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Found in: Page 469

### 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 -r is constructible.

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

See the step by step solution

## Step 1: 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.

## Step 2: Prove   is constructible

Now, assume that $\mathbf{-}\mathbit{r}$ 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" $\mathbit{r}$ is a constructible.