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Found in: Page 470

### Abstract Algebra: An Introduction

Book edition 3rd
Author(s) Thomas W Hungerford, David Leep
Pages 608 pages
ISBN 9781111569624

# Prove that r is a constructible number if and only if a line segment of length $|r|$ can be constructed by straightedge and compass, beginning with a segment oflength ${\text{1}}$.

A number $\text{r}$ is constructible if and only if the line segment of length $|\text{r}|$ can be constructed by straightedge and compass.

See the step by step solution

## Step 1: 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: Definition of constructible point

Constructible points are any points in the plane that can be created using straightedge and compass constructions. If the point $\left(\text{r,0}\right)$ is a constructible point, the real number r is said to be constructible.

## Step 3: If part

Assume a coordinate plane with the point $\text{P}\left(\text{1,0}\right)$ and the origin at $O\left(\text{0,0}\right)$

Assume that r is a constructible value with r>0.

The point $R\left(\text{r,0}\right)$ is hence constructible (as in figure 1):

OR is an r-length segment that runs through two constructible points, $\text{P}\left(\text{1,0}\right)$ and $R\left(\text{r,0}\right)$

As a result, the line segment OR can be constructed.

## Step 4: Only if part

Assume, on the other hand, that the line segment with length |r| is constructible. Let the line segment is denoted by AB, with length $|r|$, and coordinate of point A as $\left(0,0\right)$.

Draw a circle with a radius equal to the length of the segment AB with the compass point at point B.

It's important to note that the intersection of a constructible circle and a constructible horizontal axis at point B is constructible.

Furthermore, because the length of the segment AB is r, the coordinate of the point B is $\left(\text{r,0}\right)$

As a result, $\left(\text{r,0}\right)$ is a constructible point and so a constructible number.

Hence, proved.