Even Number minus Even Number is Even

Theorem

In the words of Euclid:

If from an even number an even number be subtracted, the remainder will be even.

(The Elements: Book $\text{IX}$: Proposition $24$)

Proof

Let $a$ and $b$ be even numbers.

Then by definition of even number:

$\exists c \in \Z: a = 2 c$

$\exists d \in \Z: b = 2 d$

So:

\(\ds a - b\)

\(=\)

\(\ds 2 c - 2 d\)

\(\ds \)

\(=\)

\(\ds 2 \left({c - d}\right)\)

Hence the result by definition of even number.

$\blacksquare$

Historical Note

This proof is Proposition $24$ of Book $\text{IX}$ 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 $\text{IX}$. Propositions