Arithmetical fallacies and paradoxes

Arithmetical  fallacies  and paradoxes :

Arithmetic Fallacies. I include the next few demonstrations leading to arithmetic results that are clearly impossible. I include algebraic proofs as well as arithmetic.

prove : 1 = 2

1 . Fallacy

Trying to —

         prove 1 = 2

Let us consider ,

                                a = b

Multiplying in both side by a.

a2 =  ab

Subtracting in both side by b2.

a2 – b2 =  ab – b2

(a – b)( a + b) = b (a – b )

(a – b)( a + b) = b (a – b )

a + b = b

But given as initially  a = b

2b = b  or  b = 2b

1 = 2

Which is impossible.

How and why this happening ? What is fallacy here ?

To understand what is happening here , then you must be checkout what is division property of equality?

if a=b then prove a is not equal to b

2 . Fallacy 

Let  a ≠ b  and c is the arithmetic mean of a and b .

  

Then ,

              a + b = 2c

Multiplying  in both side by ( a- b) 

  (a + b)( a- b)   = 2c ( a- b) 

a2 – b2 = 2ac – 2bc

a2 – 2ac =  b2   2bc

Add in both side  c2.          

a2  2ac + c2  =  b2   2bc +   c2 .

   ( a -c )2  = ( b – c )2

   a – c  =  b – c

here c is cancel out in both side

a  = b  but as initially  a ≠ b

Which is impossible.