Absolute values and complex numbers (1 Viewer)

[altSwift]

In the answer to the question above on line 5 they seem to substitute an abs term inside of an abs term itself. A similar thing occurs on line 2,

ie given that |x| = 1, |x - 2| can be written as |1 - 2|. However I thought that x = 1, -1, and so it can be written as either that or |-1-2|. Am I missing something here? I always thought that |x - y| cant be |x| - |y|. Thanks

 

[fan96]

On line 2 I think they are just taking the conjugate of the denominator, which doesn't matter because the absolute value function ignores the sign of the imaginary part.

As stated in the justification, , so let and we have



And because taking the conjugate is commutative with subtraction,



On line 5, they seem to just be simplifying a term inside the absolute value function.



I'm not sure why they wrote though.

 

[altSwift]

that makes much mores sense now, I had to go back over the complex conjugate properties to wrap my head around it. Thanks!!