Difference between revisions of "Three Greek problems of antiquity"
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All three constructions have been shown to be impossible. | All three constructions have been shown to be impossible. | ||
==Trisection of the General Angle== | ==Trisection of the General Angle== | ||
| − | Statement: Given an angle, construct by means of [[ | + | Statement: Given an angle, construct by means of [[straight edge]] and [[compass]] only, an angle one-third its measure. This is impossible because the non-[[algebraic number]] <math>\cos 20^{\circ}</math> would have to be constructable in this case. |
==Doubling of a Cube== | ==Doubling of a Cube== | ||
| − | Given a cube, construct by means of [[ | + | Given a cube, construct by means of [[straight edge]] and [[compass]] only, a cube with double the volume. This is impossible, because the number <math>\sqrt[3]{2}</math> would have to be constructed, and it is a non-[[algebraic number]]. |
==Squaring of a Circle== | ==Squaring of a Circle== | ||
| − | Given a circle, construct by means of [[ | + | Given a circle, construct by means of [[straight edge]] and [[compass]] only, a square with area same as that of the circle. |
| + | |||
| + | This was shown to be impossible, as the [[transcendental number]] <math>\sqrt{\pi}</math> would have to be constructable in this case. | ||
==See Also== | ==See Also== | ||
| − | [[ | + | *[[Constructions]] |
| − | + | *[[Pi]] | |
| − | [[Pi]] | + | *[[Transcendental number]] |
| − | + | *[[Irrational number]] | |
| − | [[Transcendental number]] | + | *[[Field Theory]] |
| − | |||
| − | [[Irrational number]] | ||
| − | |||
| − | [[ | ||
| − | + | [[Category:Geometry]] | |
Revision as of 13:32, 26 January 2008
The Three Greek problems of antiquity were some of the most famous unsolved problems in history. They were first posed by the Greeks but were not settled till the advent of Abstract algebra and Analysis in modern times.
All three constructions have been shown to be impossible.
Trisection of the General Angle
Statement: Given an angle, construct by means of straight edge and compass only, an angle one-third its measure. This is impossible because the non-algebraic number 
would have to be constructable in this case.
Doubling of a Cube
Given a cube, construct by means of straight edge and compass only, a cube with double the volume. This is impossible, because the number ![$\sqrt[3]{2}$](http://latex.artofproblemsolving.com/9/f/d/9fdfe572d1673f00f286d5ed753942d18f401cbb.png)
would have to be constructed, and it is a non-algebraic number.
Squaring of a Circle
Given a circle, construct by means of straight edge and compass only, a square with area same as that of the circle.
This was shown to be impossible, as the transcendental number 
would have to be constructable in this case.
