(1)

\(\alpha\beta=0\)のとき、明らかに成立。
\(\alpha\beta\ne0\)のとき、
\begin{align*} \sgn\left(\alpha\beta\right) & =\frac{\alpha\beta}{\left|\alpha\beta\right|}\\ & =\frac{\alpha}{\left|\alpha\right|}\frac{\beta}{\left|\beta\right|}\\ & =\sgn\left(\alpha\right)\sgn\left(\beta\right) \end{align*} これより与式は成り立つ。

(2)

\begin{align*} \sgn\left(-\alpha\right) & =\sgn\left(-1\right)\sgn\left(\alpha\right)\\ & =-\sgn\left(\alpha\right) \end{align*}

(3)

\begin{align*} \sgn\left(\frac{1}{\alpha}\right) & =\frac{\left|\alpha\right|}{\alpha}\\ & =\frac{1}{\sgn\alpha} \end{align*}

(4)

\begin{align*} \sgn\frac{\alpha}{\beta} & =\sgn\left(\alpha\right)\sgn\left(\frac{1}{\beta}\right)\\ & =\frac{\sgn\alpha}{\sgn\beta} \end{align*}