All About Numbers
The Distributive Property
The distributive property connects multiplication and addition: a × (b + c) = a × b + a × c. Multiplying a sum by a number gives the same result as multiplying each part separately and then adding – the multiplication “distributes” across each term inside the brackets. It is the single most-used property in algebra, forming the basis of expanding brackets and factorising expressions.
Like “commutative,” the term “distributive” was introduced by the French mathematician François Servois in his influential 1814 paper on the foundations of calculus. Servois was searching for a precise, general way to describe which mathematical operations behaved “nicely” together, and his terminology for these core properties proved so useful and clear that it has remained essentially unchanged in mathematics classrooms for over two hundred years.
The Rule
a × (b + c) = (a × b) + (a × c)
This works the same way in reverse: (a × b) + (a × c) can be factored back into a × (b + c).
Why It Works: One Rectangle, Two Pieces
Picture a rectangle with height a = 6 and width (b + c) = (10 + 3). Cutting the width into its two parts — without changing the height — splits the rectangle into two smaller ones, whose areas must add up to the area of the whole.
The whole rectangle's area is 6 × 13 = 78 — and that's exactly what you get by adding the two pieces separately: (6 × 10) + (6 × 3) = 60 + 18 = 78. The distributive property isn't a separate rule to memorise; it's just the fact that cutting a rectangle into pieces doesn't change its total area, however you slice it.
6 × (10 + 3) = (6 × 10) + (6 × 3) = 60 + 18 = 78.
Both terms share the factor 4: 4 × 7 + 4 × 5 = 4 × (7 + 5) = 4 × 12 = 48.
Real-Life Application
- Mental maths: 6 × 23 is easier as 6 × (20 + 3) = 120 + 18 = 138.
- Splitting a bill: multiplying a group rate by (adults + children) separately.
- Algebra: expanding brackets like 3(x + 5) = 3x + 15 uses this property directly.
Key Takeaways
- The distributive property: a × (b + c) = a × b + a × c.
- It connects multiplication and addition, and works in reverse for factoring.
- François Servois coined the term in 1814, alongside “commutative.”