How to Show 1=0 With Algebra... Test Your Understanding

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Ok, so obviously 1 is not equal to 0, but I will go through a simple few lines of algebra that seem reasonable at first glance, but lead to the shocking conclusion that 1=0. It is a good test of your basic understanding to show me why this is wrong... see if you can find the error and let me know!

We start with the equation

 x=1.

Now multiply both sides by x to get

 x^2 = x.

Subtracting one from both sides gives

 x^2 - 1 = x-1.

Factorize the left hand side and divide both sides by x-1.

 \frac{(x+1)(x-1)}{(x-1)} = \frac{(x-1)}{(x-1)}

The (x-1) factors cancel to leave

 x+1 = 1,

or

 x=0!!!!

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