One Equals Minus One
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Paradox
| \(\ds -1\) | \(=\) | \(\ds \sqrt {-1} \sqrt {-1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {-1 \times -1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1\) |
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 $b \ge 0$
It is specifically false when both $a$ and $b$ are negative.
 | This article, or a section of it, needs explaining. In particular: The above is subtle enough to merit some proper analysis rather than just to rely on the bald statement. This would best be documented on a separate page. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
Since $-1 < 0$, this property of square roots cannot be applied to this statement.
This is how the analysis is to go:
| \(\ds -1\) | \(=\) | \(\ds \sqrt {-1} \sqrt {-1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds i \sqrt 1 \cdot i \sqrt 1\) | as $\sqrt {-a}$ is defined as being $i \sqrt a$ for $a > 0$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {i \cdot i} \sqrt 1 \sqrt 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -1 \cdot 1\) | as $i^2 = -1$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds -1\) |
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.