# Natural Number Multiplication is Commutative/Euclid's Proof

## Theorem

In the words of Euclid:

*If two numbers by multiplying one another make certain numbers, the numbers so produced will be equal to one another.*

(*The Elements*: Book $\text{VII}$: Proposition $16$)

## Proof

Let $A, B$ be two (natural) numbers, and let $A$ by multiplying $B$ make $C$, and $B$ by multiplying $A$ make $D$.

We need to show that $C = D$.

We have that $A \times B = C$.

So $B$ measures $C$ according to the units of $A$.

But the unit $E$ also measures $A$ according to the units in it.

So $E$ measures $A$ the same number of times that $B$ measures $C$.

Therefore from Proposition $15$ of Book $\text{VII} $: Alternate Ratios of Multiplesâ€Ž $E$ measures $B$ the same number of times that $A$ measures $C$.

We also have that $A$ measures $D$ according to the units of $B$ since $B \times A = D$.

But the unit $E$ also measures $B$ according to the units in it.

Therefore $E$ measures $B$ the same number of times that $A$ measures $D$.

But we also have that $E$ measures $B$ the same number of times that $A$ measures $C$.

So $A$ measures $C$ and $D$ the same number of times.

Therefore $C = D$.

$\blacksquare$

## Historical Note

This proof is Proposition $16$ of Book $\text{VII}$ of Euclid's *The Elements*.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 2*(2nd ed.) ... (previous) ... (next): Book $\text{VII}$. Propositions