1 Equals -1

Theorem: 1 = -1

Proof:

 1 = sqrt(1) 
   = sqrt(-1 * -1) 
   = sqrt(-1) * sqrt(-1) 
 1 = -1


Also one can disprove the axiom that things equal to the same thing are equal to each other.


1 = sqrt(1)
-1 = sqrt(1)
Therefore 1 = -1


As an alternative method for solving:


Theorem: 1 = -1
Proof:
x=1
x^2=x
x^2-1=x-1
(x+1)(x-1)=(x-1)
(x+1)=(x-1)/(x-1)
x+1=1
x=0
0=1
=> 0/0=1/1=1

Here the MATHEMATICAL FALLACY is in the third step . For splitting sqrt(a*b)=sqrt(a)*sqrt(b) , the condition is tat at least one of the two numbers must be positive . But for this case Both are negative . this is called MATHEMATICAL FALLACY .