I'm used to using i for \(\sqrt{-1}\), but I'll use j since that's what most people in this forum seem to be more comfortable with.
Each step below seems reasonable, but obviously there is a problem.
\(\frac{1}{-1}\) = \(\frac{-1}{1}\)
If two numbers are equal, their square roots are equal:
\(\sqrt{\frac{1}{-1}}\) = \(\sqrt{\frac{-1}{1}}\)
The square root of a quotient is the quotient of the square roots:
\(\frac{\sqrt{1}}{\sqrt{-1}}\) = \(\frac{\sqrt{-1}}{\sqrt{1}}\)
Simplifying:
\(\frac{1}{j}\) = \(\frac{j}{1}\)
Cross-mulitiplying:
1\(^{2}\) = j\(^{2}\)
Therefore: 1 = -1
Each step below seems reasonable, but obviously there is a problem.
\(\frac{1}{-1}\) = \(\frac{-1}{1}\)
If two numbers are equal, their square roots are equal:
\(\sqrt{\frac{1}{-1}}\) = \(\sqrt{\frac{-1}{1}}\)
The square root of a quotient is the quotient of the square roots:
\(\frac{\sqrt{1}}{\sqrt{-1}}\) = \(\frac{\sqrt{-1}}{\sqrt{1}}\)
Simplifying:
\(\frac{1}{j}\) = \(\frac{j}{1}\)
Cross-mulitiplying:
1\(^{2}\) = j\(^{2}\)
Therefore: 1 = -1