Number Systems: Natural, Integer, Rational, Irrational and Real Numbers

Intuitively, a set may be thought of as a collection of well-defined objects, called its elements. In an elementary treatment, this means that it must be clear, without ambiguity, whether or not a given object belongs to the set.

When the objects under consideration are numbers, we speak of a number set. The fundamental number sets are the natural numbers, the integers, the rational numbers and the real numbers. The irrational numbers, by contrast, form the part of the real numbers that does not belong to the rationals.

Their introduction follows a definite principle: each extension makes it possible to solve problems that, in the previous set, do not always have a solution. The natural numbers allow us to count; the integers make subtraction possible without leaving the set; the rational numbers allow us to express ratios of integers; the real numbers gather rational and irrational numbers together in a single set.

The fundamental number sets are linked by the following chain of inclusions:

\[ \mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}. \]

This relation means that every natural number is also an integer, every integer is also a rational number, and every rational number is also a real number. To study the number sets is therefore to understand how numbers are organised, what properties they possess, and which operations are possible within each set.

Contents

• Why new number sets are introduced
• The natural numbers \(\mathbb{N}\)
• The integers \(\mathbb{Z}\)
• The rational numbers \(\mathbb{Q}\)
• The irrational numbers
• The real numbers \(\mathbb{R}\)
• Relations between the number sets
• Decimal representation of the real numbers
• Summary

To understand the role of the number sets it is not enough to list them: one must grasp which mathematical need leads to their introduction. The key point is that a set may be suited to certain operations but not to others.

One important property is closure under an operation. A number set is closed under a given operation if, whenever that operation is applied to elements of the set, the result still belongs to the same set.

For example, the natural numbers are closed under addition and multiplication. Indeed,

\[ 3+5=8 \]

and

\[ 3\cdot 5=15. \]

In both cases the result is again a natural number.

Subtraction, on the other hand, is not always possible if one remains within the natural numbers. Indeed,

\[ 3-5=-2, \]

but \(-2\) does not belong to \(\mathbb{N}\). In order to make subtraction possible in greater generality, the integers are then introduced.

The integers allow one to carry out additions, subtractions and multiplications without leaving the set, but they are not closed under division. For example, the equation

\[ 2x=1 \]

has no integer solution, since its solution is

\[ x=\frac{1}{2}. \]

To express ratios of this kind, the rational numbers are introduced: these are the numbers that can be written as a fraction of two integers with non-zero denominator.

Even the rational numbers, however, do not describe every mathematical quantity. The diagonal of a square of side \(1\), for instance, has length \(\sqrt{2}\), and this number cannot be written as a ratio of two integers. To include quantities of this kind as well, one passes to the set of real numbers.

The fundamental path is therefore

\[ \mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}. \]

Each extension contains the previous one and allows us to tackle problems that previously did not always have a solution. For this reason the number sets should be studied progressively: each one arises from a definite need and has properties of its own.

The natural numbers are the numbers used for counting and ordering. They serve, for instance, to indicate how many units a given quantity contains, or which position an element occupies in an ordered sequence.

We adopt the convention that zero belongs to the set of natural numbers:

\[ \mathbb{N}=\{0,1,2,3,\dots\}. \]

If one wishes to denote only the positive natural numbers, that is, those without zero, one may use the notation

\[ \mathbb{N}^{*}=\{1,2,3,\dots\}. \]

The set of natural numbers is infinite and ordered. Indeed, its elements can be arranged according to the natural order

\[ 0<1<2<3<\dots \]

and, given a natural number \(n\), the number \(n+1\) is again a natural number. The number \(n+1\) is called the successor of \(n\).

The natural numbers are closed under addition and multiplication. This means that if \(a\) and \(b\) are natural numbers, then \(a+b\) and \(a\cdot b\) are natural numbers as well:

\[ a,b\in\mathbb{N} \quad\Longrightarrow\quad a+b\in\mathbb{N} \quad\text{and}\quad a\cdot b\in\mathbb{N}. \]

For example,

\[ 4+7=11 \quad\text{and}\quad 4\cdot 7=28. \]

In both cases the result again belongs to \(\mathbb{N}\).

Not every operation, however, is always possible within the natural numbers. Subtraction may lead to a result that does not belong to \(\mathbb{N}\). For example,

\[ 3-5=-2. \]

Since \(-2\) is not a natural number, the set \(\mathbb{N}\) is not closed under subtraction.

This observation reveals the first limitation of the natural numbers: they are suited to representing non-negative whole quantities, but they do not suffice when one needs to describe differences that may be negative. For this reason the set of integers is introduced.

