Integer numbers

With natural ones we can add numbers, but we can't always subtract. We will use the whole numbers to be able to subtract any pair of natural

Although natural numbers seem obvious, mathematicians had a hard time treating negative numbers equally to positive numbers. Historically, the use of negative numbers is far after the use of fractions or even irrational positives

An example of complications from negative numbers for both Greek Diophanto and European Renaissance algebrists, an equation of the x^2+b\cdot x+c=0 (in which for us parameters b and c can be both positive and negative and the resolution method remains the same) was not unique, but should be analyzed in four cases \begin{cases}x^2+b\cdot x+c=0 \\ x^2+b\cdot x=c \\ x^2+c=b\cdot x \\ x^2=b\cdot x+c \end{cases} with b and c always positive (even more cases if we allow the possibility that b or c are worth zero) and each case had its own method of resolution

Note: the rating x^n has usual sense x^n=\underbrace{x\cdots x}_{\text{n times}}, \forall n \in\mathbb{N} and x belongs to any of the sets of numbers we consider (natural, integer, rational, and real). Also when no. 0 we have to x^0=1

Fue el matemático holandés Simon Stevin, a finales del siglo XVI, el primero que reconoció la validez de los números negativos al aceptarlos como resultado de los problemas con que trabajaba. Además, reconoció la igualdad entre la sustracción de un número positivo y la adición de un número negativo (es decir, a−b=a+(−b), con a, b > 0). Por esta razón, igual que se considera a Brahmagupta como padre del cero, Stevin es considerado como el padre de los números negativos (de hecho, Stevin hizo muchas más contribuciones al mundo de los números, en el campo de los números reales)

In \mathbb{N}_0 can add numbers, but we can't always subtract. That is why the need arises to create a new set of numbers with which to subtract any pair of natural numbers; this new set is that of integers and is denoted as \mathbb{Z}

It is totally elementary to see that this relationship is of equivalence, and that allows us to define whole numbers as the quotient set \mathbb{Z}=\mathbb{N}_0\times\mathbb{N}_0/\sim