Numbers

The numbers must be constructed with the rigor demanded by the mathematics and not only with our intuition

That is why we need to build them from concepts and primitive properties, because as Leopold Kronecker said:

God made the natural numbers; everything else is the work of man

Starting from a "basic" mathematics, there is a concept called a set and each set is made up of a collection of elements (which are unique and different from each other) which belong to the set. In the event that no element appears in the set, we will have an empty set and it is denoted by \emptyset

In case of not seeing clearly the logical necessity to introduce the numbers, it is suggested to try to answer the simple question of what is a number? and try to answer it intuitively

Egyptian unit fractions (Ahmes / Rhind Papyrus)

In this papyrus acquired by Henry Rhind in 1858 whose contents date from 2000 to 1800 BC. C. in addition to the number system, we find fractions. Only unit fractions (inverse of natural \frac{1}{20}) that are represented by an oval sign above the number, the fraction \frac{2}{3} that is represented by a special sign, and in some cases fractions of the type \frac{n}{n+1}. There are decomposition tables of \frac{2}{n} from n=1 to n=101, such as \frac{2}{5}=\frac{1}{3}+\frac{1}{15} or \frac{2}{7}=\frac{1}{4}+\frac{1}{28}, but it is not known why they did not use \frac{2}{n}=\frac{1}{n}+\frac{n+1}{n} but it seems that they were trying to use fractions unit lower than \frac{1}{n}

Being a summative system the notation is: 1+\frac{1}{2}+\frac{1}{4}. The fundamental operation is the addition and our multiplication and division was done by "duplication" and "mediation", for example 69\cdot 19=69\cdot (16+2+1), where 16 represents 4 duplications and 2 a duplication

Babylonian sexagesimal fractions (cuneiform documents)

In the cuneiform tablets of the Hammurabi dynasty (1800-1600 BC) the positional system appears, an extension of the fractions, but XXX is valid for 2\cdot 60+2, 2+2\cdot 60-1 or 2\cdot 60-1+2\cdot 60-2 with a representation based on the interpretation of the problem

To calculate they resorted to the numerous tables at their disposal: of multiplication, of inverses, of squares and cubes, of square and cubic roots, of successive powers of a given number not fixed, etc. For example to calculate a, they took their best integer approximation a_1, and they calculated b_1=\frac{a}{a_1} (one major and one minor) and then a_2=\frac{(a_1+b_1)}{2} is a better approximation, proceeding in the same way we obtain b_2=\frac{a}{a_2} and a_3=\frac{(a_2+b_2)}{2} obtaining in the Yale-7289 tablet 2 = 1; 24,51,10 (in decimal base 1.414222) as value of a_3 on the basis of a_1=1;30

They carried out the operations in a similar way to today, the division multiplying by the inverse (for which they use their inverse tables). Missing from the table of inverses are those of 7 and 11 that have an infinitely long sexagesimal expression. If they are \frac{1}{59}=;1,1,1 (our \frac{1}{9}=0,\stackrel{\frown}{1}) and \frac{1}{61}=;0,59,0,59 (our \frac{1}{11}=0,\stackrel{\frown}{09}) but did not notice the periodic development