Representation of complex of square root of negative i with unitary power.

[Leo Authersh]

Can ##sqrt(-i)## be expressed as a complex number z = x + iy with unitary power?

 

[mfb]

What does "with unitary power" mean?
The complex square root has two solutions, both of them are complex numbers, they can be expressed as z=x+iy like every complex number.

 

[Leo Authersh]

If any unit real number when rooted, powered, multiplied and divided gives a complex unit, then how can an unit imaginary number when rooted equates to a fractional complex number?

For example, ##sqrt(i)## equates to a non unit value i.e. 0.707+0.707i (approx)

 

[vanhees71]

Well, your number is ##\exp(\mathrm{i} \pi/4)=(1+\mathrm{i})/\sqrt{2}## and thus its modulus is 1 as it must be.

 

[Leo Authersh]

@vanhees71 Thank you, now I understood that 0.707+0.707i represents the x,y coordinate in the complex plan and not a value.

But shall we simply write as 0.707,0.707i so that it might not be misinterpreted as arithmetically additive?

 

[vanhees71]

The point of complex numbers is that you can calculate with them as with real numbers, because they obey all the axioms of a field concerning the fundamental arithmethics of + and ##\times##.

At the same time of course you can interpret real and imaginary part of a complex number as Cartesian coordinates in an Euclidean plane (Gauss's plane of numbers). This has great advantages, because it simplifies standard tasks like, e.g., rotations. The rotation of a vector ##\vec{x}=(x,y)## can be very easily calculated by writing ##z=x+\mathrm{i} y## and then the rotated vector is given by
$$z'=\exp(\mathrm{i} \phi) z,$$
where ##\phi \in \mathbb{R}## is the rotation angle (in radians) which you can easily check by using
$$\exp(\mathrm{i} \phi)=\cos \phi+\mathrm{i} \sin \phi$$
and multiplying out the product, splitting it again in real and imaginary part.

 

[Leo Authersh]

After from my previous two questions, I have gained some more understanding of the complex numbers and now I can see this question arises out of my misinterpretation of complex number. Thank you.

 

[fresh_42]

Have you tried to find out, where ##(x+iy)^2=-i## leads you to?
You might also want to read:
https://www.physicsforums.com/insights/things-can-go-wrong-complex-numbers/