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Perhaps the most recommendable thing, before approaching an explanation on the adequate form in which the generatriz fraction of a decimal number must be found, classified as a pure periodic unlimited Decimal, is to revise some definitions, that will allow to understand this procedure within its precise mathematical context.
In this sense, it may also be prudent to delimit this theoretical revision to four specific notions: Fractions, Decimal Numbers, Generatrix Fraction and Unlimited Pure Periodic Decimal Numbers, because the expressions and numbers directly involved in the operation are respectively, by means of which we try to establish what is the fraction equivalent to the decimal number, classified -because of the characteristics of its incomplete units- as an unlimited pure periodical decimal. Here are each of these concepts:
In this way, we can begin by saying that Mathematics considers the fraction as a mathematical expression, with which to refer to fractional quantities, which in turn constitute rational numbers. Likewise, this discipline indicates that the fraction should always be noted as the division between two integers, an expression that as a consequence will be conformed by two parts, explained in the following way:
- Numerator: this denomination will correspond to the number that constitutes the superior part of the fraction. Its mission will be to indicate how many parts of the whole the fraction represents.
- Denominator: secondly, the denominator will be found, which is then understood as the number that constitutes the lower part of the fraction. Its task is to indicate in how many parts the whole is divided, of which the fraction represents some of these, by means of the numerator.
On the other hand, it will also be useful to throw lights on the concept of the decimal number, which will be explained by Mathematics as the numerical element, by means of which non-exact or fractional quantities can be expressed, which can refer to both rational and irrational numbers. Likewise, the mathematical discipline indicates that decimal numbers are always made up of two different parts -one integer and the other decimal- which have been explained in the following way:
- Integer part: first, within the decimal number, an integer part can be distinguished, denominated by Mathematics as Units. It is made up of an integer, which may be positive, negative or even zero. As it is constituted by numbers belonging to the Decimal Numbering System, the elements of the Units 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, etc.
- Decimal part: in the second instance, within the decimal numbers, the mathematical discipline also refers to a decimal part, known as Incomplete Units. This part will always be made up of a number less than the unit, and that in the Numerical Line can be located between zero and one.
Its elements also have a positional value, distinguishing then, from left to right, between tenths, hundredths, thousandths, ten thousandths, etc. It is this part of the decimal number that is taken into account when classifying or identifying which type of decimal is the number, or whether it refers to a rational or irrational number.
Likewise, Mathematics points out that a rational number can be expressed in two different ways: either as the division or quotient of two numbers, conformed as a fraction; or on the contrary as a decimal number, which has limited or unlimited incomplete units but has periods that are repeated in them. Consequently, then, the generative fraction will be the fractional expression, constituted by two integers that divide, and from where a specific decimal number is born, referring to a rational number.
In this sense, it is important to say that the incomplete units of a decimal number representing a rational number must always be either limited, or unlimited periodic, since otherwise, the decimal number refers to a rational number, that is, a decimal number that has unlimited incomplete units, which extend to infinity, without repeating any series. Irrational numbers don´t have a generatizing fraction, since due to their characteristics it is impossible to represent them as a fraction.
Unlimited decimal number pure newspaper
Finally, it will also be important to point out that Mathematics has defined pure periodic unlimited decimal numbers as a decimal number, which refers to a rational number, which is characterized by counting in its incomplete units a number or series of numbers that repeat to infinity, and which are located immediately after the comma separating integers from incomplete units.
Generatrix fraction of a pure periodic unlimited decimal number
Once each of these concepts has been reviewed, it is probably much easier to approach an explanation of how to find the generatrix fraction from which a pure periodic unlimited decimal number has come out, which as a rational number at the end can be represented both as a decimal and as a fraction, since between these two fractions there is equivalence, since it basically refers to the same rational number or fractional quantity.
Steps to find the generatrix fraction of a pure periodic unlimited decimal number
In this order of ideas, the mathematical discipline has also pointed out that there is a method to follow, once the characteristics of the decimal number have been inspected, it has been verified that in effect it is an unlimited pure periodic decimal, and it is desired to find the generatrix fraction from which it comes, and that it will be conformed by the following steps:
1.- The decimal number will be taken and its comma will be suppressed.
2.- Only the decimal part will be taken, that is to say, the incomplete Units, and they will be annotated in the numerator of the generatrix fraction.
3.- This amount placed in the numerator will be subtracted from the integer part that originally had the unlimited periodic incomplete decimal number.
4.- In the space of the denominator it will then be necessary to place as many nines as numbers have the period that is repeated in the decimal number.
Example of how to determine the generatrix fraction of a pure newspaper unlimited decimal
However, it may be that the best way to complete an explanation about the adequate way in which the generatrix fraction of a pure unlimited periodic decimal number must be found is through a concrete example, which allows to see in practice how each one of the steps indicated by Mathematics are fulfilled. Next, the following exercise:
Find the generating fraction of the following unlimited pure periodic decimal number: 0.645664566456
October 31, 2019
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