How to Proved that 1=2 ? [Solved]

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Today i'm going to prove a interesting fact i.e. Proof. of 1=2 Which is called 'Fallacy' In mathematical Language .

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We will prove it in three different way

Proof : 1

Let,
      x = y [where, x & y both integers]
or, xy = y² [Multiplication by 'y' ]

or, xy-x² = y²-x² [Subtracting x² from both side]
or, x²-xy = x²-y² [Multiplication by (-) ]
or, x (x-y) = (x+y) (x-y)
or, x = x+y
or, x = x+x [as we put, x=y]

or, x = 2x

or, 1 = 2

Therefore , 1=2    

Proof: 2

  Let, 
        a =b  [where, a & b both integers]

or , a² = ab [Multiplication by 'a' ]

or , a² + a² = a² + ab [adding a² from both side]

or , 2a² = a² + ab

or , 2a² - 2ab = a² + ab - 2ab  [Subtracting 2ab from both side]

or , 2a² - 2ab = a² - ab

or , 2(a² - ab) = 1(a² - ab)

or , 2 = 1 [ by cancelling the (a² - ab)  from both sides]

Therefore , 1=2    

Proof: 3 (A Proof using Complex Number)

This supposed proof uses complex numbers. If you're not familiar with them, Plz comment.

The Fallacious Proof:

We know , -1/1 = 1/-1

Or , -1/1 = 1/-1 [Taking the square root of both sides]

Or , √(-1/1) = √(1/-1) [Simplifying]

Or , √-1/√1 = √1/√-1

Or , i / 2 = 1 / 2i

Or, i/2 + 3/(2i) = 1/(2i) + 3/(2i),
Or,  i (i/2 + 3/(2i) ) = i ( 1/(2i) + 3/(2i) ),

Or , (i²)/2 + (3i)/2i = i/3i + 3i/2i

Or ,(-1)/2 + 3/2 = 1/2 + 3/2,

Or , 1 = 2

Therefore , 1=2    

Note :

See if you can figure out in which step the fallacy lies. When you think you've figured it out,comment it we will tell you whether you are correct or not, and will give an additional explanation of why that step is or isn't valid.
See how many tries it takes you to correctly identify the fallacious step!

 If you want to read it in Bengali Please Click Here 

In this post we all learn How to Proved  1=2 . But , there must have a error as it is not possible . Now find the error & comment below .   If you wanted to know where the error is , you must leave a comment .


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