Perhaps the most convenient thing to do, before advancing in an explanation about unlimited non-recurrent decimal numbers, is to briefly review some definitions that will allow us to understand this type of number in its just mathematical context.
In this sense, perhaps the best thing is to focus this theoretical revision on two specific notions: the first of them, the concept of irrational Numbers, as these are closely related to the notion of unlimited non-periodic decimals. Likewise, lights should be thrown on the very definition of Decimal Numbers, in order to become aware of the nature of this class of numbers, of which the limited non-periodicals are a type. Here is each of them:
In this way, we will begin by saying that irrational Numbers will be those numbers that -different from their contraries, rational Numbers- can never be annotated or expressed in the form of a fraction, since they have an infinite decimal part, which is specifically characterized by not counting in their numbers any series or period that is repeated.
As for decimal Numbers, they have been defined by Mathematics as those numbers that will serve to express both rational and irrational Numbers, and that will be composed of two parts:
- The units: first there will be the units, which will always be made up of an integer, which can be positive, negative or even zero. At the same time, being part of the decimal system, each element within the units will have a positional value, being then from right to left the following elements: units, tens, hundreds, thousandths, units of a thousand, tens of a thousand, hundreds of a thousand, etc.
- Incomplete units: on the other hand, in the decimal numbers there will also be a second part, also known as the decimal part, which will be conformed by a smaller number of the unit, which will be located between zero and one. Likewise, the elements that make up incomplete units will have a positional value, counting from left to right. Among them, tenths, hundredths, thousandths, ten-thousandths, etc. can be distinguished.
These elements will be found within the Decimal Number united -and at the same time separated- by a comma, even though some mathematical schools also accept the use of the period. However, regardless of which of the two signs is chosen, the entire part of the decimal number – that is, the units – will always be annotated to the left of the comma, while incomplete units will be annotated after it.
Unlimited non-periodic decimals
Once each of these definitions has been reviewed, it may be much easier to approach an explanation of the numbers called Non-periodic Unlimited Decimals, which will constitute a subtype of unlimited decimals, that is, those with infinite incomplete units.
In this way, it can be said that the unlimited non-periodic numbers will be those decimal numbers that, in addition to having incomplete units that extend to infinity, in them there is no series of numbers that are repeated, that is, that their decimals are infinite and without periods. This type of numbers are also called aperiodic.
Examples of unlimited non-periodic decimals
However, it may be that the most efficient way to complete an explanation about unlimited non-periodic decimal numbers is through the exposition of a series of examples, where you can see in a practical way the structure that this type of decimal number has in its incomplete units, which are infinite and non-periodic. Here are some of them:
October 29, 2019