At the time of approaching the exposition of some examples, that allow to see in practice how any operation must be solved, that pretends to determine which is the Fraction generatriz of an unlimited periodic mixed Decimal Number, perhaps the most advisable thing is to revise of brief form the own definition of this procedure, in order to understand each one of the exercises within its precise context.
Generator fraction of an unlimited periodic mixed decimal number
Consequently, we will begin by saying that Mathematics has explained this operation, as the mathematical procedure, whose main objective is to determine which is the fraction from which comes, or is equivalent, to a rational number, expressed in decimal form, and which has the characteristics of having in its incomplete units a number that repeats itself to infinity, and that is located at a distance from the comma, being found between this symbol and this period, a number that is not repeated, and that constitutes the anteperiod, that is to say, that is to say, it is an unlimited decimal Number mixed periodic, which constitutes the anteperiod, that is to say, that it is an unlimited decimal Number mixed newspaper.
Mathematics has also indicated the steps to be followed when solving this type of operations, in order to find the correct generating Fraction of any number with these characteristics. Next, each one of these:
- The first thing that must be done, given the decimal number, is to summarize it, in order to be able to see each one of its parts correctly: its entire part, the anteperiod and the period.
- Once this has been done, and it has been determined that in reality it is an unlimited periodic mixed Decimal Number, then the Generator Fraction will begin to form, so the complete number will be written down in the space destined to the Numerator, with the suppressed comma, in other words, the whole number from its entire part to the period.
- Then, the whole number that makes up the decimal number should be subtracted from this number. The difference will finally constitute the Numerator of the decimal number.
- Finally, as the Denominator, a number composed of the following elements should be noted: first, as many nines as elements have had the anteperiod; second, as many zeros as elements have the period.
- The Generator Fraction is considered found. Therefore, it is possible to express the answer of the exercise.
Examples of how to find the Generator Fraction of an unlimited mixed periodical Decimal
Once this definition has been revised, as well as the method conceived by Mathematics to give solution to this type of operation, it will then be possible to continue with the exposition of some exercises that will serve as an example to the correct way of finding the Generator Fraction. Here are some of them:
Example 1
Find the Generator Fraction of the following number: 0,9877777
When solving this exercise, one will then begin by summarizing the number, in order to be able to examine much more easily the characteristics of its incomplete units:
At the moment of doing so, we will see how this number has an entire part, an anteperiod that is not repeated, and a period, made up of a single number. As a result, you will be faced with an unlimited periodic mixed Decimal Number. Therefore, at the time of constituting your Generatrix Fraction, you will begin by writing down in the Numerator the entire number, after suppressing its comma. As in this case the integer is equal to zero, only the anteperiod and the period will be placed:
This number should be subtracted from the number composed by the whole number and the anteperiod, as the first is equal to zero, only the second will be subtracted:
In the place of the Denominator, as many nine as the anteperiod has had will be missed: in this case, two zeros. This number will be followed by as many zeros as the period has had: that is, only one.
Example 2
Find the Generatrix Fraction of the following number: 98,45997997997…
Likewise, the first thing to be done is to summarize the number, in order to be able to see more precisely each of the elements that have this decimal number:
In doing so, you will then be able to see an unlimited periodic mixed Decimal Number, made up of an entire part, a period in which two elements are counted, and a period in which there are three elements. Once this is done, the Generator Fraction begins to be constructed by writing down in the numerator all the complete number, without the comma, and subtracting the number formed by its entire part and the anteperiod:
The next thing to do is to write down the Denominator of the Generator Fraction. For this, a total of two nines will be noted, corresponding to the two numbers that constitute the anteperiod of the number. Next, three zeros will be recorded, corresponding to the elements of the period:
Example 3
Find the Generator Fraction of the following number: 0,78888888…
As always when you are facing an exercise of this type, you should start by writing down the decimal number in summary form, in order to be able to see each of its elements clearly:
In doing so, one then has a decimal number composed of an integer part equal to zero; an anteperiod equal to 7; and a period equal to 8. In other words, it is an unlimited periodic mixed decimal number. At the moment of beginning to construct the corresponding Generator Fraction, the complete number should be noted in the numerator. Considering that the integer part is equal to 0, simply note 78:
This number will be subtracted from the value of the anteperiod, and in the Denominator will be placed a nine and a zero, equivalent to the only element that can be counted both in the anteperiod and in the period:
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