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