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.