Mathematicians' Deceptive Proofs Revealed
Delve into the world of deceptive mathematical proofs with examples showcasing common fallacies. Explore incorrect proofs like 1=0, 1+2+4+8+16=1, flawed inductive reasoning, and even a scenario where a pencil can destroy the universe through infinite speed and force. Unravel the errors in these proofs and sharpen your critical thinking skills.
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Presentation Transcript
Math Club 5/9/2011 THEY'RE ALL COMING TO GET YOU (how mathematicians fool you with their deceptive proofs)
THE MATH GAME I will present a completely incorrect proof You have to pinpoint the error in the proof
EXAMPLE I shall prove that 2 = 1. Let ? = ?. ?2= ?? 2?2= ?2+ ?? 2?2 2?? = ?2 ?? 2 ?2 ?? = 1 ?2 ?? 2 = 1 The fallacy is the last step. Since ? = ?, we are dividing both sides by zero!
ROUND 1 I shall prove that 1 = 0. Take the statement ? = 1. Take the derivative of both sides: ? ? ??1 ??? = Then 1 = 0. What went wrong?
ROUND 2 I shall prove that 1 + 2 + 4 + 8 + 16 + = 1. Let ? = 1 + 2 + 4 + 8 + . Then 2? = 2 + 4 + 8 + 16 + . 2? ? = 1 1 + 2 + 4 + 8 + = 1 What went wrong?
ROUND 3 I shall prove by induction that in any group of ? people, either they are all boys, or they are all girls. This is obviously true for ? = 1. Let ? be any group of ? + 1 people. We prove that any two people ? and ? in ? are of the same sex. Consider everyone except ?. All of them are the same sex. Also consider everyone except ?. All of them are the same sex Take any member ?, who has the same sex as ? and ?. Therefore any ? and ? in the group have the same sex. The theorem is proved. What went wrong?
ROUND 4 I shall prove that 0 = 1. 1 ??? by parts. Begin by evaluating Let ? =1 ? and d? = ??. 1 ?2?? and ? = ?. 1 ??? =? Then ?? = ? ?2?? = 1 + 1 ???. Hence ? Therefore 0 = 1. What went wrong?
ROUND 6 I can destroy the universe with a pencil! Assume that there is no friction, pencil is uniform, etc, etc. Put the pencil up to a wall and pull one end away with constant velocity while the other end slides down the wall. As your end of the pencil is distance ? from the wall, the other end is distance ? = ?2 ?2 where ? is the length of the pencil. ??? ?? ?2 ?2 Differentiating with respect to time, ?? ??= But ?? ??= ? so ?? ?? ?2 ?2 ??= So ?? infinite speed and infinite force! ?? approaches as ? ?. The pencil crashes on the ground at
