Decimal expression of rational numbers

Any rational number can be expressed in decimal base. This expression is, in a colloquial way, what most the people understand by a number with a comma.

Let's explain it with the following example,

$\dfrac{1}{2}$ can be written as $0,5$.

And then we read zero comma five instead of a half.

This expression is useful if we are referring, for example, to a price or a length, where it is necessary to get the idea of the value of the rational number.

This expression in decimal base is not always exact, for example $\dfrac{1}{3}=0,33333\ldots$ and we should write an infinite number of $3$, which would take too much time. In this case we will say that the result is a zero comma three repeating.

Whenever we say repeating, we mean that the number must be repeated infinite times.

We write it putting a bar on the repeating number, for example $\dfrac{1}{3}=0,\widehat{3}$.

The repeating decimal does not necessatry involve all the numbers behind the comma. The repeating can also be a number of more than one figure. For example $$\dfrac{1}{55}=0,018181818\ldots=0,0\widehat{18}$$

In this case the repeating is $18$ and the zero does not belong to it. We should read zero comma zero with repeating eighteen.

Given a number with repeating we can recover the expression as quotient using the following procedure.

$a$ is corresponding to the number whose comma and repeating numbers we want to remove. $b$ is the number to which we have to add the digits of the repeating number $a$. Let's say, also, that the decimal part that does not belong to the repeating has $m$ numbers and that the repeating itself has $n$ numbers. Then, our decimal expression corresponds to the quotient of $b-a$ for the number with $n$ nines followed by $m$ zeros.

It is easier to understand with some examples. Let's see that the expressions of the past examples have a correspondance with a rational number.

For the expression $0,\widehat{3}$, according to our notation; $a=0,b=3,m=0$ and $n=1$. And it corresponds to the quotient

$$\dfrac{b-a}{9}=\dfrac{3-0}{9}=\dfrac{1}{3}$$

For the expression $0,0\widehat{18}$, according to our notation: $a=0,b=018,m=1$ and $n=2$. And it corresponds to the quotient

$$\dfrac{b-a}{990}=\dfrac{18-0}{990}=\dfrac{1}{55}$$

For the expression $0,12\widehat{34}$, according to our notation: $a=12,b=1234,m=2$ and $n=2$. And it corresponds to the quotient

$$\dfrac{b-a}{9900}=\dfrac{1234-12}{9900}=\dfrac{611}{4950}$$

We can verify that the decimal expression corresponds to the original expression.

Now, we can study the rational numbers through their decimal expression. And this decimal expression is a digits sequence. We have studied that rational numbers have a correspondence with the sequences of digits that, at the end, are repeating.