## Most recent change of Fallacy

Edit made on April 20, 2009 by DerekCouzens at 07:36:33

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WW

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A Fallacy in Mathematics is an apparently correct logical argument which leads to an incorrect conclusion.

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!! Fallacy 1: 1+1=1

Let EQN:a=b

* EQN:a^2=ab

* EQN:a^2-b^2=ab-b^2

* EQN:(a+b)(a-b)=b(a-b)

* EQN:a+b=b

* EQN:a+a=a (as a=b)

Now let a=1, and hence 1+1=1.

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[[[

!! Fallacy 2: -1 = 1

* EQN:sqrt{-1}=sqrt{-1}

* EQN:sqrt{\frac{-1}{1}}=sqrt{\frac{1}{-1}}

* EQN:\frac{sqrt{-1}}{sqrt{1}}=\frac{sqrt{1}}{sqrt{-1}}

* EQN:sqrt{-1}sqrt{-1}=sqrt{1}{sqrt{1}

* Hence EQN:-1=1

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[[[> IMG:angles1.png ]]]

!! Fallacy 3: all angles are right angles

(Rouse Ball's Fallacy)

Construct a quadrilateral ABCD such that AC = BD and angle CAB is a right angle and angle DBA is obtuse.

* AB is therefore not parallel to CD,

** hence perpendicular bisectors are not parallel

** hence perpendicular bisectors intersect at some point: call it E

* Construct the perpendicular bisector of AB namely ME

* Construct the perpendicular bisector of CD namely NE

* Triangle AEM is congruent to triangle BEM (RHS)

** thus angle EAM = angle EBM

* Triangle ACE is congruent to triangle BDE (SSS)

** thus angle EAC = angle EBD

* Subtracting these angles gives angle CAB = angle DBA

Therefore all obtuse angles are right angles.

Similarly all acute angles are right angles (complement of obtuse angles) ... all angles are right angles.