Doubly complex?

Algebra Level 4

If the $\sqrt{-i}$ can be represented as $\pm \dfrac {a+bi}{\sqrt{c}}$ where $c$ is non-square and the fraction is in its lowest terms, find $a+b+c$ .

Note that $i$ is $\sqrt{-1}$

The answer is 2.

1 solution

Aareyan Manzoor
Oct 26, 2015

$-i=e^{\pi i}\times e^{\dfrac{\pi}{2}i}=e^{\dfrac{3\pi}{2}i}$ $\left (e^{\dfrac{3\pi}{2}i} \right )^{\dfrac{1}{2}}=e^{\dfrac{3\pi}{4}i},e^{\dfrac{7\pi}{4}i}$ $=\cos(\dfrac{3\pi}{4})+i\sin(\dfrac{3\pi}{4}),\cos(\dfrac{7\pi}{4})+i\sin(\dfrac{7\pi}{4})$ $=\dfrac{-1+i}{\sqrt{2}},\dfrac{1-i}{\sqrt{2}}$ $1-1+2=\boxed{2}$

Doubly complex?

The answer is 2.

1 solution

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