Question

Prove that every integer is a constructible number.

Short answer

Every integer is a constructible number.

Step-by-step solution

Prove that unit length is constructible.

We will prove this by induction method.

Let any point as the origin $\mathbf{O}\mathbf{=}\left(0,0\right)$and line passing through $\mathit{O}$ as the X-axis.

Assume length of line as a unit length .

Hence, $\left(1,0\right)$is constructible.

Prove is constructible.

Now, suppose that $\left(K,0\right)$ is constructible.

Draw a circle with centre at $\left(K,0\right)$ and radius that intersects the positive X-axis at point $\left(K+1,0\right)$.

In this way, we constructed a length $\mathit{K}\mathbf{+}\mathbf{1}$ on the positive X-axis .

Hence, $\mathit{K}\mathbf{+}\mathbf{1}$is constructible.

Prove every integeris a constructible number.

Therefore, by the principle of induction, every integer is a constructible number.

Hence proved