Commutative Property
The sum and product of two whole numbers will be the same whatever the order they are added or multiplied in, i.e., if x and y are two whole numbers, then x + y = y + x and x . y = y . x
Example: Consider two whole numbers, 2 and 7.
2 + 7 = 7 + 2 = 9
2 × 7 = 7 × 2 = 14
The commutativity property holds true for addition and multiplication.
Associative Property
When whole numbers are being added or multiplied as a set, they can be grouped in any order, and the result will be the same, i.e. if x, y, and z are whole numbers then x + (y + z) = (x + y) + z and x. (y . z)=(x . y).z
Example: Consider three whole numbers, 2, 3, and 4.
2 + (3 + 4) = 2 + 7 = 9
(2 + 3) + 4 = 5 + 4 = 9
Thus, 2 + (3 + 4) = (2 + 3) + 4
2 × (3 × 4) = 2 × 12 = 24
(2 × 3) × 4 = 6 × 4 = 24
Thus, 2 × (3 × 4) = (2 × 3) × 4
Therefore, the whole numbers are associative under addition and multiplication.
Distributive Property
If x, y, and z are three whole numbers, the distributive property of multiplication over addition is x. (y + z) = (x . y) + (x . z), similarly, the distributive property of multiplication over subtraction is x. (y – z) = (x . y) – (x . z)
Example: Consider three whole numbers, 9, 11, and 6.
9 × (11 + 6) = 9 × 17 = 153
(9 × 11) + (9 × 6) = 99 + 54 = 153
Thus, 9 × (11 + 6) = (9 × 11) + (9 × 6)
Hence, verified the distributive law of whole numbers.