The Property of Associative in Rational Numbers

The property of associative is one of the fundamental properties of rational numbers. It states that the grouping of numbers in an operation does not change the result. In other words, it does not matter how we group the numbers, the result will always be the same.

For example, let us consider the addition operation on rational numbers:

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

This equation shows that the sum of 'a' and 'b' is first calculated, and then it is added to 'c'. Alternatively, we can add 'b' and 'c' first, and then add 'a' to the result. Either way, the result will be the same.

Similarly, for multiplication, the associative property is:

(a × b) × c = a × (b × c)

This equation shows that the product of 'a' and 'b' is first calculated, and then it is multiplied by 'c'. Alternatively, we can multiply 'b' and 'c' first, and then multiply 'a' to the result. Again, the result will be the same.

The associative property holds true for all rational numbers and is an essential property in many mathematical operations.

Example:

Let us consider the following expression:

(2/3 + 1/4) + 5/6

Using the associative property, we can group the first two terms in two ways:

(2/3 + 1/4) + 5/6 = (8/12 + 3/12) + 5/6 = 11/12 + 5/6 = 41/12

Or,

(2/3 + 1/4) + 5/6 = 2/3 + (1/4 + 5/6) = 2/3 + 14/24 = 16/24 + 14/24 = 30/24 = 5/4

Both groupings give us the same result, which shows the associative property at work.

Conclusion

The associative property is an important property of rational numbers, which states that the grouping of numbers in an operation does not change the result. It is a fundamental property in many mathematical operations and holds true for all rational numbers.