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

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Discussion Overview

The discussion revolves around the representation of the complex square root of negative imaginary numbers, specifically ##\sqrt{-i}##, and the implications of expressing complex numbers in terms of unitary power. Participants explore the nature of complex numbers, their properties, and interpretations in the context of arithmetic and geometry.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions whether ##\sqrt{-i}## can be expressed as a complex number in the form ##z = x + iy## with unitary power.
  • Another participant seeks clarification on the term "with unitary power," noting that the complex square root has two solutions that can be expressed as complex numbers.
  • A participant raises a concern about the implications of rooting a unit imaginary number, suggesting that it results in a non-unit value, using ##\sqrt{i}## as an example.
  • One response identifies the modulus of the complex number ##0.707 + 0.707i## as 1, indicating it is a unit complex number.
  • A participant proposes a notation change to avoid misinterpretation of complex numbers as additive when expressed in coordinate form.
  • Another participant explains the arithmetic properties of complex numbers and their representation in the Euclidean plane, emphasizing their utility in calculations such as rotations.
  • A participant reflects on their understanding of complex numbers and acknowledges a previous misinterpretation that led to the original question.
  • One participant suggests exploring the equation ##(x + iy)^2 = -i## for further insights and provides a link to additional resources on complex numbers.

Areas of Agreement / Disagreement

Participants express varying interpretations and understanding of complex numbers and their properties. There is no consensus on the implications of unitary power or the representation of complex roots, indicating that multiple competing views remain.

Contextual Notes

Some discussions involve assumptions about the properties of complex numbers and their representations, which may not be universally agreed upon. The interpretation of complex numbers as coordinates in a plane and their arithmetic properties are also subject to differing perspectives.

Leo Authersh
Can ##sqrt(-i)## be expressed as a complex number z = x + iy with unitary power?
 
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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.
 
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)
 
Well, your number is ##\exp(\mathrm{i} \pi/4)=(1+\mathrm{i})/\sqrt{2}## and thus its modulus is 1 as it must be.
 
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@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?
 
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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.
 
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.
 

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