Odd Number minus Odd Number is Even

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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 $26$)


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

Then by definition of odd number:

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


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

Hence the result by definition of even number.


Historical Note

This proof is Proposition $26$ of Book $\text{IX}$ of Euclid's The Elements.