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.

$\blacksquare$


Also see


Sources