# Thread: Absolute values and complex numbers

1. ## Absolute values and complex numbers

question-cropped.png

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

2. ## Re: Absolute values and complex numbers

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, $|z| = |\bar{z}|$, so let $z = \alpha - \beta$ and we have

$|\alpha-\beta| = |\overline{\alpha-\beta}|$

And because taking the conjugate is commutative with subtraction,

$|\overline{\alpha-\beta}| = |\bar{\alpha}-\bar{\beta}|$

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

$\alpha \bar{\alpha} = |\alpha|^2 = 1$

I'm not sure why they wrote $|\alpha \bar{\alpha}| = 1$ though.

3. ## Re: Absolute values and complex numbers

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

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