The integers arise from the need to carry out subtractions that are not always possible within the natural numbers. For example, the subtraction \(3-5\) has no result belonging to \(\mathbb{N}\), since it yields a negative number.

The set of integers is denoted by \(\mathbb{Z}\) and consists of the natural numbers together with their opposites:

\[ \mathbb{Z}=\{\dots,-3,-2,-1,0,1,2,3,\dots\}. \]

Since every natural number is also an integer, we have the inclusion

\[ \mathbb{N}\subset\mathbb{Z}. \]

The set \(\mathbb{Z}\) is infinite both to the right and to the left. Indeed, its elements can be ordered as follows:

\[ \dots<-3<-2<-1<0<1<2<3<\dots \]

Unlike \(\mathbb{N}\), the set of integers has no least element: given any integer, there is always a smaller integer.

The fundamental property of the integers is the existence of the opposite. If \(a\) is an integer, then \(-a\) is also an integer, and one has

\[ a+(-a)=0. \]

Since every integer has an opposite, subtraction of integers is always possible. Indeed, subtracting a number amounts to adding its opposite:

\[ a-b=a+(-b). \]

For example,

\[ 3-5=3+(-5)=-2. \]

The result belongs to \(\mathbb{Z}\), so the set of integers is closed under subtraction.

Moreover, the integers are also closed under addition and multiplication. If \(a\) and \(b\) are integers, then

\[ a+b\in\mathbb{Z} \quad\text{and}\quad a\cdot b\in\mathbb{Z}. \]

For example,

\[ (-4)+7=3 \quad\text{and}\quad (-4)\cdot 7=-28. \]

In both cases the result is again an integer.

However, the integers are not closed under division. For example,

\[ 1:2=\frac{1}{2}, \]

but \(\displaystyle \frac{1}{2}\) is not an integer. Equivalently, the equation

\[ 2x=1 \]

has no solution in \(\mathbb{Z}\).

This shows the limitation of the integers: they allow one to carry out additions, subtractions and multiplications without leaving the set, but they do not suffice to represent all ratios between numbers. For this reason the rational numbers are introduced.

The rational numbers arise from the need to represent ratios of integers. Indeed, although addition, subtraction and multiplication are always possible within the integers, division does not always produce an integer.

For example, the equation

\[ 2x=1 \]

has no solution in \(\mathbb{Z}\), since its solution is

\[ x=\frac{1}{2}. \]

To include numbers of this kind, the set of rational numbers, denoted by \(\mathbb{Q}\), is introduced.

A number is rational if it can be written as a ratio of two integers with non-zero denominator. In symbols,

\[ \mathbb{Q}= \left\{ \frac{p}{q}:p,q\in\mathbb{Z},\ q\neq 0 \right\}. \]

The condition \(q\neq 0\) is necessary because division by zero is not defined.

Examples of rational numbers are

\[ \frac{1}{2}, \qquad -\frac{3}{5}, \qquad \frac{7}{4}, \qquad 0. \]

Every integer is also rational, since it can be written as a fraction with denominator equal to \(1\). For example,

\[ 5=\frac{5}{1}, \qquad -3=\frac{-3}{1}, \qquad 0=\frac{0}{1}. \]

Consequently we have the inclusion

\[ \mathbb{Z}\subset\mathbb{Q}. \]

One and the same rational number can be represented by different fractions. For example,

\[ \frac{1}{2}=\frac{2}{4}=\frac{3}{6}. \]

These fractions are different, yet they represent the same rational number. In general, multiplying numerator and denominator by the same non-zero integer, or dividing both by the same non-zero common divisor, does not change the rational number represented.

The set of rational numbers is closed under addition, subtraction and multiplication. It is also closed under division, provided the divisor is non-zero. If \(a,b\in\mathbb{Q}\) and \(b\neq 0\), then

\[ a+b\in\mathbb{Q}, \qquad a-b\in\mathbb{Q}, \qquad a\cdot b\in\mathbb{Q}, \qquad \frac{a}{b}\in\mathbb{Q}. \]

For example,

\[ \frac{1}{2}+\frac{1}{3}=\frac{5}{6} \]

and

\[ \frac{2}{3}\cdot\frac{5}{4}=\frac{5}{6}. \]

In both cases the result is again a rational number.

Nevertheless, the rational numbers do not suffice to represent every quantity that occurs in mathematics. There exist numbers that cannot be written as a ratio of two integers. A fundamental example is \(\sqrt{2}\), which represents the length of the diagonal of a square of side \(1\).

This shows the limitation of the rational numbers: they allow one to represent all ratios of integers, but not all geometric and numerical quantities. To describe these new numbers it is necessary to introduce the irrational numbers.

