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April 7th, 2014, 10:53 PM   #1
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square root of -1

What I would say is that in the first instance we took -1 as the square root of 1

In the last step we took +1 as the square root of 1

In a way the equality -1 = 1 is "TRUE" in the sense that (-1)^2 = (1)^2 but of course

Is my explanation correct? is there a better way to explain this error in reasoning?
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April 7th, 2014, 10:58 PM   #2
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The rule $\sqrt a\sqrt b=\sqrt{ab}$ applies only to nonnegative real numbers. Applying it to $-1$ isn’t valid.
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April 8th, 2014, 08:06 AM   #3
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$i\sqrt{6} = \sqrt{-2}\sqrt{3} = \sqrt{-6} = i\sqrt{6}$
The equality applies to all complex numbers.

I think Shunya's answer is more or less spot on.

Th equation $a^2 = b^2$ has two solutions $a = b$ and $a = -b$ (or more succinctly $|a| = |b|$. But $a$ cannot both positive and negative in one equation, it it either one or the other. Similarly for $b$.

You have to be careful that the roots are valid for the problem that you are solving.

Usually we write that $\sqrt{a^2} = a$, the positive root. If we want the negative root we explicitly say so, using $-\sqrt{a^2} = a$ or if both are permissible $\pm\sqrt{a^2} = a$.
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