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Fallacies

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Fallacies

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    1. Fallacies By Stephanie Sundberg Based on the article “Fallacies” in Readings for Calculus

    2. 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:

    3. The Division By Zero Fallacy Let x=1 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 x=1 in step 1, we cannot divide by 1-1=0. Whenever you see the words “Now divide by”, make sure you check for this possible fallacy!

    4. Can You Spot the Fallacy? Let a + b =12 (a–b)(a+b) = 12 (a-b) a2-b2=12a-12b a2-12a=b2-12b a2-12a +36 =b2-12b +36 (a-6)2=(b-6)2 Thus: a-6=b-6 So: a=b Since a+b=12, it follows that a=b=6 Where is the error? The square root of x2 is not x, but |x|. So, (a-6)2=(b-6)2 should be |a-6|=|b-6| This is known as the square root fallacy.

    5. 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 6’s (surprisingly, you get the right answer!). Or, how about this: solve x4=320. Answer: 80 (and here, the wrong answer.)

    6. How to Show 1=2 Using Derivatives Calculus fallacies are not as common as algebraic ones, but here is a good one: 32=3+3+3, 42=4+4+4+4 X2=x+x+…+x, where the sum has x terms. Differentiate both sides: 2x=1+1+…+1 Since the right-hand side is the sum of x 1s, its value is x, so 2x=x So 2=1 Where is the error? X2 is a sum of x’s only when x is an integer. Being discrete is different from being continuous.

    7. 0=1 See if you can find the fallacy: Use integration by parts to find I= ? dx u= v=x u’= - v’=1 I=( )x- ? x dx u v- ? u’ v I=1+ ? dx I= I+1 Now, subtract I from both sides and we have 1=0 Where is the error? We always add a +c to the end for a reason!

    8. References Fallacy Detector image: http://www.lexrex.com/images/fallacy%20detector.gif Division by zero image: http://www.mapletreelearning.com/images/sfdividebyzero3.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/netscratch.gif Laughing video: http://www.killsometime.com/Video/video.asp?ID=353 What is a fallacy image: http://www.designedlearning.com/images/TheOversightFallacy.jpg

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