phaseshift --> squareroot i = 45 degrees

[helpplease]

Hey guys,

What does a phase shift of sqrt(i)=45 degrees mean?

 

The context is that it describes distortions made by a phase of sqrt(i) = 45 degrees.

I did not understand this sentence, because I did not understand the sqrt(i) =45 degrees. 

 

Hope someone can help😇

[joigus]

The square root of a complex number is more easily obtained in terms of absolute value and argument.

Because number i (the imaginary unit) is 1_90º, and the square root of a complex number is:

z=r_theta

z^1/2=r^1/2_theta/2

You would have

i^1/2=1_45º

I hope that helps.

[Strange]

You can think of complex numbers as points on a plane where the real numbers form the X axis and the “imaginary” numbers (multiplied by i) form the Y axis. 

Values multiplied by sqrt(i) would form a line at 45º. 

To put it another way, multiplying by i is equivalent to rotating by 90º and rotating by sqrt(i) is equivalent to rotating by 45º. Note that rotating by 45º and then by 45º (i.e. rotating by 90º) is equivalent to multiplying by sqrt(i) then by sqrt(i) again; in other words multiplying by i (i.e. rotating by 90º).

(Because they can encode angles and magnitude in one (complex) number, they are very useful in things like signal processing and circuit design)

[studiot]

47 minutes ago, helpplease said:

The context is that it describes distortions made by a phase of sqrt(i) = 45 degrees.

In the context of distortion, phase shift ( of 45^o ) can be interpreted as a time delay.

The distorting component is delayed by a time equivalent to one eigth of a cycle.

More information might be available if you can expand on the context.

[helpplease]

Thank you everyone.

I think i understood your explanation. 

 

 

[HallsofIvy]

I am going to jump in here to point out that "the square root of i" is an ambiguous phrase.  Every non-zero complex number has two square roots. In the real numbers, one is positive and the other negative so we can agree that "the square root of a", if it is a real number, means the positive square root.  Since the complex numbers are not an "ordered field" we cannot make that distinction.  We cannot, in general, distinguish "the square root" of a complex number.

Edited October 2, 2020 by HallsofIvy