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Perhaps the most convenient thing to do, before giving some examples on the adequate way in which the decimal Expression of a rational number should be achieved, is to briefly review the definition of this operation, **in order to be able to understand each one of these exercises within its just mathematical context.**

## Decimal expression of a rational number

Consequently, it can be said that Mathematics has defined this operation as the procedure carried out to determine **which is the decimal number that can refer to the same fractional quantity that represents the rational number,** expressed as a fraction. Therefore, it could be said that both the decimal number and the fraction would be forms of expression of a fractional quantity.

Likewise, the mathematical discipline has pointed out that if we want to obtain the decimal expression that corresponds to a certain rational number or fraction, **simply dividing the numerator by the denominator will suffice, which will then produce a decimal number.**

However, the different mathematical sources have also pointed out that once this division is carried out, **two different decimal numbers can actually be found,** whose main difference will be the characteristic of their decimal part, as far as the number of their decimals is concerned. **Next, each of them:**

**If the division is exact:**In this way, it can happen that when dividing the numerator between the denominator an exact division is originated, that throws**as quotient a decimal number that counts on a limited number of elements in its incomplete units or decimal parts,**that is to say that it is obtained with the division a limited decimal.**If the division is inexact:**So it can also happen that when dividing the numerator between the denominator there is an inexact division, which results in a decimal number, which in its decimal parts or incomplete units has unlimited numbers that are repeated every once in a while, that is, that it is a periodic decimal, either if the part that peritates in it is located immediately after the comma (pure newspaper decimal)**or is located at a certain distance from this symbol, being between this and the period that repeats a certain part that doesn´t**(mixed newspaper decimal), being between this and the period that repeats a certain part that does not (mixed newspaper decimal).

## Note

However, it is necessary to point out that being the decimal expression of a rational number, the decimal numbers obtained can then only be either a limited decimal, or an unlimited decimal, **whether it is pure or mixed, since if on the contrary an unlimited non-periodic decimal were to be had,** then one would be faced with an irrational number, which is characterized by not being able to be expressed as a fraction, and in addition to have a decimal expression in which its incomplete units extend towards infinity, without there being in them a single series or period that is repeated.

## Examples of how to find the decimal expression of a rational number

However, the best way to complete an explanation about the correct way in which any operation should be resolved, whose purpose is to obtain the decimal expression of a rational number, **may be through the exposition of some concrete examples**, in which it is possible to see in a practical way how to proceed in this type of procedures. **Next, the following exercises:**

## Example 1

**Find the decimal expression of the next rational number:**

When complying with the requirement of this postulate,** the numerator of the fraction must be divided by its denominator in order to achieve its decimal expression:**

6 : 9 = 0,666666666666

The result will be a decimal number, which will present in its incomplete units a series of numbers that will be repeated several times, immediately after the comma, so it will be an unlimited periodic decimal, which could be expressed by simply placing the number that is repeated, wrapped by the sign that indicates it.

## Example 2

**Find the decimal expression of the next rational number:**

Similarly, in this case, in order to find the decimal expression of the number, **its numerator must be divided by the denominator:**

2 : 7 = 0,285714285714285714

This division has also yielded a decimal number in which its incomplete units are repeated every now and then, that is to say, that they make up series, being immediately after the comma, it is a pure periodic decimal. Consequently, it can also be expressed by simply writing down one of the series that are repeated,** and wrapping it with the sign that indicates that the series is repeated again to infinity:**

## Example 3

**Find the decimal expression of the next rational number:**

In the same way, when approaching this fraction, in order to obtain the corresponding decimal expression, the numerator must be divided by the denominator, **in order to comply with the request of the exercise:**

5 : 2 = 2, 5

In this case, the division that occurs between the numerator and the denominator is an exact division, so then the decimal number thrown will have in its incomplete units a specific number of elements, that is, it will be a limited decimal.

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October 31, 2019