One Equals Minus One

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Paradox

 $\displaystyle -1$ $=$ $\displaystyle \sqrt {-1} \sqrt {-1}$ $\displaystyle$ $=$ $\displaystyle \sqrt {-1 \times -1}$ $\displaystyle$ $=$ $\displaystyle \sqrt 1$ $\displaystyle$ $=$ $\displaystyle 1$

Resolution

This is a falsidical paradox arising from incorrect reasoning about the nature of square roots.

Explanation

The property:

$\sqrt a \times \sqrt b = \sqrt {a b}$

can only be used when:

$a \ge 0$ and $b \ge 0$

Since $-1 < 0$, this property of square roots cannot be applied to this statement.

This is how the analysis is to go:

 $\displaystyle -1$ $=$ $\displaystyle \sqrt {-1} \sqrt {-1}$ $\displaystyle$ $=$ $\displaystyle i \sqrt 1 \cdot i \sqrt 1$ as $\sqrt {-a}$ is defined as being $i \sqrt a$ for $a > 0$ $\displaystyle$ $=$ $\displaystyle \left({i \cdot i}\right) \sqrt 1 \sqrt 1$ $\displaystyle$ $=$ $\displaystyle -1 \cdot 1$ as $i^2 = -1$ $\displaystyle$ $=$ $\displaystyle -1$

... and all is well with the world.

$\blacksquare$