Okay, before I start my discussion, note that [itex] k \in \mathbb{R} [/itex](adsbygoogle = window.adsbygoogle || []).push({});

Edit: for anyone reading this right now, I accidentally hit 'submit post' instead of 'preview', so please ignore the thread until it has some content!!!

Okay, here is the content. In my prof's notes, he is evaluating:

[tex] \sqrt{k^2} [/tex]

and he writes:

[tex] \sqrt{k^2} = |k| [/tex]

I can understand that, because the square root sign denotes the positive square root, so in order to ensure the result is positive without making any assumptions about the intrinsic sign of k, you'd have to take its absolute value. I understand it, but it still caught me off guard!!! Maybe I'm just forgetting basic stuff, but I usually blindly write:

[tex] \sqrt{k^2} = k [/tex]

without even thinking about it. So here I am in university, taking complex analysis, and being confused by a stupid little thing like this. In this case, it's causing me even more confusion because it doesn't seem to immediately make a difference. He's actually solving a differential equation, and in this step he is solving the characteristic equation for a variable r using the quadratic formula to obtain this result:

[tex] r = -1 \pm \sqrt{-k^2} = -1 \pm i\sqrt{k^2} = -1 \pm i|k| [/tex]

That is pretty much what he has written. But the thing is, why should I even care about it? We're not just taking the positive square root. The quadratic formula accounts for both positive and negative square roots. To me, writing this:

[tex] r = -1 \pm i|k| [/tex]

seems silly because there is already a plus or minus, and the expression evaluates to:

[tex] r = -1 \pm ik [/tex] if k > 0

[tex] r = -1 \mp ik [/tex] if k < 0

so logically this covers all the bases no matter the sign of k! Regardless of whether k equals +3 or -3, I will end up with r = -1 +/- 3i

So...why can't I just write the result as:

[tex] r = -1 \pm ik [/tex]

and be done with it?

YET, it DOES make a difference to the solution of the D.E. later on whether you write k or |k|. So, strictly speaking, is |k| the ONLY correct answer to [itex] \sqrt{k^2} [/itex] when [itex] -\infty < k < \infty [/itex]?

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# Taking Square Roots

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