This false proof brings to light an assumption that we tend to take for granted. Try to see if you can figure it out, and where the logical fallacy takes place in the steps below. To start, we agree that (1)(1) = (-1)(-1)

Then,

1. $\displaystyle \frac{(1)(1)}{(1)(-1)}\ =\ \frac{(-1)(-1)}{(1)(-1)}$ - Dividing both sides by (1)(-1).

2. $\displaystyle \frac{1}{-1}\ =\ \frac{-1}{1}$ - By simplifying.

3. $\displaystyle \sqrt{\frac{1}{-1}}\ =\ \sqrt{\frac{-1}{1}}$ - Taking the positive square root.

4. $\displaystyle \frac{\sqrt{1}}{\sqrt{-1}}\ =\ \frac{\sqrt{-1}}{\sqrt{1}}$ - By simplifying.

5. $\displaystyle \frac{1}{i}\ =\ \frac{i}{1}$ - Simplifying with the imaginary number $\displaystyle i$.

6. $\displaystyle 1^2\ =\ i^2$ - By cross-multiplying.

7. $\displaystyle 1\ =\ -1$ - Simplifying.

8. $\displaystyle 2\ =\ 0$ - Adding 1 to both sides.

9. $\displaystyle 1\ =\ 0$ - Dividing both sides by 2.

10. $\displaystyle 2\ =\ 1$ - Adding one to both sides.