Half-Integer is Half Odd Integer
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Theorem
Let $r$ be a number.
Then $r$ is a half-integer if and only if $r = \dfrac n 2$ where $n$ is an odd integer.
Proof
Necessary Condition
Let $r$ be a half-integer.
Then by definition $r = n + \dfrac 1 2$ for some $n \in \Z$.
Thus:
\(\ds 2 r\) | \(=\) | \(\ds 2 \paren {n + \dfrac 1 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 n + 2 \paren {\frac 1 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 n + 1\) |
thus showing that $r$ is half of $2 n + 1$ for some $n \in \Z$.
By Odd Integer 2n + 1 it follows that $r$ is half of an odd integer.
$\Box$
Sufficient Condition
Let $k$ be an odd integer.
Then by Odd Integer 2n + 1:
- $k = 2 n + 1$
where $n \in \Z$.
Then:
\(\ds \frac k 2\) | \(=\) | \(\ds \frac {2 n + 1} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 n} 2 + \frac 1 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n + \frac 1 2\) |
thus showing that $\dfrac k 2$ is a half-integer.
$\blacksquare$