<s>[INST] Why is $1/i$ equal to $-i$? When I entered the value $$\frac{1}{i}$$ in my calculator, I received the answer as $-i$ whereas I was expecting the answer as $i^{-1}$. Even google calculator shows the same answer (Click here to check it out).
Is there a fault in my calculator or $\frac{1}{i}$ really equals $-i$? If it does then how?
 [/INST] I always like to point out that this fits well into a pattern you see when "rationalising the denominator", if the denominator is a root:
$$\frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{1}{2}\sqrt{2}$$
$$\frac{1}{\sqrt{17}} = \frac{1}{\sqrt{17}}\cdot \frac{\sqrt{17}}{\sqrt{17}} = \frac{1}{17}\sqrt{17}$$
$$\frac{1}{\sqrt{a}} = \frac{1}{\sqrt{a}}\cdot \frac{\sqrt{a}}{\sqrt{a}} = \frac{1}{a}\sqrt{a}$$
$$\frac{1}{i} = \frac{1}{\sqrt{-1}} = \frac{1}{\sqrt{-1}}\cdot \frac{\sqrt{-1}}{\sqrt{-1}} = \frac{1}{-1}\sqrt{-1} = - i.$$
In this vein, it is almost more suggestive to write 
$$\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$
$$\frac{1}{\sqrt{17}} = \frac{\sqrt{17}}{17}$$
$$\frac{1}{i} = \frac{i}{-1}.$$
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