Associative Property and Commutative Property
There are many mathematical properties that we use in statistics and probability. Out of these properties, the commutative and associative property is associated with the basic arithmetic of numbers. These properties are very similar, so can be easily mixed up. Thus it is very important to understand these properties for addition and multiplication.
The commutative property deals with the order of certain mathematical operations. For a binary operation, we can express it as a + b = b + a. On the other hand, the associative property deals with the grouping of numbers in an operation. For example, we can express it as, (a + b) + c = a + (b + c).
Commutative Property
This property states that the factors in an equation can be rearranged freely without affecting the result of the equation. However, the commutative property links itself about the ordering of operations, including the addition and multiplication of real numbers. It is also applicable to integers and rational numbers.
This equation defines the commutative property of addition:
a + b = b + a
This equation defines the commutative property of multiplication:
a * b = b * a
Sometimes these rearrangements make the process to add or multiply easier:
What is 12 × 16 × 15?
12 × 16 × 15 = (12 × 15) × 16 = 180 × 16 = 2880
Also, the numbers 12, 23, and 16 can be added together in any order without affecting the final result:
12 + 23 + 16 = 51
23 + 12 + 16 = 51
Likewise, we can multiply the numbers in any order without affecting the final result:
12 x 3 x 50 = 1800
3 x 12 x 50 = 1800
But subtraction and division are not applicable here because the order of operations is important.
Associative Property
This property states that the grouping of numbers in an operation can be changed without changing the result of the equation. We can show this with the equation a + (b + c) = (a + b) + c. It is obvious that the first pair of values in the equation will not change the result.
This equation shows the associative property of addition:
(l+v)+r=l+(v+r)
This equation shows the associative property of multiplication:
(l x v) x r = l x (v x r)
In some cases, we can simplify a calculation by multiplying or adding in a different order.
What is 19 + 36 + 14?
19 + 36 + 14 = 19 + (36 + 14) = 19 + 50 = 69
Also, take the equation 12 + 13 +25.
(12 + 13) + 25 = (25) + 25 = 50
12 + (13 + 25) = 12 + (38) = 50
Operations which are associative include the addition and multiplication of real numbers. Also, the associative property can also be applicable to matrix multiplication and function composition.
In addition, similar to a commutative property, the associative property cannot be applicable to subtraction as division operations.
Difference between Commutative and Associative
Here main difference lies with the answer to the question, “Are you changing the order of the elements, or are you changing the grouping of the elements?” If you do the reordering of the elements, then the commutative property applies. On the other hand, if the elements only regroup, then the associative property applies.
However, the presence of braces does not necessarily mean that the associative property applies. For instance:
(562 + 263) + 14 = 14 + (562 + 263)
This equation is having an example of the commutative property of the addition of real numbers. If we pay proper attention to this equation, then we can see that only the order of the elements has been changed, not the grouping. Also, to apply the associative property, we would have to rearrange the grouping of the elements as well:
(562 + 263) + 14 = (14 + 562) + 263
Solved Questions for You
Q-1: Find the missing number in this equation:
15 + (14 + 12) = 41, so (15 + 14) +12 = __________
Ans: By applying the associative property answer is 41.
Q-2: Find the missing number in this equation:
30 * ( _____ *1 5) = (30 * 7) * 15
Ans: By applying associative property answer is 7.
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