The associative property is usually found in the area of algebra and can be applied to two types of operations namely addition and multiplication. This type of mathematical property certainly does not depend on the way in which the numbers are located or grouped. This means that regardless of how the different numbers in a given operation are put together, addition and multiplication will always give the same result regardless of the order. Grouping in this way has nothing to do with the results obtained from mathematical operations.
What is an Associative Property?
The associative property is a property that usually exists in the area of algebra and can be applied to the arithmetic operations of multiplication and addition. Where the associative property indicates that when there are three numbers or more in an operation, the result will not depend on the way in which the numbers are placed.
History of Associative Properties
In the 1830s, Treaty of Algebra was published in an attempt to explain the term as an equivalent logical treatment to Euclid’s elements. Here, there are two types of algebra explained, namely arithmetic algebra and symbolic algebra. In his book, there is an explanation of symbolic algebra as the science that deals with free combinations of signs and symbols in ways defined by arbitrary laws.
Which is true is that it is very difficult to give an exact date as to when it was created. Because, people already know that for example 2+3+3+2 since ancient times. But eventually people realized that this was a general property that could be related to operations other than multiplication and addition. Then it becomes something interesting to study more deeply. It can be said that no one person made the discovery.
The Associative Property of Addition
Associative addition or the addition property tells us that changing the order in which the numbers are added will not affect the result of the addition. This is because the application of the associative property in addition does not in itself have any visible or important effect. Some doubts arise about their usefulness and importance, but having knowledge of the principles can help us to perfectly master all these operations. Especially when combined with others, such as subtraction and division. What’s more, in the division to achieve the correct use of mathematics.
The following are several ways that are quite easy to understand, for those of you who want to understand the associative nature of addition, see the full explanation below:
(a + b) + c = a + (b + c) = d
To understand more, let’s look at the example questions below:
There is a problem (7 + 5) + 4 = ?
Of course, we will work on this problem by first adding up the numbers in brackets. Only later will the result be added to the number outside the brackets.
(7 + 5) + 4 =
12 + 4 = 16
If we use the associative property then we can also do it in the following way:
7 + (5 + 4) =
7 + 9 = 16
So, (7 + 5) + 4 = 7 + (5 + 4) = 16
The Associative Property of Multiplication
Multiplication is a mathematical arithmetic operation that has various properties. One of them is the property in the case of multiplication. This shows a way of grouping factors that will not cause any kind of change in the final product of the product. Regardless of the number of factors found in the operation.
Ana has 2 toy boxes. Where each box will be filled with 3 packs of marbles. Each pack contains 4 marbles. How many marbles does Ana have? There are two ways you can count Ana’s marbles.
The first way is to count the number of packs. Then, the result is multiplied by the number of marbles in each pack.
Number of packs × number of marbles per pack
(3 packs + 3 packs ) × 4 items = (3 + 3) × 4 = (2 × 3) × 4 = 24 items
The second way is to count the number of marbles per box first, the result is multiplied by the number of boxes.
Number of squares × number of marbles = 2 × ( 4 + 4 + 4) = 2 × (3 × 4) = 24 items.
Calculation method I : (2 × 3) × 4
Calculation method II : 2 × (3 × 4).
The results of calculations in both ways are the same.
So, (2 × 3) × 4 = 2 × (3 × 4)
This multiplication method uses the associative property of multiplication. In general, the associative property of multiplication can be written:
(a × b ) × c = a × (b × c)
With a, b, and c integers.
a. (2 × 4) × 3 = 2 × (4 × 3) = 24
b. (4 × 5) × 8 = 4 × (5 × 8) = 160
c. (4 × (-3)) × 6 = 4 × (-3 × 6) = – 72
s. (5 × (-2)) × 4 = 5 × (-2 × 4) = -40
e. (-3 × 2 ) × 8 = -3 × (2 × 8) = -48
f. (-4 × (-6)) × 10 = -4 × (-6 × 10) = 240
Here we will do the operation: 5 x 4 x 2
The first step that must be taken is to group the first two numbers, in this case it will be 5 and 4. By doing this one step, we will obtain the following equation:
(5 x 4) x 2
20 x 2
Now, if we group 4 and 2 together, we will get the following result:
5 x (4 x 2)
5 x 8
As can be clearly seen in the previous operation, even if in reality the numbers are positioned differently, the result will still be the same. Another example that you can understand is as follows:
(2 x 3) x 5 = 2 x (3 x 5)
6 x 5 = 2 x 15
30 = 30
Mathematics is a basic science that plays a very important role in the development of science and technology and advances human thinking. The presence of this book is expected to add to the reference and become a reference for students in particular and those who are interested in mathematics in general.
This Logic & Mathematics book can be used as a textbook or reference to support the learning of Discrete Mathematics courses. By studying this book, students are expected to be able to improve their ability to think logically, creatively, and critically. This capability will certainly be very useful for students/readers in supporting information system developers, multimedia/game developers, and relevant competencies.
When the scores and values of the evaluation results of students are in the range of 4-5, it becomes a warning for educators to review and realign the learning strategies, evaluation strategies used during INl, to then conduct research on what kinds of strategies are actually right for students. The role of renewal of this learning can only be raised to the surface when data on the results of learning evaluations are informed in a clear and transparent manner. We no longer have to be ashamed of the evaluation data as it is, because that is for the sake of future improvement for poor results. Reflection will be interesting for educators to do to achieve the goals of high student learning achievement.
The introductory book on Mathematical Analysis for tertiary institutions can be used by students from various departments, especially those majoring in mathematics. Mathematical analysis requires students to understand more about theorems in mathematics along with their proofs and be able to solve standard problems in mathematics analytically and formally. In thinking deductively and analyzing comprehensively, students must have an understanding in developing the concepts of the material they are studying.
Some of the material from this Introduction to Mathematical Analysis is an in-depth study of calculus course material which is studied rigorously and adapted to the existing syllabus, especially in the mathematics department in college. In compiling it, the author also refers to related books as references.