Commutativity of Addition

When you add two numbers, the order doesn't matter. Whether you do a + b or b + a, you land on the very same total:

a + b = b + a

This is the commutative property of addition. It means you can always swap the two numbers around and the answer stays put. A long word for a simple, friendly idea: you can add in any order.

Picture 3 + 5 as three cats meeting five birds — and then the very same animals lined up the other way round, five birds meeting three cats:

cat cat cat + bird bird bird bird bird = 8

bird bird bird bird bird + cat cat cat = 8

Nobody arrived and nobody left — the animals just swapped sides. So the total is exactly the same: 3 + 5 = 5 + 3 = 8.

Watch both orders at once. The top number line starts at the first number and hops on by the second; the bottom one starts at the second number and hops on by the first. Each time you replay it picks new numbers — but both markers always finish on the same total.

Whichever way you line them up, you are counting the very same creatures — so the total can't change. Three monkeys joining four ducks is the same crowd as four ducks joining three monkeys: seven animals either way.

monkey monkey monkey + duck duck duck duck = duck duck duck duck + monkey monkey monkey = 7

See it: the two groups swap places

Here is the same idea with counters. The top row shows one group joined by a second group; the bottom row shows those same two groups, just with their places swapped. Count each row — the total never changes. Press Refresh for two brand-new numbers to swap.

Why it's useful: put the bigger number first

Because the order is free, you can choose the order that is easiest. The quick way to add is to count on from one number — and that is far less hopping if you start at the bigger number and count on the smaller one.

To work out 2 + 9, swap it to 9 + 2: begin at 9 and count on just two — "10, 11" — instead of starting at 2 and counting on nine. Same answer, far less work.

Imagine counting on with ducks. For 2 + 9 you would start with two ducks and add nine more, one hop at a time — nine tiring hops. Swap it round to 9 + 2 and you only hop twice. The pond ends up with the same eleven ducks either way, so always pick the order with the fewest hops.

duck duck duck duck duck duck duck duck duck + duck duck = 9 + 2 = 11 ducks

Three sums to try

Every addition can be flipped, and the total holds still:

4 + 7 = 11 and, flipped, 7 + 4 = 11 — same total.
2 + 9 = 11: swap to 9 + 2 and count on just two.
6 + 1 = 7 is the same as 1 + 6 = 7 — start at 6 and a single hop gets you there.

Swapping is allowed for some operations but not others:

It works for addition (a + b = b + a) and for multiplication (a \times b = b \times a).
It does not work for subtraction or division. Order really matters there: 5 - 3 = 2 but 3 - 5 is not the same — you can't take five away from three! So 5 - 3 \neq 3 - 5.

Adding just gathers things together, so the order of gathering never matters. But subtraction removes — and which number you start with is the whole story. If you have 5 cookies and eat 3, you have 2 left. Try it the other way and you'd need 5 cookies to eat from a plate of only 3 — impossible! That's why 5 - 3 and 3 - 5 are not the same.

cookie cookie cookie cookie cookie − 3 eaten = 2 cookies left.

Khan Academy explains the commutative law of addition here: