Is this even real?

Algebra Level 4

$\large \sqrt[i]{i}$

What is the principal brach value of the above expression to 2 decimal places?

The answer is 4.81.

3 solutions

Michael Mendrin
Sep 16, 2016

We start with the "Most Beautiful Equation" and go from there

${ e }^{ \pi i }=-1$
${ e }^{ \frac { 1 }{ 2 } \pi i }=i$
${ e }^{ \frac { 1 }{ 2 } \pi }=\sqrt [ i ]{ i } =4.810477...$

Just one query (since I wasn't sure about this). What if we instead used $e^{3i\pi}=-1$ ? Wouldn't we get a different answer?

Sharky Kesa - 4 years, 9 months ago

Anytime we're dealing with complex powers we run into branches and multiple values. Usually, we go with the "principal" value. See Complex Exponentiation It's basically a rabbit hole if one is hoping for an unique answer.

Michael Mendrin - 4 years, 9 months ago

Thanks! Sort of makes sense now.

Sharky Kesa - 4 years, 9 months ago

I think i has modulus 1 so we take argument between 0 to 2pi

Prince Loomba - 4 years, 9 months ago

How $e$ came to your mind?

And how you wrote this ? : ${e}^{\pi i}=-1$ .

During solution, i was unknown about $e$ , how you used this?

Priyanshu Mishra - 4 years, 8 months ago

Md Zuhair
Sep 18, 2016

Here we get $\large \sqrt[i]{i}$ . Now $i= e^{i \pi/2}$ [by Euler form] . hence $i^{1/i}$ = $i= e^{i \pi/2 * 1/i}$ = $e^{ \pi/2}$ = $4.810477....$

I too did the same way.

Niranjan Khanderia - 4 years, 8 months ago

Joe Potillor
Sep 18, 2016

(i)^(1/i) = e^((i (pi/2))(1/i)) = e^(pi/2) = 4.81

Is this even real?

The answer is 4.81.

3 solutions

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