Question

Prove that r is a constructible number if and only if -r is constructible.

Short answer

r is a constructible number if and only if -r is constructible.

Step-by-step solution

Prove   is constructible.

Let r is constructible number, then $\left(r,0\right)$ is constructible point.

Draw a circle with center at origin and passing through role="math" localid="1657965291636" $\left(r,0\right)$ that intersects X-axis in $\left(-r,0\right)$on the negative side.

Hence, is constructible.

Prove   is constructible

Now, assume that $\mathbf{-}\mathit{r}$ is constructible, then $\left(-r,0\right)$ is constructible.

Draw a circle with center at origin and passing through $\left(-r,0\right)$ which intersects Y-axis in $\left(r,0\right)$ on the positive side.

Therefore, role="math" localid="1657965236337" $\mathit{r}$is a constructible.