Expert Answer to "What is the square root of 2i?"

treslagosnv

Answered question

2022-02-01

What is the square root of

2 i

euromillionsna

Beginner2022-02-02Added 16 answers

Step 1

\sqrt{2 i} = {1 + i, - 1 - i}

Let us look at some details.
Let

z = \sqrt{2 i}

(Note that z are complex numbers.)
by squaring.

\Rightarrow z^{2} = 2 i

by using the exponential form

z = r e^{i θ}

\Rightarrow r^{2} e^{i (2 θ)} = 2 i = 2 e^{i (\frac{π}{2} + 2 n π)}

\Rightarrow {\begin{cases} r^{2} = 2 \Rightarrow r = \sqrt{2} \\ 2 θ = \frac{π}{2} + 2 n π \Rightarrow θ = \frac{π}{4} + n π \end{cases}

So,

z = \sqrt{2} e^{i (\frac{π}{4} + n π)}

by Eular's Formula:

e^{i θ} = \cos θ + i \sin θ

\Rightarrow z = \sqrt{2} [\cos (\frac{π}{4} + n π) + i \sin (\frac{π}{4} + n π)]

= \sqrt{2} (\pm \frac{1}{\sqrt{2}} \pm \frac{1}{\sqrt{2}} i) = \pm 1 \pm i

I kept the following original post just in case someone needs it.

{(2 i)}^{\frac{1}{2}} = {(2)}^{\frac{1}{2}} {(i)}^{\frac{1}{2}}

{(i)}^{\frac{1}{2}} = - 1

{(2 i)}^{\frac{1}{2}} = {(2)}^{\frac{1}{2}} \times - 1

{(2)}^{\frac{1}{2}} = 1.41

{(2 i)}^{\frac{1}{2}} = 1.41 \times - 1 = - 1.41

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