Duplicating cube
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The document discusses the impossibility of duplicating a cube using only straightedge and compass constructions, demonstrating that the required side length x equals 3√2, which is a non-constructible number. It defines constructible numbers and outlines the criteria for a number to be considered constructible, including relevant theorems and examples. Additionally, it explains basic operations allowed in such constructions and concludes with references.
Duplicating cube
- 1. PROBLEM THE PROBLEM OF DUPLICATING A CUBE CANNOT BE SOLVED USING ONLY STRAIGHTEDGE AND COMPASS Kavi D. Pandya SemII - 131020
- 2. 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√2 x = 3√2 = 1.2599210498948731647672106072782…..
- 3. 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
- 4. Constructible Square roots Proposition: Let ‘a’ be a constructible real number with ‘a’ > 0. Then, √a is constructible. λ λ = √a and is constructible
- 5. Constructible Number Theorem Theorem: A number tєC is constructible if and only if there exists an irreducible polynomial pєQ and an integer j≥0 such that : e.g. t2 – 4t = 0 t – 4 = 0 (No. 4 is constructible) Similarly : t3 – 2 =0 (degree of t = 3) t = 3 √2 is not-constructible ◊
- 6. Why only power of 2 The basic operations in the plane used in straightedge and compass 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.
- 7. Bibliography 1. Algebra Pure and Applied