Proving 1 = -1 with the Imaginary Unit
 

You probably remember the algebraic proof that 1 = 2 that is customarily used to challenge beginning algebra students.  A similar fallacy proof can be accomplished with the imaginary unit.

Algebraic Proof that 1 = 2
One version.		 Imaginary Unit Proof that 1 = -1
One version.

Let x = y

x - y = 0

2x - 2y = 0

2x - 2y = x - y

2(x - y) = x - y

2 = 1

The fallacy is that numbers cannot be divided by zero.  (  x - y = 0  )

1 = -1

The fallacy is that    is not an ordinary (real) square root.  There is no guarantee that is true for values of x and y that are not positive. 

The real problem is that (-1)(-1) = 1 and   but  
which is not equal to one.
The concept of does not hold when
a and b are negative numbers.

 

 

 


Roberts