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 Multiples‎ $E$ 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