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.
Problems example:
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.
Problems example:
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.
Example:
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
Problems example:
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
40
Now, if we group 4 and 2 together, we will get the following result:
5 x (4 x 2)
5 x 8
40
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
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