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

Fallacy 1: 1+1=1

Let $a=b$

  • $a^2=ab$
  • $a^2-b^2=ab-b^2$
  • $(a+b)(a-b)=b(a-b)$
  • $a+b=b$
  • $a+a=a$ (as a=b)
Now let a=1, and hence 1+1=1.

Fallacy 2: -1 = 1

  • $sqrt{-1}=sqrt{-1}$
  • $sqrt{\frac{-1}{1}}=sqrt{\frac{1}{-1}}$
  • $\frac{sqrt{-1}}{sqrt{1}}=\frac{sqrt{1}}{sqrt{-1}}$
  • $sqrt{-1}sqrt{-1}=sqrt{1}{sqrt{1}$
  • Hence $-1=1$

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.

Therefore all obtuse angles are right angles.

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

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