Intuitive understanding of why the sum of nth roots of unity is $0$

Wikipedia says that it is intuitively obvious that the sum of $n$th roots of unity is $0$. To me it seems more obvious when considering the fact that $\displaystyle 1+x+x^2+...+x^{n-1}=\frac{x^n-1}{x-1}$ and that $\{\alpha \in \mathbb{C}: \alpha^n=1\}$ is a cyclic group.

For me, it is hard to, intuitively, see this in the case that $n$ is an odd composite natural number.

Here is my reasoning thus far:

Let $n=5$. (I understand this is simple when considering the minimal polynomial).

Since the conjugate of an $n$th root of unity is also an $n$th root of unity, it is easy to see that, when written in trigonometric form, the sum (possible reordering here)

$1+\alpha^1+\alpha^2+\alpha^3+\alpha^4$

$=1+( \cos (72)+i \sin (72))+( \cos (144) + i \sin (144))+( \cos (72)-i \sin (72))+( \cos (144)-i \sin (144))$
$=1+2 \cos (72)+2 \cos (144)$.

Somehow $2 \cos (72)+2 \cos (144)=-1$, but I'm not sure this is obvious, or perhaps I'm being silly here.

What do you think? Silly?

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@quantua If $n$ is odd, would the sum still be the origin? I am confused that when summing, $n-1$ roots around the circle will cancel each others, and the sum will equal to the $1$ root left? –  user2468 May 13 '11 at 21:52
A physical obvious interpretation. If you think of $n$ forces $\vec{f}_{i}$ applied to the origin $O$, arranged in such way that the angles between them is the same, and with equal magnitudes, then the resulting force $\vec{F}=\sum_{i=1}^{n}\vec{f}_{i}$ is zero. –  Américo Tavares May 13 '11 at 22:28
Ahh equilibrium of forces! –  Jason Smith May 13 '11 at 22:43

Consider the cube roots of unity. You can write them in the plane with $x$ being the real part and $y$ being the imaginary part. They give three vectors all with unit length, equally spaced around the unit circle. Now let's add the two vectors that are not 1 (ie, the two that face to the left) and you get $-1$ as shown in the second picture below (the light vectors are gone and their sum is the left-pointing vector which represents $-1$). Of course you can add any two vectors you like, but it will always give you a vector that is the negative of the remaining one. Adding the final two will give 0. This happens for any number of vectors when they are a complete set of all roots of unity.

Note: This is far from a rigorous proof, but I'm trying to appeal to your intuition. The picture isn't nearly so nice for 5-ths roots of unity.

To address your edited question, yes, $2\cos(72) + 2\cos(144) = -1$. It's not obvious, so I don't think you're being silly. It comes from the general formula

$$\sum_{k=1}^n \cos \frac{2 \pi k}{n} = 0$$

which, with a tiny amount of manipulation, gives you the formula above. This identity gives another proof that the $n$-th roots of unity sum to 0: the $\sin$ values will cancel each other out leaving a cosine sum equivalent to the above. The normal proof for this formula uses complex numbers, but you can also obtain a more elementary proof using trig identities. However, I would argue that your request for an intuitive understanding is better served by the respondents using "center of mass" type arguments than this trigonometry stuff.

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Thanks, thats what I needed. It makes total sense now! –  Jason Smith May 13 '11 at 22:21
Ok, so Im not sure I get this in the situation $n=5$. Lets say $x_1=cos(72)+isin(72)$ and $x_4=cos(-72)+isin(-72)=cos(72)-isin(72)$. Will $x_1+x_4=1$? I mean if $x_1=<a,b>$, then $x_1+x_4=2a$ which seems far less than 1. –  Jason Smith May 13 '11 at 22:42
@Jason: No, you won't get 1. For $x_1+x_4$ you'll get something like $.62$. But for $x_2+x_3$ you'll get about $-1.62$ which gives the $-1$ needed to cancel out the $1$ and obtain 0. Unfortunately, except for the $n=3$ example I gave, you don't get unit vectors along the way. I'll expand my answer for odd $n > 3$ later when I get time. –  Fixee May 14 '11 at 1:27
$2*\cos(72) + 2* \cos(144)$ is a transcendental real number with value approximately $-0.1922$. –  Pete L. Clark May 18 '11 at 17:28
@Pete L. Clark: I'm not sure I follow. You are interpreting 72 and 144 as radians? From the context of the question it's clear he means degrees, in which case $2\cos(72)+2\cos(144)$ is $-1$ as stated. –  Fixee May 18 '11 at 18:22

The $n$ roots of unity of order $n$ (all of them, not just the primitive ones) are equally spaced around the unit circle. If their sum was nonzero, it would have an argument (angle relative to the $x$ axis) which would be a violation of symmetry. Therefore, by symmetry, the sum must be 0.

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Why does this argument fail for $n=1$? –  sdcvvc Sep 30 '12 at 17:14
The only number that is invariant under at least one non-trivial rotation of the plane is 0. But all numbers are invariant under no rotations at all; the case $n=1$ corresponds to no (non-trivial) rotations. –  Alon Amit Oct 3 '12 at 21:41

If $\rm\:n>1\:$ then $\rm\:x^n - b\ x^{n-1} + \cdots\:$ has root sum $\rm\:= b\:,\:$ hence $\rm\:x^n - 1\:$ has root sum $= 0\:.$

At the heart of the matter is the following addition law:

$$\rm\ \ \ (x^k-{\color{red} r}\ x^{k-1}+\cdots\:)\ (x^n - {\color{red} s}\ x^{n-1}+\cdots\:)\:\ =\ \ x^{k\:+\:n} - ({\color{red}{r+s}})\ x^{k\:+\:n-1} + \cdots$$

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a.k.a Vieta. –  Guess who it is. May 14 '11 at 3:20

Appealing to physical intuition, it is just the center of mass.

Think of the roots of unity as the coordinates of $n$ identical balls equally spaced around a circle in $\mathbb{R}^2$. What is the location of the center of mass? It is the center of the circle. Consequently, adding the roots of unity gives the center of the unit circle, which is $0$.

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I shave with Occam's razor. I admire the intuition and simplicity of this comment. Dead on. –  ncmathsadist May 13 '11 at 23:15
Note that this is closely related to the symmetry argument. The reason the center of mass has to be the center of the circle is because it couldn't be anything else by symmetry. –  Qiaochu Yuan May 14 '11 at 3:57

Here's an algebraic version of Alon's comment, basically your initial argument hopefully written more-intuitively: the $n$ roots are $1, \zeta, \zeta^2, \ldots, \zeta^{n-1}$ where $\zeta$ is a primitive $n$th root ($\zeta^n=1$). Let $S$ be the sum of all of these: $S=1+\zeta+\zeta^2+\ldots = \Sigma_{i=0}^{n-1} \zeta^i$. Then $\zeta S = \zeta+\zeta^2+\ldots+\zeta^n = \zeta+\zeta^2+\ldots+\zeta^{n-1}+1 = S$; rotating so that the zero'th root becomes the first, the first becomes the second, etc. - that is, multiplaying by $\zeta$ - can't change the sum.

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There's another intuitive explanation I haven't seen mentioned yet. If you start at the origin and draw a regular $n$-gon with side length 1, starting by going from $(0,0)$ to $(1,0)$ and then proceeding counterclockwise, then the vector corresponding to the $k$th side is exactly $\zeta^{k-1}$. But since the $n$-gon ends out back at the origin, the sum of all these vectors is exactly $0$. Thus the sum of all $n$th roots of unity must be zero.

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I came here to give this explanation but I see I was beaten to it. Good job. –  Michael Lugo May 14 '11 at 2:36