\mathbb{Z}$

When the need to distinguish between some values and others from a reference position appears is when negative numbers come into play. For example, when from level 0 (sea level) we differentiate above sea level or deep sea. Or in the case of temperatures below zero or positive. So we can be at an altitude of 700m, $+700$, or dive to 10m deep, $-10$, and it can be about 25 degrees $+25$, or 5 degrees below 0, $-5$.

To denote negative numbers we add a minus sign before the number.

In short, the set formed by the negative integers, the number zero and the positive integers (or natural numbers) is called the set of integers.

They are denoted by the symbol $\mathbb{Z}$ and can be written as:

$$\mathbb{Z}=\{\dots ,-2,-1,0,1,2,\dots \}$$

We represent them on a number line as follows:

An important property of integers is that they are closed under addition, multiplication and subtraction, that is, any addition, subtraction and multiplication of two integers results in another integer. Note that the quotient of two integers, for instance $3$ and $7$, is not necessarily an integer. Thus, the set is not closed under division.

Rational numbers $\mathbb{Q}$

Rational numbers are those numbers which can be expressed as a division between two integers. The set of rational numbers is denoted as $\mathbb{Q}$, so:

$$\mathbb{Q}=\{{\displaystyle \frac{p}{q}}|p,q\in \mathbb{Z}\}$$

The result of a rational number can be an integer ($-{\displaystyle \frac{8}{4}}=-2$) or a decimal (${\displaystyle \frac{6}{5}}=1,2$) number, positive or negative. Furthermore, among decimals there are two different types, one with a limited number of digits which it's called an exact decimal, (${\displaystyle \frac{88}{25}}=3,52$), and another one with an unlimited number of digits which it's called a recurring decimal (${\displaystyle \frac{5}{9}}=0,5555\dots =0,\hat{5}$).

We call them recurring decimals because some of the digits in the decimal part are repeated over and over again. If just repeating digits begin at tenth, we call them pure recurring decimals ($6,8888\dots =6,\hat{8}$), otherwise we call them mixed recurring decimals ($3,415626262\dots =3,415\widehat{62}$).

Note that every integer is a rational number, since, for example, $5={\displaystyle \frac{5}{1}}$; therefore, $\mathbb{Z}$ is a subset of $\mathbb{Q}$. In the same way every natural is also an integer number, specifically positive integer number. Thus we have:

$$\mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}$$

The rational numbers are closed not only under addition, multiplication and subtraction, but also division (except for $0$).

Irrational numbers $\mathbb{I}$

We have seen that any rational number can be expressed as an integer, decimal or exact decimal number.

However, not all decimal numbers are exact or recurring decimals, and therefore not all decimal numbers can be expressed as a fraction of two integers.

These decimal numbers which are neither exact nor recurring decimals are characterized by infinite nonperiodic decimal digits, ie that never end nor have a repeating pattern.

Note that the set of irrational numbers is the complementary of the set of rational numbers.

Some examples of irrational numbers are $\sqrt{2},\pi ,\sqrt[3]{5},$ and for example $\pi =3,1415926535\dots$ comes from the relationship between the length of a circle and its diameter.

Real numbers $\mathbb{R}$

The set formed by rational numbers and irrational numbers is called the set of real numbers and is denoted as $\mathbb{R}$.

Thus we have:

$$\mathbb{R}=\mathbb{Q}\cup \mathbb{I}$$

Both rational numbers and irrational numbers are real numbers.

One of the most important properties of real numbers is that they can be represented as points on a straight line. We choose a point called origin, to represent $0$, and another point, usually on the right side, to represent $1$.

A correspondence between the points on the line and the real numbers emerges naturally; in other words, each point on the line represents a single real number and each real number has a single point on the line. We call it the real line. In the next picture you can see an example: