The Properties of Addition and Multiplication

The idea in one sentence

Addition and multiplication don’t care about order or grouping, and multiplication distributes over addition — which is permission to rebuild any calculation into an easier one.

Commutative: order doesn’t matter

Three rows of eight is the same array as eight rows of three, seen from the side. This is why the times tables are half the size they look, and why 2 + 98 can be read as 98 + 2 and finished instantly.

Subtraction and division are not commutative. 7 − 3 is not 3 − 7. Half the point of naming the property is knowing which operations lack it.

Associative: grouping doesn’t matter

So 25 × 17 × 4 can be regrouped as 17 × (25 × 4) = 17 × 100 = 1700. The hard-looking multiplication was never necessary; the 25 and the 4 were waiting to be put together.

Together, commutativity and associativity say that in a chain of all-additions or all-multiplications, you may do them in any order you like. That is the licence behind almost every mental arithmetic trick.

Distributive: the important one

Six lots of 23 is six lots of 20 plus six lots of 3, giving 120 + 18 = 138. Split the awkward number into friendly pieces, multiply each, add up.

Draw it as a rectangle 6 tall and 23 wide, cut into a 6 × 20 block and a 6 × 3 block. The area doesn’t change when you draw a line through it, and that picture is the whole proof.

This property is why the column method for multiplication works, why area models work, and later why \((x + 2)(x + 5)\) expands the way it does. It is the same fact every time.

The identities and the zero

Adding 0 and multiplying by 1 change nothing, which sounds too obvious to name until you use it: rewriting 3/4 as 6/8 is multiplying by 1 in disguise, and “borrowing” in subtraction relies on rewriting a number as an equal one.

Multiplying by 0 gives 0, always. That one is a property of multiplication, not a special case to memorise: zero groups of anything is nothing.

Using them on purpose

• 48 + 27 + 52 → put the 48 and 52 together for 100, then add 27.
• 7 × 99 → 7 × 100 − 7 = 693.
• 15 × 12 → 15 × 10 + 15 × 2 = 180.
• 36 ÷ 4 hard? 36 = 40 − 4, so the answer is 10 − 1 = 9.

None of these are separate tricks. They’re all the same three properties.

Common mix-ups

• Assuming subtraction and division play along. They don’t: (12 − 5) − 3 ≠ 12 − (5 − 3).
• Distributing only over the first term, so 6 × 23 becomes 120 + 3.
• Distributing multiplication over multiplication. 6 × (4 × 5) is not 24 × 30.
• Thinking these are named rules to recite. They’re permissions to rearrange.

Check yourself

Work out 6 × 23 by splitting the 23, then find 25 × 17 × 4 by regrouping.

6 × 20 + 6 × 3 = 120 + 18 = 138. And 25 × 4 = 100 first, so 17 × 100 = 1700.