Decimal expression of a rational number

Perhaps the best thing to do before explaining the correct way to obtain the decimal expression of a rational number is to briefly review some definitions, which may help to understand this procedure in its just mathematical context.


Fundamental definitions

In this sense, it may also be pertinent to delimit this conceptual revision to two specific notions: Decimal numbers and rational numbers, since these are the numerical elements directly involved in the procedure of ensuring that the same quantity, which has previously been expressed in the form of a fraction, can also be written as a decimal number. Below are each of these definitions:

Decimal numbers

In this way, we will begin by saying that Mathematics has conceived Decimal Numbers as those numerical elements, through which the expression of non integer quantities is achieved, whether they constitute a rational or irrational number. Likewise, decimal Numbers are understood as those numerical elements, which are composed by two different parts, one integer and the other decimal, each of which has been explained in turn in the following way:


Integer part: also known as Units, this part of the decimal number will be constituted -always and without exception- by an integer, which may be positive, negative, or even zero. As it is constituted by numbers belonging to the Decimal Numbering System, the numerical elements that make up the whole part of the decimal number will have a positional value, being possible to count in them, from right to left, the units, tens, hundreds, units of a thousand, tens of a thousand, hundreds of a thousand, etc.

Decimal part: on the other hand, decimal numbers will also have a decimal part, called Incomplete Units, which will be formed by a number less than the unit, and that in the Numerical Line is between zero and 1. In its elements there is also a positional value, being able to order its elements, from left to right, such as tenths, hundredths, thousandths, ten-thousandths, etc.

Both parts are separated -and at the same time joined- by a comma, even though there are some mathematical currents that also prefer the use of the point. Independently of the sign that is preferred, to the right of this one the incomplete units -decimal part- will have to be annotated while to the left of the comma -or the dot- the units, or whole part of the decimal number will have to be arranged then.

Rational numbers

Secondly, it will also be necessary to throw lights on the definition given by Mathematics with respect to rational numbers, which can then be understood as those numbers, not integers, whose expression is represented as the quotient between two integers, which then gives rise to a fraction, which will have a numerator and a denominator. According to the different mathematical sources, this type of numbers are called rational, precisely because they do not represent an entire quantity, but a portion, ration or fraction of it.

Likewise, Mathematics points out that rational numbers will be the numerical elements that will make up the numerical set Q, which equally contains both the set of integers (Z) and the set of natural numbers (N). At the same time, the numerical set Q is included in the set of real numbers (R).

Decimal expression of a rational number

Once each of these definitions has been reviewed, it is certainly much simpler to approximate the mathematical procedure known as decimal expression of a rational number, which basically consists of determining how to express a given fraction as a decimal number, that is, a number made up of an integer part and a decimal part, which are separated by a comma.

As for this procedure, Mathematics says that it is totally possible, since both the fraction and the decimal number can be used to express the same fractional quantity. The fraction from which the decimal number that will be found comes is also known as the generatrix fraction, since the decimal expression comes from it or is generated.

How to make the decimal expression of a rational number

Also, the mathematical discipline has indicated that the adequate way to obtain the decimal Expression of a rational number will be making the division that is declared in the fraction, that is to say, dividing the number that serves as numerator between the number that acts as denominator. This operation must result in a decimal number. The following is a concrete example of how this operation should be solved:

Cases that occur in the decimal expression of a decimal number

However, when performing a procedure aimed at obtaining the number or decimal expression corresponding to a given fraction or rational number, one should be aware that two possible cases may occur, depending on the characteristics of the decimal found when dividing the numerator by the denominator. Next, each one of them:

A limited decimal

On the one hand, it can happen then that when dividing the numerator by the denominator, the division is exact, producing then a limited decimal number, that is, that it has a precise number of decimal parts or incomplete units. An example of this type of cases will be the following:

A periodic decimal

However, it can also happen that the division between numerator and denominator does not produce an exact division, but a decimal number that counts in its incomplete units some series of numbers that are repeated, that is to say that they are periodic, either if the series that is repeated is immediately after the comma of the decimal (pure newspaper) or at a certain distance, after some numbers that are not repeated (mixed newspaper). An example of this type of decimal expressions of rational numbers will be the following:

In general, this type of number is expressed by writing down once the series that is repeated, and placing on it the symbol that indicates that this number is repeated several times in incomplete units:

Note

The decimal expression of a rational number can only lead to a limited decimal or an unlimited periodic decimal, precisely because they are expressions of rational numbers, since if it were an unlimited non-periodical number the expression would correspond to an irrational number, that is, a number where its incomplete units, or decimal part, extend to infinity without ever producing any series of numbers or periods that are repeated.

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Decimal expression of a rational number
Source: Education  
October 31, 2019


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