Distributive Property
The distributive property of multiplication over addition (or subtraction) states that the product of a sum (or difference) is equal to the sum (or difference) of the products.
Examples
7 × 36 = 7 × (30 + 6) = 7 × 30 + 7 × 6 = 210 + 42 = 252 7 × 36 = 7 × (40 – 4) = 7 × 40 – 7 × 4 = 280 – 28 = 252Property
The distributive property can simplify calculations. It is especially helpful in simplifying mental math.
An operation denoted by ⊗ distributes over an operation denoted by ⊕ if, regardless of the numbers a, b and c, we have :
a ⊗ (b ⊕ c) = (a ⊗ b) ⊕ (a ⊗ c). This property is called the distributive property.
Specifically, the operation is left-distributive if : a ⊗ (b ⊕ c) = (a ⊗ b) ⊕ (a ⊗ c).
It is right-distributive if : (a ⊕ b) ⊗ c = (a ⊗ c) ⊕ (a ⊗ c).
Examples
- For the set of real numbers, multiplication distributes over addition : 12 × (3 + 10) = (12 × 3) + (12 × 10) = 36 + 120 = 156
- For the set of real numbers, multiplication distributes over subtraction : 25 × (20 − 5) = (25 × 20) − (25 × 5) = 500 − 125 = 375
- In set theory, the following equalities are obtained for the subsets A, B and C of U : A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) this means that the intersection of sets is left-distributive over the union of sets. It can also be shown that the union of sets distributes over the intersection of sets : A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)