helpplease Posted August 26, 2020 Share Posted August 26, 2020 Hey guys, What does a phase shift of sqrt(i)=45 degrees mean? Â 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.Â Â Hope someone can helpđ Link to comment Share on other sites More sharing options...

joigus Posted August 26, 2020 Share Posted August 26, 2020 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_{90Âș}, 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_{45Âș} I hope that helps. 1 Link to comment Share on other sites More sharing options...

Strange Posted August 26, 2020 Share Posted August 26, 2020 You can think of complex numbers as points on a plane where the real numbers form the X axis and the âimaginaryâ numbers (multiplied by i) form the Y axis.Â Values multiplied by sqrt(i) would form a line at 45Âș.Â To put it another way, multiplying by i is equivalent to rotating by 90ÂșÂ and rotating by sqrt(i) is equivalent to rotating by 45Âș. Note that rotating by 45Âș and then by 45Âș (i.e. rotating by 90Âș) is equivalent to multiplying byÂ sqrt(i) then byÂ sqrt(i) again; in other words multiplying by iÂ (i.e. rotating by 90Âș). (Because they can encode angles and magnitude in one (complex) number, they are very useful in things like signal processing and circuit design) 1 Link to comment Share on other sites More sharing options...

studiot Posted August 26, 2020 Share Posted August 26, 2020 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. 1 Link to comment Share on other sites More sharing options...

helpplease Posted August 27, 2020 Author Share Posted August 27, 2020 Thank you everyone. I think i understood your explanation.Â Â Â Link to comment Share on other sites More sharing options...

HallsofIvy Posted October 2, 2020 Share Posted October 2, 2020 (edited) I am going to jump in here to point out that "the square root of i" is an ambiguous phrase.Â 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.Â Since the complex numbers are not an "ordered field" we cannot make that distinction.Â We cannot, in general, distinguish "the square root" of a complex number. Edited October 2, 2020 by HallsofIvy Link to comment Share on other sites More sharing options...

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