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:

\(\displaystyle 2r\) \(=\) \(\displaystyle 2 \left({n + \dfrac 1 2}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle 2 n + 2 \left({\frac 1 2}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle 2 n + 1\)

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:

\(\displaystyle \frac k 2\) \(=\) \(\displaystyle \frac {2 n + 1} 2\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {2 n} 2 + \frac 1 2\)
\(\displaystyle \) \(=\) \(\displaystyle n + \frac 1 2\)

thus showing that $\dfrac k 2$ is a half-integer.

$\blacksquare$