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I understand that when adding 2 complex numbers say, (a + bi) and (c + di), you add the real parts together and add the imaginary parts together say, (a + c) + (bi + di) = (a + c) + (b + d)i, but my question is, is it possible to then return to the normal complex numbers by adding the real part to the imaginary part?

For example, (a + bi) + (c + di) = (a + c) + (bi + di) And then turn it back to the original equation by adding the real and imaginary parts, (a + bi) + (c + di). If this is possible, then why?

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  • $\begingroup$ Yes, equality means you can go in both directions. $\endgroup$
    – angryavian
    Sep 11, 2016 at 4:50
  • $\begingroup$ Not clear what you are asking. You can indeed calculate $z=z_1+z_2$ by adding the real and imaginary parts, respectively. But if you only know $z$ then there is no way to "turn it back" and determine which $z_1+z_2$ it was originally calculated from. $\endgroup$
    – dxiv
    Sep 11, 2016 at 4:57

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Yes, you can indeed do this. Addition of complex numbers is defined as:

$$(a+bi) + (c+di) = (a+c) + (b+d)i$$

Since this is an equality, you can start on the left and say it is equal to the right, and you can also start on the right and say it is equal to the left.

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  • $\begingroup$ Golden answer! I would upvote if I could but do not know why I can't. $\endgroup$
    – plee
    Sep 11, 2016 at 4:52
  • $\begingroup$ I think you need to have 15 or more reputation before you can vote. (See: Privileges) You should be able to accept the answer by clicking the little checkmark, though! $\endgroup$
    – Christian
    Sep 11, 2016 at 4:55

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