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Q12E

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Abstract Algebra: An Introduction
Found in: Page 470
Abstract Algebra: An Introduction

Abstract Algebra: An Introduction

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 a line segment of length |r| can be constructed by straightedge and compass, beginning with a segment oflength 1.

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

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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 (r,0) is a constructible point, the real number r is said to be constructible.

Step 3: If part

Assume a coordinate plane with the point P(1,0) and the origin at O(0,0)

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

The point R(r,0) is hence constructible (as in figure 1):

OR is an r-length segment that runs through two constructible points, P(1,0) and R(r,0)

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 (0,0).

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 (r,0)

As a result, (r,0) is a constructible point and so a constructible number.

Hence, proved.

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