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How to Proved that 1=2 ? [Solved]

Hello guys, how are u ?
Mathematics is our live . We can't get a step without it . Now know more about it with fun .
Today i'm going to prove a interesting fact i.e. Proof. of 1=2 Which is called 'Fallacy' In mathematical Language .

We will prove it in three different way

Proof : 1

      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

        = [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!

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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Anonymous said...

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HSC English Previous Year Question said...

লাইনটি আমি বের করে দিতে পারব যদি আপনি বলতে পারেন x=y কিভাবে ধরলেন । ধন্যবাদ ।

Nilotpol Bedi said...

Actually 'X' & 'Y' both are unknown variable .A possible relation between two unknown variable is , they may be equal or not .Most of the cases , we consider that they are not equal - As a result some people thinks that two unknown variable always be different . No need to see that , the value of 'X' may be same with the value of 'Y' as both of them chosen arbitrarily . With our primary mathematical concept sometimes we put a result 'X' before doing the calculation & in the end we calculate the value of 'X' .
More Clearly Can we not say that 5=5,6=6,7=7,9=9 e.t.c.
Is your conception clear ?
So,there are no fault in the first line , now find the fault as U promise .
Thanks for Comment . Visit regularly to get more Content .