Even Number minus Odd Number is Odd

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In the words of Euclid:

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

(The Elements: Book $\text{IX}$: Proposition $25$)


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

Then by definition of even number:

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

and by definition of odd number:

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


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

Hence the result by definition of odd number.


Historical Note

This proof is Proposition $25$ of Book $\text{IX}$ of Euclid's The Elements.