Negative ^ Irrational

What is the value of $(-1)^{\sqrt{2}}$

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
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Comments

Let $(-1)^{\sqrt{2}}= A$

Then on taking the natural logarithm on both sides:

$\sqrt{2}\ln(-1) = \ln(A)$ Which implies:

$\sqrt{2}\ln(i^2) = \sqrt{2}\ln(e^{i\pi})=\sqrt{2}i\pi= \ln(A)$

Or, $A = e^{i\pi\sqrt{2}} = \cos(\pi\sqrt{2}) + i \sin(\pi\sqrt{2})$

So, the result is a complex number.

Note: $i = \sqrt{-1}$ $e^{i\theta}= \cos(\theta) + i\sin(\theta)$

Karan Chatrath - 1 year, 11 months ago

Not quite right. $i^2$ can be equal to $e^{3i \pi}$ as well. So the answer could also be $\cos(3\pi \sqrt2) + i \sin(3\pi \sqrt2)$ .

Pi Han Goh - 1 year, 11 months ago

Good observation. The answer can be generalised for odd multiples of $\pi$ . However, my intent was to only show that the result of raising a negative number to an irrational exponent is complex. Generalisation was not the intent of the comment.

The uniqueness of the resulting complex number is something to ponder on. Is there a better way of approaching such a computation?

Karan Chatrath - 1 year, 11 months ago

@Karan Chatrath – I really don't know if there's a better way.

Pi Han Goh - 1 year, 11 months ago

@Karan Chatrath – If you were to generalize it, how about e^ki(pi)=.... for k is an odd integer. Forgive me, I do not know how to use the math symbols on this yet.

Dacota Sprague - 1 year, 11 months ago

@Dacota Sprague – Yes, you are correct, this is how I would generalise it.

Karan Chatrath - 1 year, 11 months ago

@Dacota Sprague – Try SearchOnMath.

Pi Han Goh - 1 year, 11 months ago

@Karan Chatrath – I think this answer is the principal one.

Ruilin Wang - 1 year, 10 months ago

@Ruilin Wang – Hello! Could you explain why?

Karan Chatrath - 1 year, 10 months ago

@Karan Chatrath – Oh. I just meany that because this is the one without any constants, so it should be the principal one, like the principal cube root of unity is 1.

Ruilin Wang - 1 year, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$