Even Number minus Odd Number is Odd
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Theorem
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 $25$)
Proof
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:
- $\exists d \in \Z: b = 2 d + 1$
So:
\(\ds a - b\) | \(=\) | \(\ds 2 c - \left({2 d + 1}\right)\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \left({c - d}\right) - 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \left({c - d - 1}\right) + 1\) |
Hence the result by definition of odd number.
$\blacksquare$
Historical Note
This proof is Proposition $25$ 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