Odd Number minus Odd Number is Even

Theorem

In the words of Euclid:

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

(The Elements: Book $\text{IX}$: Proposition $26$)

Proof

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

Then by definition of odd number:

$\exists c \in \Z: a = 2 c + 1$

$\exists d \in \Z: b = 2 d + 1$

So:

\(\ds a - b\)

\(=\)

\(\ds 2 c + 1 - \left({2 d + 1}\right)\)

\(\ds \)

\(=\)

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

Hence the result by definition of even number.

$\blacksquare$

Historical Note

This proof is Proposition $26$ 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