What purpose do complex numbers serve in applied mathematics?

Well here's an example I know of:

Think of measuring two populations: Population A, 236 people, 48 of them children. Population B, 1234 people, 123 of them children. You might say that the fraction of children in population A is 48/236 while the fraction of children in population B is 123/1234, and that 48/236 (approx. 0.2) is much less than 123/1234 (approx. 0.1), so population A is a much younger population on the whole.

Now observe that I have used fractions, non-integer numbers, in a problem where they have no physical relevance. You can't measure populations in fractions; you can't have "half a person", for example.

Also, for solving many Differential Equations,Trigonometric Equations Improper Integrals and Polynomial Equations.The square root of -1 is imaginary,but many real-life phenomena,if expressed in mathematical form,such as in Quantum Mechanics and Relativity,make no sense, and become simpler to understand if some variables are imagined to be imaginary(sorry,bad pun).

Hope that helps

because of the fact the Queen scornfully advised Alice, " Even you should be waiting to paintings out that only real #s can serve a real purpose while imaginary #s can only serve an imaginary purpose ! "