1. Addition of two complex numbers is very easy to do.
2. Addition of two complex numbers (a + ib) and (c + id) is simply (a +c) + i(b + d).
3. But do not consider it like simple algebra of real numbers. At a very subtle level it is different from the normal algebra.
4. Consider this addition as filling up two containers of different shape as shown in the below figure.
5. The left container shows the addition of real parts and the right container shows the addition of imaginary parts.
6. Adding two complex numbers here is simulated by adding two different liquids (blue liquid and yellow liquid) in the containers.
7. For this situation, the above representation of addition only tells you about the increase in weight of the two containers.
8. But what about the rise in the level of the liquids? For the container representing the real part, it is straight forward to measure the increase in the level of liquids.
9. What about the rise in the level of the liquids in the container for imaginary parts? Well, that will change in different situations and it all depends upon from where1 and at what angle2 the liquids were poured in the container.
10. With respect to complex numbers, it means that there are quantities in case of complex numbers that change very differently even if the addition looks simple.
11. These quantities are the 1absolute value and the 2argument of the complex numbers.
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