Absent properties in fractions division

Perhaps most pertinently, before reviewing what mathematical properties are impossible to meet in the Fractions Division, is to take into account some definitions, indispensable for understanding each of the absences of these laws, in their context Specific.


Fundamental definitions

In this sense, it may also be prudent to delimit this conceptual review to two specific notions: fractions and fractions division, as these are respectively the expression and operation in which the absence of some mathematical laws occurs, such as commutative and associative. Here are each of these definitions:

Fractions

In this way, it will begin by saying that Mathematics has defined fractions as one of the two types of expressions with which fractional numbers count, that is, they are mathematical elements used to represent non-integer or non-exact amounts . Thus, the different sources agree that fractions will be composed always and without exception by two elements, which have been explained as follows:

  • Numerator: first, fractions will have the Numerator, which will be understood as the element by which it expresses how many parts of the whole have been taken, or represents the fraction. Your place will always be the top of the expression.
  • Denominator: on the other hand, the fraction will also be composed of the Denominator, which will unalterably occupy the bottom of the fraction, and which will serve to indicate in how many parts the whole was divided, from which some, expressed, expressed by the Fraction Numerator.

Fractional division

Likewise, it will be necessary to throw lights on the concept of Fraction Division, a procedure that will be assumed as a mathematical operation in which the main objective is to determine how many times one fraction is included in another, or what is the same: which one is the ratio obtained from dividing a fraction that acts as a dividend and another that it exercises as a divider.

With regard to the correct way to solve such an operation, the different mathematical sources also point out that the correct form will be through cross multiplication, which then forces to multiply the numerator of the operation by the denominator of the second expression, and the denominator of the first fraction by the numerator of the fraction that serves as a divider. This procedure may be expressed mathematically as follows:

Absent properties in fractions division

Once these definitions have been reviewed, it may be much easier to approach an explanation of what mathematical properties aren´t present, or to which the Fractions Division doesn´t respond.

In this order of ideas, Mathematics notes that there are two specific laws that are not found in this operation: Commutative Property and Associative Property. Here is a brief explanation of why they are not met within this operation:

Non-commutative property

First, it will then be necessary to point out that in the Fractions Division it will be impossible to comply with the Comcomative Property. It should be said that this Mathematical Law expressly refers that in an operation, the factors that constitute it may vary or modify its order, without this resulting in different results, that is, that the order of the factors does not alter the product.

However, in the Fractions Division this is not the case, so any change in the order of fractions based on which the fractions division is established will lead to different results. The absence of this property may be expressed mathematically as follows:

Non-associative property

Similarly, associative property will be one of the mathematical properties that cannot be fulfilled in the Fractions Division. In this regard, the Mathematics notes that associative property will be the one that occurs whenever there is an operation between three or more factors, and they have the freedom to exercise different associations, without this referring to an alteration in the results Obtained.

However, in the Fractions Division this does not happen, but on the contrary, each new association produces different results, hence it is said that associative property is one of the mathematical laws absent in this operation. The inability of the Fractions Division to comply with this mathematical property can be expressed as follows:

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Absent properties in fractions division
Source: Education  
September 30, 2019


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