The real numbers

One, two, three... ten, eleven, twelve...

We all know how to count. Numbers are used in all countries and in all languages ​​of the world to be able to quantify. It is something UNIVERSAL and without them humanity would not be what it is today.

They are everywhere... All technology uses them to function...

But... where do they come from?

Throughout history, human beings have used different ways to count and express quantities. For example, sticks, knots, fingers, among other objects were used.

The first signs of expressing numbers on a surface as a symbol date back to 400 BC in Mesopotamia. Clay tablets have been found with symbols made with a stick expressing the first numbers.

Symbols to express numbers were later also used in Ancient Greece and Ancient Rome. However, it was not until the 19th century when the German mathematician Richard Dedekind began to formalize the concept of numbers. Dedekind's work led mathematician Giuseppe Peano to publish the famous Peano postulates, which precisely defines the natural numbers, from which integers, rational and real numbers arise.

In Europe, real numbers have been used since the 17th and 18th centuries. However, its construction was not systematized. It was not until the 19th century, when Richard Dedekind and Georg Cantor formalized the construction of real numbers through different means. Nowadays, we can define real numbers through Dedekind cuts or through Cauchy sequences.

In this section we will try to define these numbers and some of their properties in a simple way so that everyone can understand it. In later sections, for the bravest, we will delve into mathematical formalism and formalize the concept of a real number.

Real numbers are made up of the set of natural, integer, rational and irrational numbers. Seen from a geometric point of view, any point on the real line corresponds to a real number.

Simply speaking, any number that we use in everyday life, whether it is a normal number, a negative number or a number with a decimal, are also real numbers.

These types of numbers have properties that allow us to add, subtract, multiply and divide them. Furthermore, they have a specific order. For example, 2 is always smaller than 3 or 11.2 is larger than 11.11. What seems obvious, mathematically, the order of the numbers, has to be defined.

There are numbers that do not belong to the set of real numbers. For example, complex numbers. However, we will detail these types of numbers in future publications since their use is not obvious and we do not use them in our day-to-day lives.

The set of integers together with addition and multiplication form an algebraic structure known as a field (in the next section we will go into more detail about this). This defines some properties of the set of reals for addition and other properties for multiplication.

Let us remember that real numbers can also be subtracted, but the subtraction operation is actually adding the opposite element of a certain number. The same thing happens with division. Division consists of multiplying a number by the inverse element of another number.

Associative property: Given 3 real numbers a, b and c, the following property is satisfied:

(a+b)+c = a+(b+c)

(1+2)+3 = 1+(2+3) = 6

Commutative property: Given 2 real numbers a and b, the following holds: a+b=b+a. Example with numbers: 2+3 = 3+2 = 5

Neutral element: There is an element e that for any real number a holds: a + e = a. In the case of real numbers, the number e is 0 since if we add any real number to zero it returns the same real number.

Symmetrical element: For any real number a there is an opposite element that if we add them together gives the neutral number. Applied in the case of the sum of the reals:

a + (-a) = 0

3 + (-3) = 3 – 3 = 0

In this case the real number a is 3 and its opposite is (-3). Added together it gives us the neutral number which is 0.

Associative property: Given 3 real numbers a, b and c, the following property is satisfied:

(a·b)·c = a·(b·c)

(1·2)·3 = 1·(2·3) = 6

Commutative property: Given 2 real numbers a and b, the following holds: a·b=b·a. Example with numbers: 2·3 = 3·2 = 6. The order of the factor does not alter the product.

Neutral element: There is an element e that for any real number a holds: a · e = a. In the case of real numbers, the number e is 1 since if we multiply any real number by 1 it returns the same real number.

Symmetrical element: For any real number a there is an inverse element that if we multiply them gives the neutral number. Applied in the case of the multiplication of reals:

a · 1/a = 1

3 1/3 = 1

Distributive property: For 3 real numbers a, b, c the following property is satisfied: a·(b+c) = (a·b) + (a·c)

Too easy? Are you a math expert?

Well let's make it a little more difficult!

As we have seen, real numbers have well-defined properties for addition and multiplication. These two internal operations provide the set of reals with an algebraic structure called a body. Furthermore, the body of reals has the property of being orderly and complete.

Ordered means that there is a total order of its elements and that they are compatible with the two operations: addition and multiplication.

The concept of completeness of the reals is a little more complex. We could imagine that it is a body that does not have “holes”. For example, if we take the real line and select any random point, that number will also be real.

On the other hand, the body of irrationals is not complete since between two rationals there may be an irrational (which is therefore not a rational number) leaving gaps between two rational numbers.

Formally, we could define the completeness of a field as the property that any Cauchy sequence on that field is convergent on that field.

Is it difficult to understand?

We had already said that now came the difficult part!

Now that we have understood a little about the algebraic structure of the set of real numbers, it is now time to look at its construction.

You already know what real numbers are, where they come from and their properties. But... Why are they so important?

Real numbers are the basis of any mathematical operation that we may encounter in our daily lives. That you can add your shopping list or receive a WhatsApp on your mobile phone is thanks to the fact that these numbers exist and that their operations are well defined.

The next time you take a plan or make purchases with your credit card, think that it is thanks to the MAGIC OF REAL NUMBERS.