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Q1E

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Abstract Algebra: An Introduction

Found in: Page 469

Book edition 3rd

Author(s) Thomas W Hungerford, David Leep

Pages 608 pages

ISBN 9781111569624

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Short Answer

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.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Prove is constructible.

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

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

Hence, is constructible.

Step 2: Prove is constructible

Now, assume that $- r$ is constructible, then $(- r, 0)$ is constructible.

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

Therefore, role="math" localid="1657965236337" $r$ is a constructible.

Most popular questions for Math Textbooks

Is it possible to construct with straightedge and compass an isosceles triangle of Perimeter $8$ and Area $1$ ?

Squaring the Circle Given a circle of radius r, show that it is impossible to construct by straightedge and compass the side of a square whose area is the same as that of the given circle. You may assume the nontrivial fact that $⊓$ is not the root of any polynomial in Q[x] .

Let $A$ be a constructible point not on the constructible line $L$ . Prove that the line through $A$ parallel to $L$ is constructible. [Hint: Use Exercise $17$ to find a constructible line M through $A$ , perpendicular to $L$ . Then construct a line through $A$ perpendicular to $M$ .]

Consider a rectangular box with a square bottom of edge and height . Assume the volume of the box is 3 cubic units and its surface area is 7 square units. Can the edges of this box be constructed with straightedge and compass?

Let $C$ be a constructible point and $L$ a constructible line. Prove that the linethrough $C$ perpendicular to $L$ is constructible. [Hint: The case when $C$ is on was done in Example 1. If is not on $L$ and $D$ is a constructible point on $L$ , the circle with center $C$ and radius $CD$ is constructible and meets $L$ at the

constructible points $D$ and $E$ . The circles with center $D$ , radius $CD$ and center

$E$ , radius $CE$ intersect at constructive points $C$ and $Q$ . Show that line $CQ$ is

perpendicular to $L$ .]

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