 Errors in “Does 1 = 2?” Proofs

1st Proof

 x = 1 set x equal to 1 x2 = x multiply both sides by x x2 - 1 = x - 1 subtract 1 from both sides (x - 1)(x + 1) = x - 1 separate left side into factors x + 1 = 1 divide both sides by (x - 1) 1 + 1 = 1 substitute 1 for x 2 = 1

Division by (x-1) is not allowed since x=1 and therefor x-1=0 and division by zero is an undefined operation.

2nd Proof

 (-1)(-1) = 1 the square of -1 is 1 -1 = 1/-1 divide both sides by -1 -1/1 = 1/-1 identity operation; for all real (or complex) x, x = x/1 i/1 = 1/i take the square root of both sides (i=sqrt(-1)) i/2 = 1/2i divide both sides by two i2/2 = i/2i multiply both sides by i -1/2 = 1/2 substitute -1 for i2 and 1 for i/i -1/2 + 3/2 = 1/2 + 3/2 add 1 1/2 (3/2) to both sides 1 = 2

There are two square roots for -1, i and -i and for 1, 1 and -1.

3rd Proof

 x2 = x+x+x+...+x (x times) definition of x2; x not equal to zero 2x = 1+1+1+...+1 (x times) take derivative of both sides;  derivative of xn = nxn-1 2x = x x = 1+1+1+...+1 (x times) 2 = 1 divide both sides by x (x not equal to zero)

Only continuous functions have derivatives.  x2 as defined above is valid only for non-negative whole numbers and is therefor not a continuous function.