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Perhaps the best thing to do before moving towards an explanation of the definition of Unlimited Decimals is to briefly review some concepts that will make it possible to understand these elements within their precise context.
In this sense, it may be best to limit this theoretical revision to three specific notions: Rational Numbers, Irrational Numbers and Decimal Numbers, since these are some of the numerical elements that may or may not be expressed by Unlimited Decimals, and whose characteristics will help to understand their nature and characteristics. Here are some of them:
In the first place, rational Numbers will be found, elements that can be understood as the expression of the quotient existing between an integer and a natural number. This quotient usually takes the form of a decimal number, which can be both limited and unlimited. In some cases, on the contrary, it is simply an integer.
For their part, irrational Numbers will be those which, due to the infinity of their decimal part, don´t have the capacity to be expressed in the form of a fraction, but will always be expressed in the form of a decimal, since their decimal part will be infinite, without any of its elements being repeated.
Finally, it will also be important to throw lights on the concept of Decimal Numbers, which have been described by Mathematics as a number composed of two parts: the first of them, consisting of an integer, which may be a positive integer, negative integer or even zero; secondly, this type of number will have a decimal part, which will be made up of a number less than the unit, and between 0 and 1.
These two elements will be joined – and at the same time separated – by a comma, even though some mathematical traditions also accept the use of a period. Also, the whole part -called units- will always be found to the left of the comma, while the part -known as incomplete units- that corresponds to decimals will be annotated to the right of this symbol. Decimal numbers will then serve to give expression to both rational and irrational numbers.
Unlimited decimal numbers
Once each of these definitions has been revised, it is certainly much easier to approach the concept of Unlimited Decimal Numbers, which will constitute one of the two types of decimal numbers that exist, and whose main characteristic will be to have an infinite decimal part, that is, that the incomplete units of the decimal number, annotated to the right of the comma are infinite.
However, these incomplete units may or may not be repeated, which will also give rise to the two subtypes of unlimited decimals that exist: non-periodic decimals and periodic decimals, each of which will have the following definition:
- Unlimited non-periodic decimal numbers: on the one hand, there will be an unlimited type of decimal that will have incomplete units, that will extend infinitely, without any period being repeated, that is to say, neither a number of infinite form, nor a series of numbers. Consequently, this kind of unlimited decimals will be used to express rational numbers. For example:
- Unlimited periodic decimal numbers: secondly, you will find unlimited newspapers, which will also have an infinite part, but which will have in it series of numbers that are repeated every certain period. In them there will also be two subtypes, whose main difference will be the distance that separates the first number of this period from the comma of the decimal number. The two types of unlimited periodic decimal numbers will be the following:
- Pure Periodicals: First, the pure periodic unlimited decimal numbers will be those that will have the first number of the period repeating in their complete units immediately after the comma. For example:
- Mixed newspapers: on the other hand, mixed newspapers will also be found, which will have a period which is not close to the comma, and is also preceded by some numbers which are contiguous to this sign, and which are not repeated. By having in its decimal part numbers that are repeated and others that are not, these numbers also receive the name of semi-periodic numbers. Some examples would be the following:
October 29, 2019
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