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squaring the circle


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One of the problems that the Greek geometers attempted (like the duplication of the cube and the trisection of an angle) was to find a construction, with ruler and pair of compasses, to obtain a square whose area was equal to that of a given circle. This is equivalent to a geometrical construction to obtain a length of √π from a given unit length. Now constructions of the kind envisaged can give only lengths that are algebraic numbers (and not even all algebraic numbers at that: for instance, ∛2 cannot be obtained). So the proof by Lindemann in 1882 that π is transcendental established the impossibility of squaring the circle.

Subjects: Mathematics.


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