Associative
In mathematics, the term “associative” refers to a property that certain operations possess
In mathematics, the term “associative” refers to a property that certain operations possess. An operation is said to be associative if the grouping of numbers or variables within the operation does not affect the final result.
Formally, let’s consider an operation denoted by “*”, and three elements a, b, and c. The operation “*” is associative if (a * b) * c = a * (b * c) for any values of a, b, and c.
To better understand the concept, let’s explore some examples:
1. Addition:
Consider the operation of addition, “+”. For any three numbers a, b, and c, the operation is associative. This can be demonstrated as follows:
(a + b) + c = a + (b + c)
For instance, let’s take a = 2, b = 4, and c = 6:
(2 + 4) + 6 = 6 + 6 = 12
2 + (4 + 6) = 2 + 10 = 12
As you can see, regardless of how we group the numbers, the final result remains the same. This property makes addition associative.
2. Multiplication:
Multiplication, “*”, is another operation that is associative. Let’s demonstrate this using the same values:
(a * b) * c = a * (b * c)
Using a = 2, b = 4, and c = 6:
(2 * 4) * 6 = 8 * 6 = 48
2 * (4 * 6) = 2 * 24 = 48
The final result is the same, confirming the associative property for multiplication.
It’s worth noting that not all operations are associative. A well-known example is subtraction.
Consider the operation of subtraction, “-“:
(a – b) – c ≠ a – (b – c)
Using a = 6, b = 4, and c = 2:
(6 – 4) – 2 = 2 – 2 = 0
6 – (4 – 2) = 6 – 2 = 4
As you can see, the results are different when we group the numbers differently. Therefore, subtraction does not possess the associative property.
In summary, the associative property is a fundamental property of certain operations, such as addition and multiplication, where the grouping of the numbers or variables within the operation does not impact the final result.
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