On 2003-01-21 03:09, Marcos wrote:
Quote:
On 2003-01-20 18:16, Muad'Dib wrote:
Hooray for sco08y! He is correct.
All of these "proofs" have the common error of sneaking in a division or multiplication by zero, thus making everything after them meaningless.
Not all of them. Geometric proof that 1 = 2:
Take a circle and a semicircle of the same radius. Map each point on the semicircle to two diametrically opposite points on the circle. This sets up a 1:2 correspondence between points on the semicircle and points on the circle.
Draw a line tangent to one endpoint of the semicircle and offset it so it no longer touches the semicircle. For each point on the semicircle, construct a tangent line that intersects the offset line. Each line will intersect the offset line at exactly one point, giving a 1:1 correspondence between points on the semicircle and points on the offset line.
Draw a line tangent to the circle and parallel to the offset line from the semicircle. For each point on the circle, draw a line tangent to the circle that intersects the original tangent line. Each line will intersect the tangent line at exactly one point, setting up a 1:1 correspondence between points on the circle and points on the line.
For each point on the offset line, construct a line perpendicular to that line intersecting the circle tangent line. This sets up a 1:1 correspondence between points on the offset line and points on the circle tangent line.
These three 1:1 correspondences can be used to construct a 1:1 correspondence between points on the circle and points on the semicircle. Since there is also a 1:2 correspondence between points on the semicircle and points on the circle, 2 must equal 1.
Q.E.D.
As a circle can be considered as a polygon of infinitely many sides, taking "every point on a semicircle" would likewise result in an infinite number of points. Since infinity does not follow the rules for real numbers, 2x infinity = infinity. The argument is invalid when applied to a regular polygon with any finite number of sides.
<font size=-1>[ This Message was edited by: Pyromancer on 2003-01-21 04:08 ]</font>