ASSOCIATIVE PROPERTY: ADDITION, MULTIPLICATION, EXAMPLES, EXERCISES - MATHS - 2024

The associative property of addition represents the associative character of the addition operation in various mathematical sets. In it, three (or more) elements of said sets are related, called a, b and c, such that it is always true:

a + (b + c) = (a + b) + c

In this way it is guaranteed that, regardless of the way of grouping to carry out the operation, the result is the same.

Figure 1. We use the associative property of addition many times when doing arithmetic and algebraic operations. (Drawing: freepik Composition: F. Zapata)

But it should be noted that the associative property is not synonymous with the commutative property. That is, we know that the order of the addends does not alter the sum or that the order of the factors does not alter the product. So for the sum it can be written like this: a + b = b + a.

However, in the associative property it is different, since the order of the elements to be added is maintained and what changes is the operation that is executed first. Which means that adding first (b + c) and adding a to this result does not matter than starting adding a with by to the result adding c.

Many important operations such as addition are associative, but not all. For example, in the subtraction of real numbers it happens that:

a - (b - c) ≠ (a - b) - c

If a = 2, b = 3, c = 1, then:

2– (3 - 1) ≠ (2 - 3) - 1

0 ≠ -2

Associative Property of Multiplication

As was done for addition, the associative property of multiplication states that:

a ˟ (b ˟ c) = (a ˟ b) ˟ c

In the case of the set of real numbers, it is easy to verify that this is always the case. For example, using the values ​​a = 2, b = 3, c = 1, we have:

2 _˟ (3 _˟ 1) = (2 _˟ 3) _˟ 1 → 2 _˟ 3 = 6 _˟ 1

6 = 6

Real numbers fulfill the associative property of both addition and multiplication. On the other hand, in another set, such as that of vectors, the sum is associative, but the cross product or vector product is not.

Applications of the associative property of multiplication

An advantage of operations in which the associative property is fulfilled is to be able to group in the most convenient way. This makes resolution much easier.

For example, suppose that in a small library there are 3 shelves with 5 shelves each. In each shelf there are 8 books. How many books are there in all?

We can carry out the operation like this: total books = (3 x 5) x 8 = 15 x 8 = 120 books.

Or like this: 3 x (5 x 8) = 3 x 40 = 120 books.

Figure 2. One application of the associative property of multiplication is to calculate the number of books on each shelf. Image created by F. Zapata.

Examples

-In sets of natural, integer, rational, real and complex numbers, the associative property of addition and multiplication are fulfilled.

Figure 3. For real numbers, the associative property of addition is fulfilled. Source: Wikimedia Commons.

-For polynomials they also apply in these operations.

-In the cases of subtraction, division and exponentiation operations, the associative property does not hold for real numbers or polynomials.

-In the case of matrices, the associative property is fulfilled for addition and multiplication, although in the latter case, commutativity is not fulfilled. This means that, given the matrices A, B and C, it is true that:

(A x B) x C = A x (B x C)

But… A x B ≠ B x A

The associative property in vectors

Vectors form a different set than real numbers or complex numbers. The operations defined for the set of vectors are somewhat different: there are addition, subtraction and three types of products.

The sum of vectors fulfills the associative property, as do numbers, polynomials, and matrices. As for the scalar products, scalar by vector and cross that are made between vectors, the latter does not fulfill it, but the scalar product, which is another kind of operation between vectors, does fulfill it, taking into account the following:

-The product of a scalar and a vector results in a vector.

-And when scalarly multiplying two vectors, a scalar results.

Therefore, given the vectors v, u and w, and additionally a scalar λ, it is possible to write:

- Sum of vectors: v + (u + w) = (v + u) + w

-Scalar product: λ (v • u) = (λ v) • u

The latter is possible thanks to the fact that v • u is a scalar, and λ v is a vector.

However:

v × (u × w) ≠ (v × u) × w

Factorization of polynomials by grouping of terms

This application is very interesting, because as it was said before, the associative property helps to solve certain problems. The sum of monomials is associative and this can be used for factoring when an obvious common factor does not appear at first glance.

For example, suppose you are asked to factor: x ³ + 2 x ² + 3 x +6. This polynomial has no common factor, but let's see what happens if it is grouped like this:

The first parenthesis has a common factor of ax ²:

In the second the common factor is 3:

Exercises

- Exercise 1

A school building has 4 floors and each has 12 classrooms with 30 desks inside. How many desks does the school have in total?

Solution

This problem is solved by applying the associative property of multiplication, let's see:

Total number of desks = 4 floors x 12 classrooms / floor x 30 desks / classroom = (4 x 12) x 30 desks = 48 x 30 = 1440 desks.

Or if you prefer: 4 x (12 x 30) = 4 x 360 = 1440 desks

- Exercise 2

Given the polynomials:

A (x) = 5x ³ + 2x ² -7x + 1

B (x) = x ⁴ + 6x ³ -5x

C (x) = -8x ² + 3x -7

Apply the associative property of addition to find A (x) + B (x) + C (x).

Solution

You can group the first two and add the third to the result:

A (x) + B (x) = + = x ⁴ + 11x ³ + 2x ² -12x +1

Immediately the polynomial C (x) is added:

+ = x ⁴ + 11x ³ - 6x ² -9x -6

The reader can verify that the result is identical if it is solved by the option A (x) +.

References

1. Jiménez, R. 2008. Algebra. Prentice Hall.
2. Math is Fun. Commutative, Associative and Distributive Laws. Recovered from: mathisfun.com.
3. Math Warehouse. Definition of Associative Property. Recovered from: mathwarehouse.com.
4. Sciencing. Associative & Commutative Property of Addition & Multiplication (With Examples). Recovered from: sciencing.com.
5. Wikipedia. Associative property. Recovered from: en.wikipedia.org.