Commutative property in the sum of fractions

It is likely that the most accurate way to address an explanation about commutative property in the sum of fractions is to pre-review some definitions, which will be useful in understanding this mathematical property within its context Precise.


Fundamental definitions

In this regard, it may also be prudent to delimit this revision to two specific notions: first ly xout ingestry the concept of fraction, since it will be necessary in understanding the mathematical expression involved in the operation of sum that gives rise to this mathematical property, the definition of which must also be brought to chapter. Here’s each one:

The fraction

In this way, it will begin by saying then that the Mathematics designates the Fraction as one of the two expressions with which you can count a fractional number, explained in turn as the numerical element by which a non-integer or non-exact number is represented.

Likewise, this discipline indicates that the fraction will be presented as a division between two integers, which are arranged on top of each other. The numeric element that occupies the top position will be named Numberer, and will be responsible for pointing out which part of the whole represents the fraction.

For its part, the number located at the bottom will be called Denominator, and you will aim to account for the whole of which part has been taken. Examples of fractions will be as follows:

Sum of fractions

As for the sum of fractions this will be understood as a mathematical operation, in which it is a question of finding the total that would yield the addiction of the respective values of two or more fractions, which would play the role of additions. However, Mathematics also points out that the way to solve this type of operation will fundamentally depend on whether or not the fractions match their denominator, thus having these two possibilities:

  • Fractions with equal denominator: in case the fractions involved in a sum have equal denominators, then the sum operation must be solved simply by the sum of the values with which the numerators count.
  • Fractions with different denominator: on the contrary, if fractions that function as sums do not have equal denominators, then it will be necessary, before advancing in the sum of fractions, to find the common denominator, so that once calculated, it will be necessary, before advancing in the sum of fractions, to find the common denominator, so that once calculated, it will be then proceed to the sum of the respective numerators, obtained in the operations carried out for the determination of this common denominator.

Commutative property in the sum of fractions

With these definitions in mind, it may then be much easier to address a definition on commutative property present in this mathematical operation.

In this order of ideas, it will be necessary to say that like any addition operation, the succession in fractions complies with the Commutative Law, which indicates that always and without any exception the additions of this operation can exchange their respective positions, post that the “order of factors will not alter the product”. This property can be expressed mathematically as follows:

Example of Commutative property in the sum of fractions

However, a concrete example may still be needed to see in practice how whenever the order is actually altered or modified in the sum of fractions, the same results will be obtained. For this, a sum will be taken as an example between fractions of different denominator, in order to show the most complex case, and as regardless of this the Comcomtive Property is still fulfilled without exception, as shown below:

In the following operation, check the commutative property:

To comply with this approach, it will be necessary to begin by resolving the operation in the above order. Taking into account that for this the common denominator must then be calculated, before proceeding to the sum:

Now having two fractions of equal denominator, the sum can be resolved, causing the numerator values to find a total:

The resulting fraction may be simplified, so it will be necessary to divide each element of the expression by a common divider:

Upon reaching this result, it cannot be further simplified, so it will be taken as the final product of the operation. Once this is done, you must then reverse the order of the original sums, to check whether doing so you get equal result in the sum of fractions:

Being still fractions of different denominator, the first step in solving this sum will be to find the common denominator. Taking into account that multiplication also responds to the commutative law there is no risk of different results:

At this point, it can be observed that the same factors have been obtained after the calculation of the common denominator. The sum operation can also be continued:

Likewise, the result should be simplified by dividing each term by a common denominator:

Equal results have been obtained, so in this case the Commutative Property is considered to have been verified in the sum of fractions.

Commutative property in the sum of fractions
Source: Education  
September 21, 2019


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