About Commutative Properties
Introduction
The commutative property is applicable to addition and multiplication arithmetic. This implies that the outcome of adding or multiplying two numbers remains unchanged if the order or position of the numbers are changed. For instance, 3 + 2 equals 5, and 2 + 3 equals 5. The sum is unaffected by the sequence in which two numbers are added. The idea behind multiplication is the same. For subtraction and division, the commutative property does not apply because the outcomes are entirely different when the order of the numbers change.
Rapid Recall
Solved Examples
Example 1: Verify the equation a + b = b + a if a = 12 and b = 8.
Solution:
Provided that, a = 12 and b = 8
LHS = a + b = 12 + 8 = 20 ……(1)
RHS = b + a = 8 + 12 = 20 ……(2)
From equation (1) and (2), we get;
LHS = RHS
Hence, proved.
Example 2: Prove that A.B = B.A, if A = 5 and B = 6.
Solution:
We know that A = 5 and B = 6.
A.B = 5.6 = 30 ……(1)
B.A = 6.5 = 30 ……(2)
From equation (1) and (2), we get;
LHS = RHS
A.B = B.A
Hence, proved.
Example 3: 24 chairs are arranged such that there are three chairs in each row and eight in each column. What are the other ways the same number of chairs can be arranged differently?
Solution:
We can determine the different number of chairs that can be used by finding the factors of 24. They are listed as follows.
1 \(\times\) 24 = 24
2 \(\times\) 12 = 24
3 \(\times\) 8 = 24
4 \(\times\) 6 = 24
We can consider one factor as the row and the other factor as the column of the number of chairs.
We can find four more ways in which the chairs can be arranged using the commutative property of multiplication.
24 \(\times\) 1 = 24
12 \(\times\) 2 = 24
8 \(\times\) 3 = 24
6 \(\times\) 4 = 24
Hence, these are the eight combinations as to how the chairs can be arranged.
Yes, the commutative number applies to all types of numbers when they are added or multiplied.
When adding or multiplying numbers, the orders of the numbers can be switched without changing the value of the result. This is the commutative rule. The associative property is with reference to the grouping of numbers. If a, b and c are three numbers, according to the associative law, a + (b + c) is the same as (a + b) + c. Likewise, the associative rule of multiplication says a × (b × c) is the same as (a × b) × c. So, the laws are not the same.
Yes, the commutative property of multiplication is used to solve real-life problems including money, measurements, and so on. The commutative property, along with other properties, make calculations easier.