The rational numbers allow one to represent all ratios of integers, but they do not exhaust every number that occurs in mathematics. There exist quantities that cannot be expressed by means of a fraction with integer numerator and denominator.

A fundamental example is given by \(\sqrt{2}\). This number represents the length of the diagonal of a square of side \(1\). Indeed, by the Pythagorean theorem, if \(d\) denotes the diagonal, then

\[ d^2=1^2+1^2=2, \]

so that

\[ d=\sqrt{2}. \]

The number \(\sqrt{2}\), however, is not rational. Let us show this by contradiction.

Suppose that \(\sqrt{2}\) were rational. Then there would exist two integers \(p\) and \(q\), with \(q\neq 0\), such that

\[ \sqrt{2}=\frac{p}{q}. \]

We may moreover assume that the fraction is in lowest terms, that is, that \(p\) and \(q\) have no common divisors other than \(1\) and \(-1\).

Squaring both sides, we obtain

\[ 2=\frac{p^2}{q^2}, \]

whence

\[ p^2=2q^2. \]

Hence \(p^2\) is even. Since the square of an odd integer is odd, \(p\) must also be even. There then exists an integer \(k\) such that

\[ p=2k. \]

Substituting into the equality \(p^2=2q^2\), we obtain

\[ (2k)^2=2q^2, \]

that is,

\[ 4k^2=2q^2. \]

Dividing by \(2\), it follows that

\[ q^2=2k^2. \]

Therefore \(q^2\) is even. Again, since the square of an odd integer is odd, \(q\) must also be even.

We have thus shown that \(p\) and \(q\) are both even. This is impossible, because we had chosen the fraction \(\displaystyle \frac{p}{q}\) to be in lowest terms. The contradiction shows that \(\sqrt{2}\) is not rational:

\[ \sqrt{2}\notin\mathbb{Q}. \]

Numbers such as \(\sqrt{2}\), that is, numbers that cannot be written as a ratio of two integers, are called irrational numbers.

In the context of the real numbers, the set of irrational numbers is denoted by

\[ \mathbb{R}\setminus\mathbb{Q}. \]

Examples of irrational numbers are

\[ \sqrt{2}, \qquad \sqrt{3}, \qquad \pi. \]

The irrational numbers show that the rational numbers are not sufficient to describe all geometric and numerical quantities. For this reason it is necessary to consider a larger set, able to contain both the rational and the irrational numbers: the set of real numbers.

The set of real numbers contains all the rational numbers and all the irrational numbers. It is denoted by the symbol \(\mathbb{R}\).

In particular, every rational number is also real, so that

\[ \mathbb{Q}\subset\mathbb{R}. \]

Numbers that are real but not rational include

\[ \sqrt{2},\qquad \sqrt{3},\qquad \pi. \]

We can therefore distinguish two classes of real numbers:

• the rational numbers, which can be written as a fraction of two integers with non-zero denominator;
• the irrational numbers, which cannot be written in this form.

In symbols, the set of irrational numbers is denoted by

\[ \mathbb{R}\setminus\mathbb{Q}. \]

Hence the real numbers are formed by the union of the rational and the irrational numbers:

\[ \mathbb{R}=\mathbb{Q}\cup(\mathbb{R}\setminus\mathbb{Q}). \]

Moreover, a real number cannot be both rational and irrational at once. Indeed, the two sets have no elements in common:

\[ \mathbb{Q}\cap(\mathbb{R}\setminus\mathbb{Q})=\varnothing. \]

The real numbers can be represented on the number line. On this line one places the natural numbers, the integers, the rationals and the irrationals as well.

For example, the numbers

\[ -2,\qquad 0,\qquad \frac{1}{2},\qquad \sqrt{2},\qquad 3 \]

are all real numbers and can be ordered on the number line.

The set \(\mathbb{R}\) thus allows one to work, within a single setting, with integers, fractions, terminating decimals, recurring decimals and infinite non-recurring decimals.

In this sense, the real numbers constitute the largest of the number sets studied in this introduction:

\[ \mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}. \]

Having introduced the main number sets, we can now describe more precisely the relations that connect them.

To say that one set is contained in another means that every element of the first set is also an element of the second. For example, every natural number is also an integer; hence the set of natural numbers is contained in the set of integers:

\[ \mathbb{N}\subset\mathbb{Z}. \]

In the same way, every integer is also a rational number, since it can be written as a fraction with denominator \(1\). Indeed, if \(n\in\mathbb{Z}\), then

\[ n=\frac{n}{1}. \]

Therefore we have the inclusion

\[ \mathbb{Z}\subset\mathbb{Q}. \]

Finally, every rational number is also a real number, so that

\[ \mathbb{Q}\subset\mathbb{R}. \]

Combining these relations, we obtain the fundamental chain of number sets:

\[ \mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}. \]

This chain is to be read from left to right: on passing from one set to the next, the numbers already introduced are not lost, but new ones are added.

