Visual representation of the roots of unity

The property that if $\omega$ is a $k$-th root of unity, then ${\omega }^i$ and ${\omega }^{i+k/2}$ are additive inverses may seem a little abstract — this chapter introduces a visual that makes this concept easier to remember.

Recall that the $k$-th roots of unity are generated by taking the primitive $k$-th root of unity $\omega$ and raising it to successive powers. For example, if $k=4$, we compute the 4-th roots of unity as

• ${\omega }^0=1$
• ${\omega }^1=\omega$
• ${\omega }^2={\omega }^2$
• ${\omega }^3={\omega }^3$

If we were to keep going, the exponents would wrap around modulo 4:

• ${\omega }^4=1$ (by definition of being a 4-th root of unity)
• ${\omega }^5={\omega }^4\omega =1\cdot \omega =\omega$
• ${\omega }^6={\omega }^4{\omega }^2=1\cdot {\omega }^2={\omega }^2$
• ${\omega }^7={\omega }^4{\omega }^3=1\cdot {\omega }^3={\omega }^3$
• ${\omega }^8={\omega }^4{\omega }^4=1$
• ${\omega }^9={\omega }^4{\omega }^4\omega =\omega$

and so on.

Note that the exponent “wraps around” at each multiple of four. Thus, for any ${\omega }^i$ and ${\omega }^j$, ${\omega }^i{\omega }^j={\omega }^{i+j(\mathrm{mod}k)}$. Since $k=4$ in our example, we have that ${\omega }^i{\omega }^j={\omega }^{i+j(\mathrm{mod}4)}$

Now recall that addition modulo $k$ can be represented as a “clock.” Here, our clock consists of the “hour markers” 0, 1, 2, 3, which are the exponents of $\omega$

One way to think about this is

“adding two numbers $i$ and $j$ on the clock is equivalent to multiplying the roots of unity that have exponents $i$ and $j$ i.e. ${\omega }^i$ and ${\omega }^j$.”

If we take any root of unity and add 1 to the exponent, this is equivalent to multiplying that root of unity by ${\omega }^1$ (or just $\omega$). For example, multiplying ${\omega }^2$ by $\omega$ is the same as adding one to the exponent to get ${\omega }^3$.

Therefore, multiplying a $k$-th root of unity by $\omega$ or adding 1 to the exponent is the same as taking $1/k$-th step around the circle.

For example, if we multiply ${\omega }^2$ by $\omega ,$ we get ${\omega }^3$, which equals moving forward 1/4-th step:

We can think of generating the roots of unity by starting with ${\omega }^0$ and repeatedly adding 1 to the exponent to produce ${\omega }^1,{\omega }^2,{\omega }^3$, and so on until we reach ${\omega }^k$ at which point the results will wrap around modulo $k$. This is exactly the same as taking $k$ steps around the circle.

Visualizing congruences with the unit circle

Let’s expand this circle to include more congruences.

If we multiply ${\omega }^3$ and ${\omega }^2$, we will get ${\omega }^5$, which is congruent to $\omega$, exactly as the chart suggests:

Another way to think about this is starting at ${\omega }^2$ and taking 3 steps forward:

Points k/2 steps apart

Since it takes $k$ steps to “walk around” the circle, $k/2$ steps takes you from a point to the opposite point.

Now observe in the circle of $k=4$ that opposite points are additive inverses of each other (they sum to zero). Recall that $-1\equiv {\omega }^{k/2}$. Since k = 4, ${\omega }^2=-1.$

In the $k=4$ example, we have that

• ${\omega }^0+{\omega }^2\equiv 1+(-1)=0$
• ${\omega }^1+{\omega }^3\equiv \omega +(-\omega )=0$

Note that we are now adding the roots of unity together, not multiplying them, so the addition of exponents rule does not apply! Don’t confuse ${\omega }^0+{\omega }^2$ with ${\omega }^0\cdot {\omega }^2$! Roots of unity are finite field elements, and fields have 2 operations: addition and multiplication.

Examples with other values of k

If the circle is partitioned into $k$ segments, then taking $k/2$ steps takes you to the opposite side. In each of the cases shown here, we see that opposite points are additive inverses.

k = 8

k = 6

k = 16

Summary

To remember that ${\omega }^i+{\omega }^{i+k/2}$ are additive inverses (their sum is zero), we draw a circle with $k$ points where each step is a multiplication by $\omega .$ The opposite points will be additive inverses.

The circle diagram will also be very useful for visualizing subgroups of the roots of unity as well as square roots — we will show those visualizations in the upcoming chapters.