Crazy Fallacy in Proving 2 =1

by Charles Gonzales Gagui - CleVer Vibration

Did anyone tricked you of fallacious proving of an ambiguous equality? One good example is below:

This is a total classic fallacy can be proven that you can also believe into and make other people curious. You can try this to your friends, colleagues and professors or even anyone you come across with.

Let us start

Proving 2=1

Let x and y be equal to non-zero quantities

x=y

Multiply through by x

x² = xy

Subtract y²

x²-y²=xy-y²

Factor both sides

(x-y)(x+y)=y(x-y)

Divide both sides by (x-y)

(x+y)=y

Observing that (x=y)

(y+y)=y

Combine like terms on the left

2y=y

Divide both sides by y

2=1

Here is another one

Let a=b

Then, multiply both sides by a

a²=ab

Add a² on both sides

a²+a²=a²+ab

2a²=a²+2ab

Subtract –2ab

2a²-2ab=a²+ab-2ab

2a²-2ab=a²+-ab

Factor left side by 2

2(a²-ab)=a²-ab

And divide both sides by a²-ab

This will give you

1=2

Have you identified the deception here? The fallacy are in the arguments

Divide both sides by (x-y)

and on the second case

divide both sides by a²-ab

Here, we are canceling off both sides by a quantity, we know that they must have a same quantity equal to both sides. So, we are dividing each side by the same quantity and it will still have equal two sides, because the equation are equal.

Nevertheless, division only makes sense when the number we are dividing is a non-zero variable.

In this case,

(x-y) and a²-ab is zero

observing that x=y and a=b, 

then the two equations are zero

This is called fallacies based on division by zero.

Proving this to you,

Having assumptions

0x1=0 and

0x2=0

Then

0x1=0x2

Dividing by zero gives

(0/0)x1=(0/0)x2

So, this will also give us

1=2

As you can see, it only hides the actual value of zero by forming variables as (x-y) and (a²-ab).

This is why when we define any number divided by zero, it is called undefined.

Please see reference for more mathematical fallacies.


References:
http://en.wikipedia.org/wiki/Invalid_proof