Fallacy

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.

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.

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Fallacy

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Fallacy 1: 1+1=1

Fallacy 2: -1 = 1

Fallacy 3: all angles are right angles