What are the real numbers?

Numbers have different properties and can be classified into various groups. One of these groups, with wide applications in various branches of mathematics, are the real numbers. To understand them better, let’s first see what the different types of numbers are.

The numbers

The first thing we learn about numbers is how to use them to count; we start with matching them with our fingers to do simple operations. Thus, our ten fingers are the base of the decimal system. From there we count quantities as large as we can and note that the numbers are infinite. And so, adding zero (0) when we have nothing to count, the natural numbers are formed.

With the natural numbers we do arithmetic operations and when we subtract another number from a number, we have to introduce the negative numbers. So, adding the negative numbers to the natural ones, we obtain the set of integers.

Among the arithmetic operations we perform with numbers is division. And we find that there are cases in which when dividing one number by another, the result is not an integer; In many cases, this division result can only be accurately represented by the division expression itself, that is, a fraction. This is how the set of rational numbers is constructed, in which all numbers are written as a fraction and the integers have the number 1 as the denominator.

It was the ancient civilizations that observed that there were numbers that could not be represented as fractions. When working with geometric figures, they found the number pi, the relationship between the radius and the length of a circle, a number that cannot be expressed as the quotient between two integers. It is also the case of the square root of the number 2 (that is, the number that multiplied by itself would give the number 2 as a result). And there are many numbers that emerge in various branches of knowledge that are not part of the set of rational numbers. These numbers, which cannot be exactly represented as the quotient of two whole numbers, are called irrational numbers. The set of rational and irrational numbers constitutes, then, the set of real numbers.

The real numbers are part of an even larger set of numbers: the complex numbers. This extension of the set of real numbers arises when we want to calculate the square root of a negative number; Since the product of two negative numbers is always positive, there is no real number that multiplied by itself is negative. Then the imaginary number i is defined , which represents the square root of -1, and the set of complex numbers arises.

decimal representation

All numbers can be expressed in decimal form; For example, the rational number 1/2 can be expressed in decimal form as 0.5. Unlike the rational number 1/2, which can be exactly represented by a single decimal place, other rational numbers have an infinite number of decimal places and do notThey can be expressed exactly with the decimal representation. This is the case of the number 1/3; Its decimal representation is 0.33333…, with an infinite number of decimal places. These rational numbers are called periodic decimal numbers, since in all cases there is a sequence of numbers that is repeated infinitely many times. In the case of the number 1/3 that sequence is 3; in the case of the number 1/7, its decimal form is 0.1428571428571…, and the sequence that is repeated infinitely is 142857. Irrational numbers are not periodic decimal numbers; there is no sequence that is repeated infinitely many times in its decimal representation.

Visual representation

The real numbers can be visualized by associating each of them to one of the infinitely many points along a straight line, as shown in the figure. In this graphic representation is located the number pi, whose value is approximately 3.1416, the number e , which is approximately 2.7183, and the square root of the number 2, approximately 1.4142. From the number 0 to the right the positive real numbers are located in increasing form, and to the left the negative ones increasing their absolute value in that direction.

Some properties of real numbers

Real numbers behave like integers or rational numbers, with which we are more familiar. We can add, subtract, multiply, and divide them in the same way; the only exception is the division by the number 0, an operation that is not possible. The order of the additions and multiplications is not important, since the commutative property still holds, and the distributive property applies in the same way. In the same way, two real numbers x and y are ordered in a unique way, and only one of the following relations is correct:

x = y , x < y or x > y

The real numbers are infinite, just like the integers and the rational numbers. In principle this is obvious since both the integers and the rational are subsets of the real numbers. But there is a difference: in the case of integers and rational numbers it is said that they are countably infinite numbers; instead, the real numbers are infinite innumerable.

A set is said to be countable or countable when each of its components can be associated with a natural number. The association is obvious in the case of integers; in the case of rational numbers it can be seen as the association with a pair of natural numbers, the numerator and the denominator. But this association is not possible in the case of real numbers.

Sources

• Arias Cabezas, Jose Maria, Maza Saez, Ildefonso. Arithmetic and Algebra . In Carmona Rodríguez, Manuel, Díaz Fernández, Francisco Javier, eds. Mathematics 1. Bruño Editorial Group, Limited Company, Madrid, 2008.
• Carlos Ivorra. Logic and Set Theory . 2011.