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Where $a=ix$ and $b=-ix$ then what is: $$|a+b|^2$$

$$|b-a|^2$$

And then is this equality true?

$$|a+b|^2=|a|^2+|b|^2$$ because it seems $a+b=0$!

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Whatever makes you think $|a+b|^2=|a|^2+|b|^2$? – Gerry Myerson Jun 16 '12 at 12:22
up vote 2 down vote accepted

You have $a+b=0$ and $b-a=-2ix$ so $|a+b|=0$ and $|b-a|=2|x|$ Since $|a+b|^2=0$ and $|a|^2 + |b|^2 = 2|x|^2$ you have not this équality in general. The general formula about complexe numbers is : $$|a+b|^2=|a|^2 + |b|^2 + 2 {\mathcal R}e(\overline ab)$$

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