Question

Let $a,c$ be constructible numbers with $c\ne 0$. Prove that $a/c$is constructible.[Hint: The case when$a>0,c>0$ was done in the proof of Theorem$15.1$ .]

Short answer

The value of$\text{a/c}$ is constructible.

Step-by-step solution

Conceptual Introduction

Drawing precise lines, line segments, shapes, circles, and other figures with a ruler, compass, or protractor is known as geometric construction.

Step 2:Properties of constructible.

$\left(1\right)$We know that if$\text{a,c}$is constructible number with $a>0,c>0$, then$\text{a/c}$is constructible.

$\left(2\right)$ A number $\text{r}$is constructible if $\text{-r}$ is constructible.

If both have same signs.

There are two possible cases:

Case 1^st: if $\text{a,c}$both have same signs.

In this case $\text{a/c}$ is positive and hence it is constructible by the result of $\left(1\right)$.

If both have opposite signs.

Case 2^nd: If $\text{a,c}$ have opposite signs.

In this case$\text{a/c}$is negative and hence$\text{-a/c}$ is constructible. So, by the result of $\left(1\right)$$\text{-a/c}$is constructible.

Hence,$\text{a/c}$is constructible.