Unlimited periodic mixed decimals

Perhaps the best thing to do before explaining Unlimited Decimal Numbers in mixed periodicals is to briefly review some definitions that will allow us to understand this type of decimal numbers in their precise mathematical context.


Fundamental definitions

Consequently, it may also be convenient to delimit this theoretical revision to three fundamental notions: Rational Numbers, Decimal Numbers and Unlimited Decimal Numbers, since these are the three types of numbers directly related to the concept of mixed newspapers. Next, each one of them:

Rational numbers

In the first instance, we will begin by saying that rational Numbers have been defined by Mathematics as a type of number, characterized by being the written expression of the quotient between an integer and a natural number, which in turn are written in the form of a fraction.

As for its form, this quotient can be an integer, or also a decimal number, where the part of the decimals can be constituted by unlimited numbers, or on the contrary extend infinitely, with repetitions of its elements. These numbers constitute the numerical set Q, and in it they are found as subsets Z (integers) and N (natural numbers).

Decimal numbers

Likewise, the mathematical discipline has also thrown lights on the definition of Decimal Numbers, which will be understood as those numerical elements used to represent both rational and irrational numbers, that is, those numbers that can never be represented as a fraction, because their decimal part is infinite, and in addition there is no series of numbers to be repeated.

On the other hand, the different sources coincide in pointing out that the Decimal Numbers are made up of two parts: the first of them, called Units, and that it will always be made up of an integer, whether it is positive, negative or even zero (0); as well as a second, which is called Incomplete Units, which in general will be made up of a number less than the unit, and included in the Numerical Line between 0 and 1.

Both elements of the decimal Numbers will be separated -and at the same time joined- by a comma, with the Units always and without exception to be noted to the left of this symbol, while the incomplete Units should be placed to the right. In some mathematics schools the use of the dot is also accepted.

Unlimited Decimal Numbers

Finally, it will also be pertinent to approach the concept of Limited Decimal Numbers, which have to be seen as one of the two different types of Decimal Numbers that exist, and whose main difference lies in the finiteness of their incomplete Units. Therefore, the unlimited decimals will be those decimal numbers that present infinite incomplete units, whether they extend to infinity without presenting repetitions of some groups of numbers, or if they present them.

Mixed periodic decimal numbers

In fact, mixed newspapers can be identified as one of the two different groups that are counted within Unlimited Decimal Numbers, being then defined as those decimals where incomplete Units tend to infinity, having in them numbers that make up series, which are repeated every certain period, hence they are also known as newspapers.

However, in mixed newspapers there is an additional characteristic, which is that the first number that makes up the series of numbers that are repeated infinitely in their incomplete Units is at a certain distance from the comma. Likewise, between the comma of the decimal number and the first number of the series to be repeated there are some numbers that are not repeated. Therefore, being a periodic number, where there will be incomplete units constituted by numbers that are repeated and others not, these numbers are known as mixed newspapers, or even as semiperiodic decimal numbers.

Examples of mixed newspaper decimals

However, perhaps the best way to complete an explanation of Mixed Decimal Numbers is through the presentation of some examples, which allow us to see in a practical way how the structure of the incomplete Units of these numbers is configured, as can be seen in the following cases:

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Unlimited periodic mixed decimals
Source: Education  
October 29, 2019


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