For example:

• \(5\) is a natural number, hence it is also an integer, a rational and a real number;
• \(-3\) is an integer, hence it is also rational and real, but it is not natural;
• \(\displaystyle \frac{2}{5}\) is a rational and real number, but it is not an integer;
• \(\sqrt{2}\) is a real number, but it is not rational.

The irrational numbers deserve particular attention. They do not form a set situated between \(\mathbb{Q}\) and \(\mathbb{R}\); rather, they constitute the part of the real numbers that does not belong to \(\mathbb{Q}\).

In symbols, the set of irrational numbers is

\[ \mathbb{R}\setminus\mathbb{Q}. \]

Every real number is therefore either rational or irrational. The two possibilities are mutually exclusive:

\[ \mathbb{Q}\cap(\mathbb{R}\setminus\mathbb{Q})=\varnothing. \]

Moreover, their union yields the whole set of real numbers:

\[ \mathbb{R}=\mathbb{Q}\cup(\mathbb{R}\setminus\mathbb{Q}). \]

In conclusion, the number sets are not separate and independent of one another: they are organised according to relations of inclusion. Understanding these relations makes it possible to determine precisely to which sets a number belongs and which properties may be used when working with it.

Every real number can be written in decimal form. The decimal representation describes a number by means of an integer part and a fractional part, separated by the decimal point.

For example,

\[ \frac{1}{2}=0.5, \qquad \frac{1}{4}=0.25, \qquad \frac{1}{3}=0.333\dots \]

The decimal form makes it easy to distinguish rational numbers from irrational numbers.

Terminating decimals

A decimal is said to be terminating if, after the decimal point, it has a finite number of digits. For example,

\[ 0.5, \qquad 1.25, \qquad -3.75 \]

are terminating decimals.

Every terminating decimal is rational, since it can be written as a fraction whose denominator is a power of \(10\). Indeed,

\[ 0.5=\frac{5}{10}=\frac{1}{2}, \qquad 1.25=\frac{125}{100}=\frac{5}{4}. \]

Recurring decimals

A decimal is said to be recurring if, from a certain point on, a digit or a group of digits repeats indefinitely.

For example,

\[ 0.333\dots \]

is recurring, since the digit \(3\) repeats without end. Likewise,

\[ 1.272727\dots \]

is recurring, since the group of digits \(27\) repeats indefinitely.

Recurring decimals are rational. For example,

\[ 0.333\dots=\frac{1}{3}, \qquad 1.272727\dots=\frac{14}{11}. \]

In general, a number is rational if and only if its decimal representation is terminating or recurring.

Infinite non-recurring decimals

A decimal is said to be infinite and non-recurring if it has infinitely many digits after the decimal point and is not eventually periodic, that is, from no point on does its fractional part become periodic.

Infinite non-recurring decimals are irrational. For example,

\[ \sqrt{2}=1.414213562\dots \]

and

\[ \pi=3.141592653\dots \]

are irrational numbers: their decimal representation neither terminates nor is eventually periodic.

We may therefore summarise the situation as follows:

• terminating decimals are rational;
• recurring decimals are rational;
• infinite non-recurring decimals are irrational.

This distinction is very useful for recognising the nature of a number. For example,

\[ 0.75=\frac{3}{4} \]

is rational, whereas

\[ \sqrt{3}=1.732050807\dots \]

is irrational.

Some real numbers admit two decimal representations. For example,

\[ 1=0.999\dots \]

This peculiarity does not alter the classification above, but it shows that decimal notation must be interpreted with care.

We now summarise the main features of the number sets studied.

The fundamental inclusions among the number sets are:

\[ \mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}. \]

This means that every natural number is also an integer, every integer is also rational, and every rational number is also real.

The irrational numbers, on the other hand, do not belong to \(\mathbb{Q}\) but belong to \(\mathbb{R}\). In symbols:

\[ \mathbb{R}=\mathbb{Q}\cup(\mathbb{R}\setminus\mathbb{Q}) \quad\text{and}\quad \mathbb{Q}\cap(\mathbb{R}\setminus\mathbb{Q})=\varnothing. \]

The number sets thus allow numbers to be organised progressively. Each extension preserves the numbers already introduced and makes it possible to represent new quantities or to carry out operations that previously were not always possible.