# An interesting way to explain commutative property of multiplication

Five times three is equal to three times five due to the commutative property of multiplication. Is there any interesting way to explain why this is so?

• five baskets of three oranges is the same amount of oranges as three baskets of five oranges? Sep 22 at 19:06

$${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$$

$$5+5+5=15=5\times3$$ and $$3+3+3+3+3=15=3\times5$$

Basically, if $$15$$ units were the length of a rod/sausage/stick then if you divide it into $$5$$ pieces of equal length (each of those pieces will be $$3$$ units long) via $$4$$ equi-spaced cuts OR into $$3$$ pieces of equal length (each piece being $$5$$ units long) via $$2$$ equi-spaced cuts, it doesn't matter. They'll all add up to give back the original length of $$15$$ units (obviously).

Bear with me for a second:

Multiplication commutes in $$\mathbb Z$$ because it commutes inside $$\mathbb R$$, and $$\mathbb R$$ has commutative multiplication because $$\mathbb R\times\mathbb R$$ is a Pappian plane.

In fact, the commutativity of multiplication in a division ring $$D$$ is characterized by Pappus's theorem holding in $$D\times D$$.

To me, this is an interesting reason for the commutativity of $$\mathbb Z$$ because it's rooted in the actual geometry of the real plane. Sorry if it is too obscure.

There are elementary pictorial or computational demonstrations (indeed, that is how I think the other answers are turning out so far) but I feel like they are not satisfying enough to be interesting, however useful they may be.