Mathematical properties present and absent in fractions division

Perhaps the best thing, prior to addressing an explanation of each of the properties inherent in the Fractions Division, is to briefly review some concepts, which will allow us to understand the laws present in this operation, as well as those that aren´t found in it.

Fundamental definitions

In this sense, it will also be necessary to focus this review on two precise notions: fractions and fractions division, as these are respectively the expression and mathematical operation on the basis of which the properties that are their properties are existing conferred by this discipline. Here’s each one:


In this way, it can be said that fractions have been explained by mathematical discipline as one of the two possible forms of expression with which fractional numbers count, that is, that fractions will be used to represent fractional or non-whole amounts.

On the other hand, the different sources choose to indicate that the fraction is an expression consisting of two elements, each of which have been described as follows:

Numerator: First, the Mathematics mentions the Numerator, who will be placed at the top of the expression. Your task will be to point out how many parts of the whole have been taken, or represent the fraction.

Denominator: With respect to the Denominator, the various authors point out that the denominator will occupy the lower level of the fraction, pointing on how many parts the whole is divided, of which the numerator represents only a fraction.

Fractional division

Likewise, it will be helpful to review the concept of Fraction Division, which is then seen as a mathematical operation, by which it seeks to calculate how many times a fraction is contained, which serves as a divider, within another expression of the same kind, which in turn serves as a dividend.

As for the correct way in which this type of operations should be carried out, in order to obtain the quotient, the Mathematics indicates that the best method will be that of cross multiplication, where the numerator of the first expression is multiplied by the denominator of the second, while the denominator of the second fraction is multiplied by the numerator of the other expression involved. This operation may be represented mathematically as follows:

Properties of the Fractional Division

Once these definitions have been revised, it may certainly be much easier to approach each of the properties present or absent in the Fractions Division, which should also be studied as they exist or not in this operation, as is shown below:

Existing properties in fractions division

In the case of mathematical properties that can be found in the Fractions Division, Mathematics indicates that two of them can be distinguished:

  • Internal Property: this property will indicate that whenever a Division is made involving two or more fractions, the result will always be and if I exception another fraction, hence it receives the name Internal Property, since the ratio obtained will also belong to fractional numbers.
  • Distributive property: secondly, the Mathematics will point out that the Fractions Division may also speak of Distributive Property, which is always given in relation to the sum, provided that this operation takes the place of the diviso, otherwise Property will be impossible.
  • Thus, whenever a sum of fractions is divided by another fraction, this mathematical law dictates that the operation can be solved, dividing each one by the fraction that divides them both, and then sum the respective results. This Property may be expressed as follows:

Non-existent properties in fractions division

On the other hand, it will also be relevant to explain which mathematical properties aren´t found or not possible in the Fractions Division, and which according to the different sources, would then be the following:

  • Commutative ownership: according to what the various authors point out, the Fractional Division may not speak of Commutative Property, since any division has been raised, any change or investment between their respective factors will result in a change in the outcome. This Law may be expressed as follows:

  • Associative property: likewise, mathematics indicates that once a division has been proposed between three or more fractions, they will not have the possibility of establishing different relationships with each other, without this not resulting in a change to the quotients obtained, so it is then claimed that in the Fractions Division it isn´t possible for associative Property to happen. This mathematical law shall have the following expression:


Mathematical properties present and absent in fractions division
Source: Education  
September 30, 2019

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