Root of unity problem for wiki

Relevant wiki: Roots of Unity

The 7 $^\text{th}$ roots of unity can be found with:

$z=e^{2k\pi i/7}\text{ for }k=1,2,3,4,5,6,7$

The goal is to maximize $a+b$ , so the angle $\large\frac{2k\pi}{7}$ should either be in the first quadrant of the complex plane , on the Positive Real Axis, or on the Positive Imaginary Axis.

The only angles that meet this criteria are $2\pi$ and $\large\frac{2\pi}{7}$ .

Using Euler's Formula : $e^{2\pi i}=\cos(2\pi)+i\sin(2\pi)=1+0i$ $e^{2\pi i/7}=\cos{\left(\frac{2\pi}{7}\right)}+i\sin{\left(\frac{2\pi}{7}\right)}\approx 0.62349+0.78183i$

It is clear that the root of unity that corresponds to the angle $\large\frac{2\pi}{7}$ maximizes the value of $a+b$ .

The maximum value is $a+b\approx 0.62349+0.78183=\boxed{1.40532}$ .

$a + b$ will be maximized when the 7th root is in the first quadrant. Thus we only have to check for these two values. Using euler's formula: \[\begin{align} e^{2\pi i} &= cis(2 \pi) = 1 + 0i & \implies a + b = 1

\\ e^{2\pi i / 7} &= cis\bigg(\frac{2 \pi}{7}\bigg) \approx 0.6234 + 0.7818i & \implies a + b \quad \approx 1.405 \end{align}\] The greater value is obtained at the second root with $a + b = \boxed{1.405}$

We can have $7$ values of $z$ . So, using roots of unity, we have that: $\implies z = e^{(2\pi/7) \cdot i}\, e^{(4\pi/7) \cdot i}\, e^{(6\pi/7) \cdot i}\, \ldots, e^{(14\pi/7) \cdot i}$

Now test all of these values, and use Euler's Formula, to see that our largest value will be around: $\approx 1.405$