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I am taking a MOOC and in handouts there is the following expression:

$\left\lvert\alpha\right\rvert = \left\lvert\alpha_r + \alpha_c\right\rvert = \sqrt{\alpha^2_r+\alpha_c^2} = \sqrt{(\alpha_r+i\alpha_c)(\alpha_r-i\alpha_c)} = \sqrt{\alpha\bar\alpha} = \left\lvert\bar\alpha\right\rvert$

My question is how did they got from $\sqrt{\alpha^2_r+\alpha_c^2}$ to $\sqrt{(\alpha_r+i\alpha_c)(\alpha_r-i\alpha_c)}$? I am pretty sure that it should be $\sqrt{(\alpha_r\pm\alpha_c)^2 \pm 2\alpha_r\alpha_c}$.

Also in the handouts expression is missing some parentheses, so it looks like this: $\sqrt{\alpha_r+\alpha_c)(\alpha_r-\alpha_c}$

Can anyone clarify this for me?

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  • $\begingroup$ What is a MOOC? $\endgroup$ – Dietrich Burde Nov 5 '17 at 19:52
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    $\begingroup$ @DietrichBurde Massive Open Online Course $\endgroup$ – Raskolnikov Nov 5 '17 at 19:56
  • $\begingroup$ Massive open online course $\endgroup$ – Randall Nov 5 '17 at 19:57
  • $\begingroup$ looks like there is a typo $\endgroup$ – mercio Nov 5 '17 at 19:57
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    $\begingroup$ Thank you. I found it here. Hope it was not too massive, this course. $\endgroup$ – Dietrich Burde Nov 5 '17 at 19:58
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When dealing with complex numbers $s=x+iy$ we can factor expressions such as $x^2+y^2=(x-iy)(x+iy)$ Due to $(i)(-i)=1$. It is essentially like a difference of 2 squares.

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$\alpha$=$\alpha_r+i\alpha_c$ so the i was left out by mistake .Put it back in and you are fine.

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    $\begingroup$ I'd say this was a massive mistake. $\endgroup$ – Dietrich Burde Nov 5 '17 at 20:01
  • $\begingroup$ Sure was but when you get used to a subject ,sometimes you can suspect that it was their mistake .However you should in my opinion always find out when you don't see why some technical detail holds ,so you are to be commended for pursuing it .Good luck $\endgroup$ – StuartMN Nov 5 '17 at 20:08
  • $\begingroup$ that was actually my mistake. $\endgroup$ – YKY Nov 5 '17 at 20:15

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