TITLE: How can people understand complex numbers and similar mathematical concepts?
QUESTION [6 upvotes]: In mathematics, how does something like complex numbers apply to the real world? Why do complex numbers exist? How can we comprehend addition of complex numbers? For example, addition of natural numbers can be understood as putting together two apples and two oranges makes four fruits. How can we apply this thinking to complex numbers?

REPLY [3 votes]: There are lots of ways to develop intuition with complex numbers and they've been mentioned above, so I'll try to say something different. I don't think it matters whether or not complex numbers "exist", it simply matters that they are very useful and therefore worth studying. In fact, its hard to say whether any numbers really exist. For example, what is the number $5$? There isn't a tree somewhere in Iceland where $5$'s grow and are shipped all over the world for commercial use. Instead, we have a concept of $5$ that has been universally agreed upon in some sense and can be represented by $5$ of something, for example the following represents $5$ in some sense: $* * * * *$. 
My point is, to me natural numbers don't even "exist" in any meaningful sense. They just turn out to be concepts that seem to occur in nature and we've come up with a useful way to characterize them. You could even argue that complex numbers occur "in nature" in the sense that we can use them to describe certain physical laws. However, that is less satisfying to me, since there are plenty of mathematical constructions with no physical interpretation. Instead, I like to think that complex numbers are simply a mathematical construction that turn out to be very useful in solving different types of problems.