Definition of rational numbers


In order to define the rational numbers we will start by defining the natural numbers. The natural numbers are $$$1,2,3,4,5,\ldots$$$

The natural numbers are those which can be written as the addition of ones.

For example, the number $$5$$ is natural since $$5=1+1+1+1+1.$$

The set of natural numbers is so-called $$\mathbb{N}$$.

The formalism that we have just exposed can be interesting in a context where rigorousness in the definitions is required. In our context, we will leave this formalism aside so as to focus on the most intuitive point of view.

The natural numbers allow us to define the addition of two numbers in a "natural" way. We can also define subtraction as a natural operation, but we will easily observe that the subtraction of two natural numbers is not necessarily natural.

For example, $$3-5$$ does not give a natural number.

To be able to define the subtraction of natural numbers we define the integers, which consist of adding to the natural numbers the necessary numbers in order to be able to define the subtraction.

As we know, the integers are $$$\ldots,-3,-2,-1,0,1,2,3,\ldots$$$

So we can say that the natural numbers are the integers with a positive sign. We called the set of integers $$\mathbb{Z}$$. According to these notations we have $$\mathbb{N} \subseteq \mathbb{Z}$$.

The integers have another operation, which is multiplication. The result of the product of integers is another integer.

We might now define the division of integers. That is not always possible because the division of integers does not necessarily give an integer. For example, $$\dfrac{1}{2}$$ is not an integer.

To be able to define the division of integers, it is necessary to consider the rational numbers.

The rational numbers involve the quotient of integers: two rational numbers are equal if the crossed product is the same.

So, $$\dfrac{a}{b}$$ is equal to $$\dfrac{c}{d}$$ as long as $$ad=cb$$.

In the previous definition it is necessary that both $$b$$ and $$d$$ are not null, so the rational number is definite.

The integer $$n$$ is associated with the rational one $$\dfrac{n}{1}$$.

This expression is not used if it is not necessary, so we usually only write $$n$$. The set of rational numbers is called $$\mathbb{Q}$$. According to the previous comment $$\mathbb{Z}\subseteq \mathbb{Q}.$$

We can observe that the definite operations of addition and subtraction for integers can be extended to rational numbers.

$$$\dfrac{a}{b}+\dfrac{c}{d}=\dfrac{ad+bc}{bd}$$$ $$$\dfrac{a}{b}-\dfrac{c}{d}=\dfrac{ad-bc}{bd}$$$

The operations of multiplication and division must be solved according to this rule;

$$$\dfrac{a}{b}\cdot\dfrac{c}{d}=\dfrac{ac}{bd}$$$ $$$\dfrac{a}{b} : \dfrac{c}{d}=\dfrac{ad}{bc}$$$