Fallacies

1
Fallacies

• By Stephanie Sundberg
• Based on the article Fallacies in Readings for
  Calculus

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What is a Fallacy?

• Fallacies are arguments that look correct but are
  actually absurd. They can be used to illustrate
  some important mathematical points.
• Some are easy to spot, and others are more
  difficult. The following fallacy is one of the
  most common

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The Division By Zero Fallacy

• Let x1
• Multiply both sides of the equation by x
  
• x2 x
• Subtract x x2 x 0
• Factor x (x 1 ) 0
• Divide by x - 1 x 0

• The conclusion is that 1 0.
• Where is the error?
• The error occurs with division by x-1.
• As we defined x1 in step 1, we cannot divide by
  1-10.
• Whenever you see the words Now divide by, make
  sure you check for this possible fallacy!

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Can You Spot the Fallacy?

• So ab
• Since ab12, it follows that ab6
• Where is the error?
• The square root of x2 is not x, but x. So,
  (a-6)2(b-6)2 should be
  a-6b-6
• This is known as the square root fallacy.
  

• Let a b 12
• (ab)(ab) 12 (a-b)
• a2-b212a-12b
• a2-12ab2-12b
• a2-12a 36 b2-12b 36
• (a-6)2(b-6)2
• Thus a-6b-6

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A note on Howlers

• Fallacies (which illustrate important
  mathematical points) are not the same as Howlers,
  which are simply illegal (and funny) operations.
• Here is a simple Howler Simplify 16/64 by
  canceling the 6s (surprisingly, you get the
  right answer!).
• Or, how about this solve x4320. Answer 80 (and
  here, the wrong answer.)

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How to Show 12 Using Derivatives

• Calculus fallacies are not as common as algebraic
  ones, but here is a good one
• 32333, 424444
• X2xxx, where the sum has x terms.
  Differentiate both sides
• 2x111

• Since the right-hand side is the sum of x 1s, its
  value is x, so
• 2xx
• So 21
• Where is the error?
• X2 is a sum of xs only when x is an integer.
  Being discrete is different from being
  continuous.

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01

• See if you can find the fallacy
• Use integration by parts to find I ? dx
• u vx
• u - v1
• I( )x- ? x dx
• u v- ? u v

• I1 ? dx
• I I1
• Now, subtract I from both sides and we have 10
• Where is the error?
• We always add a c to the end for a reason!

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References

• Fallacy Detector image
• http//www.lexrex.com/images/fallacy20detector.gi
  f
• Division by zero image
• http//www.mapletreelearning.com/images/sfdivideby
  zero3.gif
• Dudley, Underwood (1993). Readings For Calculus,
  Vol 5. The Mathematical Association of America.
• Head scratching gif
• http//www.rsc-ni.ac.uk/pictures/Technical/netscra
  tch.gif
• Laughing video
• http//www.killsometime.com/Video/video.asp?ID353
  
• What is a fallacy image http//www.designedlearni
  ng.com/images/TheOversightFallacy.jpg