1 December 2023 In mathematics and linear algebra, many properties and theorems are applied that significantly help resolution and simplification of a mathematical problem. In early stages, we are taught some of the simpler properties. Among these we find the commutative, distributive and associative property. Well, today we will focus on the last one, since they have great relevance when solving an algebraic addition problem.

So today you will have the opportunity to learn all the most fundamental and important aspects of this property. You will notice that it can be useful both in the field of mathematics and in other scientific fields, such as physics and chemistry. In addition, it applies to any type of term within an equation or polynomial, from integers to terms with dependent and independent variables.

## Concept and definition of the associative property

If we start from the associative term, this is a variant of the word association, which indicates associating an element with others as long as they have the same characteristics. For example, if you have several baskets of fruits, by association, what you do is gather those fruits that are the same in one place in order to classify them and have an order.

With this definition and example, we can define the associative property as the procedure that someone does in order to group like terms. Now, this is something that is linked to all mathematical operations, whether you are dealing with addition, subtraction, multiplication or division. Of course, the associative property becomes more complicated when the equation has many terms and with complex terms to manipulate. Now, it is possible that teachers tell you that this property is linked to addition and multiplication, but with practical examples, you will notice that this is not always the case.

The simplest and most basic way to apply this property is before an algebraic sum. That is, where there are terms that are adding and subtracting in the same polynomial or equation. What is done is grouped within a parenthesis those terms with a positive sign, and, in another parenthesis, the terms with a negative sign.

Now, this property allows us to move the position of the numbers and terms in such a way that there is no inequality and that the resolution of the problem, not be affected. Of course, this as long as the corresponding signs are respected. For this reason, one must be very careful when there are negative signs, since changing this position term must be done taking said operation sign into account.

## What is the associative property used for?

In the first instance, the associative property allows facilitate the resolution of any mathematical problem. That is to say, that, through this, what you can do is a grouping, in such a way that you have everything ordered and you solve the equation step by step without getting confused with the terms, variables and signs of operation.

It must be taken into account that this property is, in a certain way, the basis of others, since after having made the association of terms, the distributive or commutative property can be used. For example, in the first case, after making the grouping, you can distribute the multiplication or division among the terms wereto of parentheses and those who are within it.

Now, making a more precise and detailed summary of this property, we can say the following:

• allows the grouping of terms and elements.

• You can change the position of terms within an equation or polynomial.

• It facilitates the power to apply mathematical properties and fundamental theorems of calculus.

• It is possible to apply the distributive property, either with the operative signs, or with other terms.

• It can be applied to both addition, subtraction and multiplication operations.

It is important to be very careful when trying to apply this property to subtraction. If the correct association is not made, this can lead to the final result not being correct. For example, imagine we have the following math problem: 10 – 5 +1 – 5. Thanks to the associative property we can rewrite this as follows: (10 + 1) + (-5 -5). Although it can also be written this way: (10 + 1) – (+5 +5).

Whichever way you choose, it will be the right one. In case you are wondering how this is possible, it is all due to the use of the operation signs that are used, as well as the distributive property that is executed once the problem has been solved. Both in the first and in the second case, the central sign multiplies everything that is inside the second parenthesis. Keep in mind that if you want to factor out the negative sign, all the elements that are inside the negative parentheses become a sum. The equivalence is maintained thanks to the fact that at the end, you will do a multiplication of signs or, in other words, you will apply the distributive property of the minus sign.

## What are the advantages and disadvantages of using the associative property?

As for the advantages of this propertywe can mention the following:

• It makes it easier to solve operations step by step and by parts within an equation.

• Helps facilitate grouping of terms.

• It allows to have a greater order.

• It gives the person or student the option of ability to apply, with greater precisionthe commutative property in case of being in a multiplication.

• It also allows the implementation of the distributive property once the mathematical operations have been solved.

• It helps to identify what are the like terms and the independent terms.

• It can be applied to both integers, natural, rational and irrational numbers.

• The basis of this property can be applied to the field of complex numbers.

With all of the above, does the associative property have any disadvantages? The reality is that no. It is simply a way of facilitate the problem-solving work, either in mathematics, linear algebra, chemistry and physics. The only thing that could be considered as a disadvantage is that it cannot always be applied, since it is necessary that there are more than two terms. That is, if you have an equation or a one-term polynomial. Therefore, more than two elements have to exist in order to implement this property and the concept behind it.

## What are the characteristics of the associative property?

The first characteristic is that it allows associate two or more elements or terms that have the same structure or shape. For example, if you have the following algebraic expression: 2x + 7y +5z -8y -x. What we can do with this by applying the associative property is the following: (2x – x) + (7y – 8y) + 5z.

What happens with this simple example is that we have put everything together in such a way that now the math problem has an easier way to solve it. If you notice, the last term has been left without parentheses. Not having another element with which to pair it, it is not necessary to put it inside this resource, but leave it as it is. Now, regarding the operation inside the first parenthesis, we find that the result is positive. For the second, the result is negative and the third, there is no change. Thus, the final result is as follows: x – y + 5z.

Another of the characteristics of this property is that after having done the corresponding process, you can implement other properties like the distributive in case there are terms adding or subtracting. You can also use the commutative property for situations where there is a multiplication. We can say that the order of the factors, at a mathematical level, does not alter the product, as long as the operative signs are respected.

### Difference Between the Associative and Commutative Property

The main difference is that, through the associative property, what you do is group terms that are similar and that are adding or subtracting. It does not matter if it is an integer, real or fractional number, you can group them inside parentheses respecting the corresponding signs. However, in the case of the commutative, what is done is change the position of elements in a multiplication. For example:

• If we take the expression 2x +4y -11x -11 +4y +15 as an example, the application of the associative property would be as follows: (2x -11x) + (4y – 4y) + (15 – 11).

• Of all the examples of the commutative property, the plus would be if we have the following expression: X20Y + XZ11. Applying the property, we could write it as follows: 20XY + 11XZ.

### What is the relationship between the associative and distributive property?

The most direct relationship that exists between these two properties is that the first in a certain way gives way to the application of the second. That is, you first apply an associative procedure and then implement the distributive property. Of course, this as long as there are terms and elements that are multiplying numbers and variables inside parentheses. For example, if we have 4(2x -11x) + (4y – 4y) + 10(15 – 11), what remains is to solve the multiplication in the terms outside the parentheses. In a way, it’s a way to simplify a result. Mathematical.

## Examples of the associative property in mathematics

Throughout the article we have been giving some examples where you can see and understand what this property is about. Either way, you can consider the following examples: