In a book I am reading, I'm following an equation that has the line:
$$ \frac{1}{\sqrt{-\frac{a}{b}}} = \sqrt{\frac{-b}{a}} = i\sqrt{\frac{b}{a}}$$
but while I was working ahead I did:
$$ \frac{1}{\sqrt{ -\frac{a}{b}}} = \frac{1}{i \sqrt{\frac{a}{b}}} = -i\sqrt{\frac{b}{a}}$$
Which is correct? Both?
Answer
The second one is correct. Implicit in the assumptions in the first is using an identity like
$$\frac{1}{\sqrt x} = \sqrt{\frac 1 x}.$$
Although this is correct for $x \in \mathbb{R}^+$, it does not extend to negative or to complex numbers. There are quite a few false proofs based on the premise that $\sqrt{ab} = \sqrt a \cdot \sqrt{b}$ holds unconditionally!
No comments:
Post a Comment