REAL NUMBERS: HISTORY, EXAMPLES, PROPERTIES, OPERATIONS - MATHS - 2025

The real numbers constitute the numerical set that includes the natural numbers, the integers, the rational and the irrational. They are denoted by the symbol ℝ or simply R and their scope in science, engineering and economics is such that when speaking of “number”, it is almost taken for granted that it is a real number.

Real numbers have been used since ancient times, although they were not given that name. Already from the time when Pythagoras developed his famous theorem, numbers arose that could not be obtained as quotients of natural numbers or integers.

Figure 1. Venn diagram showing how the set of real numbers contains the other number sets. Source> Wikimedia Commons.

Examples of numbers are √2, √3, and π. These numbers are called irrational, as opposed to rational numbers, which do come from quotients of whole numbers. It was therefore necessary a numerical set that encompasses both classes of numbers.

The term "real number" was created by the great mathematician René Descartes (1596-1650), to distinguish between the two kinds of roots that can arise from solving a polynomial equation.

Some of these roots can be even roots of negative numbers, Descartes called these "imaginary numbers" and those that were not, were real numbers.

The denomination persisted over time, giving rise to two large numerical sets: the real numbers and the complex numbers, a larger set that includes real numbers, imaginary numbers, and those that are part real and part imaginary.

The evolution of real numbers continued its course until in 1872, the mathematician Richard Dedekind (1831-1936) formally defined the set of real numbers through the so-called Dedekind cuts. The synthesis of his work was published in an article that saw the light that same year.

Examples of real numbers

The table below shows examples of real numbers. This set has as subsets the natural numbers, the integers, the rational and the irrational. Any number of these sets is, in itself, a real number.

Therefore 0, negatives, positives, fractions, and decimals are real numbers.

Figure 2. Examples of real numbers are natural, integer, rational, irrational, and transcendent. Source: F. Zapata.

Representation of real numbers on the real line

Real numbers can be represented on the real line R, as shown in the figure. It is not necessary that the 0 is always present, however it is convenient to know that the negative reals are to the left and the positive ones to the right. That is why it is an excellent point of reference.

On the real line, a scale is taken, in which the integers are found:… ​​3, -2, -1, 1, 2, 3…. The arrow indicates that the line extends to infinity. But that is not all, in any considered interval, we will also always find infinite real numbers.

The real numbers are represented in order. To begin with, there is the order of the integers, in which the positives are always greater than 0, while the negatives are less.

This order is kept within the real numbers. The following inequalities are shown as an example:

a) -1/2 <√2

b) e <π

c) π> -1/2

Figure 3.- The real line. Source: Wikimedia Commons.

Properties of real numbers

-Real numbers include natural numbers, integers, rational numbers, and irrational numbers.

-The commutative property of addition is fulfilled: the order of the addends does not alter the sum. If a and b are two real numbers, it is always true that:

a + b = b + a

-The 0 is the neutral element of the sum: a + 0 = a

-For the sum the associative property is fulfilled. If a, b and c are real numbers: (a + b) + c = a + (b + c).

-The opposite of a real number to is -a.

-The subtraction is defined as the sum of the opposite: a - b = a + (-b).

-The commutative property of the product is fulfilled: the order of the factors does not alter the product: ab = ba

-In the product the associative property is also applied: (ab).c = a. (Bc)

-The 1 is the neutral element of the multiplication: a.1 = a

-The distributive property of multiplication is valid with respect to addition: a. (b + c) = ab + ac

-Division by 0 is not defined.

-Any real number a, except 0, has a multiplicative inverse of ^-1 such that aa ^-1 = 1.

-If a is a real number: a ⁰ = 1 and a ¹ = a.

-The absolute value or modulus of a real number is the distance between said number and 0.

Operations with real numbers

With the real numbers you can do the operations that are done with the other number sets, including addition, subtraction, multiplication, division, empowerment, radication, logarithms and more.

As always, the division by 0 is not defined, neither do the logarithms of negative numbers or 0, although it is true that log 1 = 0 and that the logarithms of numbers between 0 and 1 are negative.

Applications

The applications of real numbers to all kinds of situations are extremely varied. Real numbers appear as answers to many problems in exact science, computer science, engineering, economics, and social science.

All kinds of magnitudes and quantities such as distances, times, forces, intensity of sound, money and many more, have their expression in real numbers.

The transmission of telephone signals, the image and sound of a video, the temperature of an air conditioner, a heater or a refrigerator can be controlled digitally, which means transforming physical quantities into numerical sequences.

The same happens when making a banking transaction over the Internet or consulting instant messaging. The real numbers are everywhere.

Exercise resolved

We are going to see with exercises how these numbers work in common situations we encounter on a daily basis.

Exercise 1

The post office only accepts packages for which the length, plus the girth measurement, does not exceed 108 inches. Therefore, for the displayed package to be accepted, it must be fulfilled that:

L + 2 (x + y) ≤ 108

a) Will a package that is 6 inches wide, 8 inches high, and 5 feet long make it through?

b) What about one that measures 2 x 2 x 4 ft ³ ?

c) What is the highest acceptable height for a package whose base is square and measures 9 x 9 inches ² ?

Answer to

L = 5 feet = 60 inches

x = 6 inches

y = 8 inches

The operation to solve is:

L + 2 (x + y) = 60 + 2 (6 + 8) inches = 60 + 2 x 14 inches = 60 + 28 inches = 88 inches

The package is accepted.

Answer b

The dimensions of this packet are smaller than packet a), so they both make it through.

Answer c

In this package:

x = L = 9 inches

It must be observed that:

9+ 2 (9 + y) ≤ 108

27 + 2y ≤ 108

2y ≤ 81

and ≤ 40.5 inches

References

1. Carena, M. 2019. Pre-University Mathematics Manual. National University of the Litoral.
2. Diego, A. Real numbers and their properties. Recovered from: matematica.uns.edu.ar.
3. Figuera, J. 2000. Mathematics 9th. Degree. CO-BO editions.
4. Jiménez, R. 2008. Algebra. Prentice Hall.
5. Stewart, J. 2006. Precalculus: Mathematics for Calculus. 5th. Edition. Cengage Learning.