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

It was the Dutch mathematician Simon Stevin, at the end of the 16th century, who first recognized the validity of negative numbers by accepting them as a result of the problems he worked with. In addition, it recognized the equality between subtracting a positive number and adding a negative number (i.e., a-b-a+(-b), with a, b > 0). For this reason, just as Brahmagupta is considered to be the father of zero, Stevin is regarded as the father of negative numbers (in fact, Stevin made many more contributions to the world of numbers, in the field of real numbers)

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