My Second Problem!

Algebra Level 2

Taking into account that − 1 \sqrt{-1} = i = i , what does i = ? \sqrt{i}= ?

2 1 \frac{2}{\sqrt{1}} + 1 2 \frac{1}{\sqrt{2}} i 1 2 \frac{1}{\sqrt{2}} + 1 2 \frac{1}{\sqrt{2}} 1 2 \frac{1}{\sqrt{2}} + 1 2 \frac{1}{\sqrt{2}} i 1 2 \frac{1}{\sqrt{2}} i

Firts Last by Firts Last , 30, USA

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1 solution

Star Chou
Jul 14, 2018

Since e i x = cos ⁡ ( x ) + i sin ⁡ ( x ) e^{i x} = \cos(x)+i \sin(x) , putting x = π 2 x = \frac{\pi}{2} then obtain e i π 2 = i e^{ i \frac{\pi}{2} } = i . So

\sqrt{i} = \sqrt{e^{ i \frac{\pi}{2}} = e^{ i \frac{\pi}{4}} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i.

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