Question 409611: What is the point of an imaginary number?
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! I think you mean "point" as in "What is the purpose of an imaginary number" instead of "point" on the complex plane.
Anyway, complex numbers (including imaginary numbers, note that the two sets are different) have interesting properties and can be used in problems that might not normally involve a complex number solution. Complex numbers, along with the Taylor series for  can be used to prove Euler's identity  . Anyway, here are a few problems I have encountered in which the best solutions involve complex numbers, even though it might seemingly involve a different solution. I'll let you try them on your own.
1. How many subsets of {1, 2, 3, ..., 2000} are there such that the sum of all elements in the subset is a multiple of 5?
2. Let ABCDEFGHI be a regular 9-sided polygon inscribed in a unit circle. Prove that  .
3. Let ABC be a triangle, and ABD, ACE, BCF be equilateral triangles constructed outside ABC. Show that  .
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