Given two complex numbers, $a+bi$ and $c+di,$ the sum is obtained by adding the separate components:

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

This is a case of taking advantage of the associative and commutative properties to rearrange the expression and get:

$(a+bi)+(c+di)~=~a+c+bi+di$

Then we look at $bi+di$ and take out a common factor of $i.$

Plotting the two complex numbers on an Argand Diagram, we can think of them each as a vector, so geometrically, adding complex numbers is the same as adding 2D vectors.

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