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I came across a few problems in which three vectors in a triangle(which were also solutions to a cubic equation) or $6$ complex numbers - (heptagon - $7$ roots of unity) added up to $0$. When does this happen? Is it only when the numbers are solutions to an equation?

When do complex number vectors add up to $0$?

NoLand'sMan
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  • Your question is quite vague. Certainly one can say that three complex numbers add up to zero if they are the solution of the equation $z_1 + z_2 + z_3 = 0$. – Lee Mosher Sep 20 '19 at 03:56
  • There is nothing mysterious to this. They add up to 0 if and only if their sum is 0. Some such sums are kinda cool because the vanishing is a result of a symmetry. A simple example (there of planar vectors, but we can choose to view them as complex roots of unity also). – Jyrki Lahtonen Sep 20 '19 at 04:01
  • Vieta relations do mean that $z_1+z_2+\cdots+z_n=0$ if and only if the numbers $z_i,i=1,2,\ldots,n,$ are the zeros of a polynomial of degree $n$ without a term of degree $n-1$. – Jyrki Lahtonen Sep 20 '19 at 04:04

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Every finite set of complex numbers $\{z_1,z_2,\ldots,z_n\}$ is the set of roots of an equation: $$ (z - z_1)(z - z_2)\cdots(z - z_n) = 0. $$

To get a set of $n$ complex numbers whose sum is $0,$ begin with any set of $n-1$ complex numbers $\{z_1,z_2,\ldots,z_{n-1}\}.$ Then set $$ z_n = -(z_1+z_2+\cdots+z_{n-1}). $$

There are certain highly symmetric sets of complex numbers that form regular polygons around $0$ in the complex plane, and those have interesting equations. But you don't need any kind of symmetry to have the numbers add up to $0.$ They just need to add that way.

David K
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