Non-commutative ownership in the subtraction of fractions

It is likely that the best thing, before moving forward on a definition of non-commutative Property in the Fraction Sunder, is to take into account some definitions, which will allow us to understand this Mathematical Law within its precise context.


Fundamental definitions

In this sense, it may also be relevant to delimit this theoretical revision to two specific notions: the first, the definition of Fractions itself, since this will allow to have the nature of the expressions on the basis of which the Operation of The Subtraction of Fractions, the definition of which will also be brought to chapter, as it is the mathematical procedure in which the Non-Commutative Property takes place. Here are each of these definitions:

Fractions

In this way, it will begin by saying then that fractions are understood by Mathematics as one of the two expressions that can have fractional numbers, so then it is assumed that the fraction realizes an amount not accurate or not whole, hence its name, since it represents a part or fraction of the quantity.


Likewise, mathematical discipline has indicated that the fraction will be constituted as a sustained division between two integers, each of which occupies its own position, and fulfills its own task, as can be seen below:

  • Numerator: On the one hand, the Mathematics indicates that the numerator will correspond to the number that occupies the top of the fraction. This number is intended to point out what part of the whole, which represents the fraction, of which this numerator is part.
  • Denominator: on the contrary, the number that is occupying the bottom of the fraction will then be understood as the denominator, which will fulfill the function of indicating in how many parts the whole is divided.

Rest of fractions

In another order of ideas, it will also be necessary to cast lights on the definition of Fractions Subtraction, which will be explained by Mathematics as the operation that occurs when a subtraction, which acts as a minuendo, suppresses in it the exact amount that indicates a second fraction, which explains how to subtract, in order to generate a result, which is seen as the difference.

However, it is important to note that Mathematics also indicates that there is not a single method for resolving a Fraction Sunder, but that the method applied will depend directly on the relationships of homogeneity or heterogeneity between the different fractions, then having two precise cases:

  • If fractions are of the same denominator: if the fractions participating in the subtraction have the same denominator, that is, they are homogeneous, the solution to the operation will be assumed by a single denominator, and subtracting the values of the numerators.
  • If fractions are different denominator: however, fractions may also not match their numerators or denominators, which will require the need to undertake operations that allow the fractions to be homogenized, for which the fractions are apply the cross multiplication of numerators by denominators, while the common denominator will be obtained by multiplying the numerator values of each fraction, as seen below:

Non-commutative property in the FractionS Subtraction

With these definitions in mind, it is perhaps certainly much easier to address an explanation of one of the properties that can be found in the Fraction Sunder: non-commutative property.

In this regard, the Mathematics has pointed out that in this operation this mathematical law takes place, which indicates that in the Subtraction of fractions any modification that is made in relation to the order of the elements, that is, of the fractions, will involve direct changes in the difference obtained, since “the order of factors does alter the product”, a situation that can be expressed mathematically as follows:

Example of Non-Commutative Property in fraction sunder

However, a specific example may still be needed to demonstrate why whenever a subtraction operation reverses or alters the places of fractions, changes will occur in the result, as can be seen below:

Check the Non-Commutative Property in the following Fraction Subtraction:

In order to comply with what has been indicated by the postulate of this exercise, it will be necessary to resolve this subtraction in the two possible orders:

First order:

Second order:

Therefore, by obtaining different results, which even result in inverse fractions, it is then considered that the Non-Commutative Property has been checked in the subtraction of fractions, since by varying the order of its factors, it is obtained totally different results. This relationship, or mathematical property, in this specific case, can then be expressed as follows:

Consequently, in a Fraction Sunder, the order of the factors cannot be changed without this leading to an alteration in the result or final difference.

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Non-commutative ownership in the subtraction of fractions
Source: Education  
September 22, 2019


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