# 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

- 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Miscellaneous Problems: $128$

- For a video presentation of the contents of this page, visit the Khan Academy.