Odd Number minus Even Number is Odd

From ProofWiki
Jump to navigation Jump to search


In the words of Euclid:

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

(The Elements: Book $\text{IX}$: Proposition $27$)


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

Then by definition of odd number:

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

and by definition of even number:

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


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

Hence the result by definition of odd number.


Historical Note

This proof is Proposition $27$ of Book $\text{IX}$ of Euclid's The Elements.