One of my favorite places where imaginary numbers pop up is in solving recurrence relations[0]. For instance, the Fibonacci numbers are an example of a recurre

One of my favorite places where imaginary numbers pop up is in solving recurrence relations[0]. For instance, the Fibonacci numbers are an example of a recurrence relation and you can obtain the closed form solution by solving the characteristic polynomial. However, you can also get a closed form solution involving imaginary numbers when the roots of the polynomial are imaginary, even though the sequence only contains real numbers! From that you can express the result completely with real numbers if you use the Euler transformation of complex numbers. [0] https://math.unm.edu/~jvassil/Recurrence%20Relations.pdf https://math.unm.edu/~jvassil/Recurrence%20Relations.pdf