PROBLEM

THE PROBLEM OF DUPLICATING A CUBE CANNOT BE SOLVED USING ONLY STRAIGHTEDGE AND COMPASS

Kavi D. PandyaSemII - 131020

Duplicating a cube

Existing cube : side ‘A’ and volume ‘A3’

New cube : side ‘xA’ and volume ‘2(A)3’

Determine x :

(xA)3 = 2(A)3

(x)3(A)3 = 2(A)3

x3 = 2

Therefore : x = 3√2x = 3√2 = 1.2599210498948731647672106072782…..

Constructible numbers

DEFINITION: A real number ‘x’ is said to be constructible by straightedge and compass if a segment of length |x| can be obtained starting from our unit segment by using a finite sequence of straightedge and compass construction.

x = 3√2 = 1.2599210498948731647672106072782…..x is non-terminating number

Constructible Square roots

Proposition: Let ‘a’ be a constructible real number with ‘a’ > 0. Then, √a is constructible.

λ

λ = √a and is constructible

Constructible Number Theorem

Theorem: A number tєC is constructible if and only ifthere exists an irreducible polynomial pєQ and an integer j≥0 such that :

e.g. t2 – 4t = 0t – 4 = 0 (No. 4 is constructible)

Similarly : t3 – 2 =0(degree of t = 3)

t = 3 √2 is not-constructible ◊

Why only power of 2

The basic operations in the plane used in straightedge andcompass constructions are as follows:

(1) to draw a line through two given points(2) to draw a circle with centre at a given point and radius equal to the distance between two other given points(3) to mark the point of intersection of two straight lines(4) to mark the points of intersection of a straight line and a circle(5) to mark the points of intersection of two circles

Any straightedge and compass construction starts from given points, lines, and circles and involves a finite sequence of steps of these kinds to obtain some other points, lines, or circles.

Bibliography

1. Algebra Pure and Applied