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Perhaps the most recommendable thing, before approaching an explanation on the correct form in which the Fraction generatriz of an unlimited periodic mixed decimal number must be found, is to revise of brief form some definitions, that will allow to understand this mathematical procedure, within its precise context.
In this sense, it may also be useful to focus this theoretical review on three specific notions: Decimal Numbers, Unlimited Decimal Numbers, Mixed Periodicals, and Generatrix Fraction, as these constitute respectively the numbers and expressions directly involved in determining which fraction has generated, or corresponds to, a specific unlimited periodic decimal. Here are each of these concepts:
In this way, we will begin by saying that Decimal Numbers have been described by different mathematical sources, as those numerical elements, through which fractional quantities are given written expression, which in turn constitute rational and irrational numbers. Also, the mathematical discipline has pointed out that in the Decimal Numbers two different parts can be distinguished, one integer and another decimal, which have been explained in the following way:
- Integer Part: in the first place, within the Decimal Number there will be a part, known by the name of Units, and constituted by an integer, which can be either positive or negative, or even zero. Since it is constituted by numbers belonging to the Decimal Numbering System, in the Units of the decimal number there will be a positional value, so it will be possible to identify, from right to left, the units, tens, hundreds, units of a thousand, tens of a thousand, etc., that construct the whole number that makes up this part of the decimal number.
- Decimal Part: on the other hand, in the Decimal Number there can also be a decimal part, which is called by Mathematics as Incomplete Units. These will be conformed by a number always smaller than the unit, and that can be located, in the Numeric Line, between 1 and 0. In this part of the decimal number it will also be possible to speak of positional value, being in it on its side, and from left to right, the tenths, hundredths, thousandths, ten-thousandths, etc.
Both parts of the Decimal Number, that is, the Units and the Incomplete Units, are separated -and at the same time joined- by a comma, even when there are mathematical currents that incline for the use of the point. Regardless of the sign chosen to constitute the decimal number, to the right of the decimal number, incomplete Units should always be noted, that is, the decimal part of the number, while the whole part, the Units, will be placed to the left.
Unlimited decimal number periodical mixed
Secondly, it should also be noted that there are several types of decimal numbers, which are classified according to the characteristics of their incomplete units, according to whether they are limited or unlimited.
Within the unlimited decimal numbers, that is, those decimal numbers that have infinite incomplete Units, you will find a subtype known as mixed periodic unlimited decimal numbers, which basically will be described by Mathematics as those decimal numbers that have incomplete units in which you can see a period that is repeated several times in the number, but that – unlike the unlimited pure periodic decimal – the first number of this period is at a certain distance from the comma, with some numbers between this sign and the series or period which will not be repeated, and which in turn are known as the anteperiod.
In fact, the existence within the incomplete Units of the unlimited mixed periodical decimals of a period and an anteperiod is what has made these numbers also known as mixed periodicals, impure or even as unlimited semiperiodical decimal numbers.
Finally, it is also advisable to pause for a moment on the definition given by Mathematics on the Generatrix Fraction, which has been seen as the fraction from which a specific decimal number comes, is generated or corresponds, as long as it is constituted by a rational number, that is to say, that it has limited incomplete units or periods that are repeated, since otherwise it will be in front of a decimal number that refers to an irrational number, which by nature has the impossibility of being expressed in the form of a fraction, that is to say, that it does not have a generative Fraction.
It is also important to remember that a fraction is basically the expression of a quotient between two whole numbers, which make up respectively each of the parts that have a fraction: a numerator, an element that constitutes the upper part of the fraction, and that indicates how many parts of the whole have been taken; and a denominator, which makes up the lower part of the expression, at the same time indicating into how many parts the whole of which the fraction represents only some parts, indicated by the numerator.
Generatrix fraction of an unlimited mixed newspaper decimal
Once each of these concepts has been reviewed, it is perhaps much easier to explain the correct way in which any operation should be resolved, the objective of which is to determine which is the Generator Fraction of any decimal that has an anteperiod and a period in its incomplete units, i.e., that is, an unlimited periodic mixed decimal.
In this order of ideas, Mathematics has raised a series of steps that must be followed when facing an operation of this type, and that basically can be enumerated in the following way:
- In the first place, once the number on which the Generator Fraction must be found has been given, its incomplete units should be reviewed to determine what type of decimal number it is, and if it actually corresponds to a rational number.
- Done this, and found that in the incomplete units of the number the existence of an anteperiod and a period can be found, it is understood that it is a rational number, and that therefore it does have a generative fraction.
- In order to determine the Generator Fraction, the decimal number must be taken and its comma must be suppressed, to be placed later in the numerator of what will be the Generator Fraction. That is to say, that in the numerator all the number is placed, without commas, having then its whole part, its anteperiod and its period.
- Next, this number that has been placed in the numerator will be subtracted by a number constituted, in order, by the whole part and the anteperiod. This subtraction will originate the definitive numerator.
- On the other hand, the Denominator will be conformed by a number that will be constituted by as many nines as numbers have the whole part of the original decimal number, followed by as many zeros as elements has had the anteperiod of the unlimited periodic mixed decimal number, on which the generative Fraction has been calculated.
Example of a Generator Fraction of an Unlimited Periodic Mixed Decimal
However, the best way to conclude an explanation of how to proceed whenever the Generator Fraction of an unlimited mixed newspaper is to be determined may be through the exposition of a concrete example that allows us to see in practice how each of the steps indicated by the mathematical discipline are fulfilled, as can be seen in the exercise shown below:
Find the Generator Fraction of the following number: 234,987564564564…
- Once you have the decimal number, you can see in its incomplete units how there is a part, located after the comma, which isn´t repeated, and is followed by a part that does several times, and that as indicated by the suspension points that follow it, this repetition extends to infinity. In conclusion, the decimal number provided is an unlimited periodic mixed decimal, where there is an anteperiod and a period, therefore it does have a generatrix Fraction, because it is a rational number.
- However, before proceeding to find this generative Fraction, it will be necessary to express the number in a summarized form, that is, only with its entire part, its anteperiod and the series of numbers that constitute the period that is repeated in it, to which a sign will be added in the upper part, which will then indicate that this number is repeated infinitely in the decimal number:
3. Once this is done, the necessary steps can then be taken to find the generatrix fraction corresponding to this unlimited mixed newspaper decimal. To do this, you should start by taking the complete number, suppressing the comma, and consider it as the first number to be noted in the numerator:
4. Once again, the decimal number will be taken, from which the comma has been deleted, and only the part constituted by the whole part and the anteperiod will be taken:
5. This number obtained will then be subtracted from the first number recorded in the numerator:
6. On the other hand, the denominator must be conformed by a number constituted by as many nines as the elements have had the whole part, and as many zeros as numbers have had the anteperiod:
7. Once this expression is obtained, it is considered that the Generator Fraction of the unlimited periodic mixed decimal number, which was given in the exercise, is found:
October 31, 2019