Rational numbers

The extent of the integers, \mathbb{Z}, to rational numbers \mathbb{Q}, has a clear parallel with the extent of \mathbb{N}_0 to \mathbb{Z}. As we could not subtract in \mathbb{N}_0, invented a new type of numbers to achieve it. Now we find the problem that we can't always divide into \mathbb{Z}, and invented a new type of numbers to achieve it

To define \mathbb{Z} we took pairs of natural numbers, and applied an equivalence ratio. The definition of \mathbb{Q} follow the same steps, but one in an even clearer way: rational numbers will be pairs of integers, which correspond to the numerator and denominator of each fraction; equivalence classes must also be taken to identify fractions representing the same number

As the denominator of a fraction cannot be null, instead of taking \mathbb{Z}x\mathbb{Z}, the set is taken:

\mathbb{Z}x(\mathbb{Z}\backslash\left\{0\right\})=\left\{(a, b) | a, b \in \mathbb{Z} \quad b \not= 0\right\}

and in it we define the relationship \sim given by:

(a, b)\sim (c, d) \Longleftrightarrow a\cdot d = b\cdot c

that can be easily demonstrated, that it is of equivalence

(a, b) and (c, d) will end up being, respectively, the rational \frac{a}{b} and \frac{c}{d}; you still can't talk about equality \frac{a}{b}=\frac{c}{d} (for these fractions do not yet exist), but if it made sense it would be equivalent to saying that a\cdot d = b\cdot c, which is what we're using to define the equivalence ratio

Now we define \mathbb{Q} as the quotient set: