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A Fallacy in Mathematics is an apparently correct logical argument which leads to an incorrect conclusion. COLUMN_START^ [[[ !! 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. ]]] COLUMN_SPLIT^ [[[ !! 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 ]]] COLUMN_END [[[> 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.