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Find the exponential form of the complex number

e^17πi/60+e^27πi/60+e^37πi/60+e^47πi/60+e^57πi/60
with proof.

Could someone help me on this and type out the solving process and the proof please?

Thanks!

Aug 13, 2024

#2
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We want to find the exponential form of the complex number:

$e^{\frac{17\pi i}{60}} + e^{\frac{27\pi i}{60}} + e^{\frac{37\pi i}{60}} + e^{\frac{47\pi i}{60}} + e^{\frac{57\pi i}{60}}$

### Step 1: Recognize the terms as roots of unity

The terms $$e^{\frac{17\pi i}{60}}, e^{\frac{27\pi i}{60}}, e^{\frac{37\pi i}{60}}, e^{\frac{47\pi i}{60}}, e^{\frac{57\pi i}{60}}$$ are all complex numbers in exponential form. Notice that these exponents can be expressed as:

$\frac{17\pi i}{60}, \frac{27\pi i}{60}, \frac{37\pi i}{60}, \frac{47\pi i}{60}, \frac{57\pi i}{60}$

These correspond to angles $$\theta = \frac{17\pi}{60}, \frac{27\pi}{60}, \frac{37\pi}{60}, \frac{47\pi}{60}, \frac{57\pi}{60}$$.

Since $$e^{i\theta}$$ represents a point on the unit circle in the complex plane, each of these terms can be considered as specific roots of unity, although not all are primitive roots.

### Step 2: Consider the sum of the angles

The sum can be expressed as:

$S = e^{\frac{17\pi i}{60}} + e^{\frac{27\pi i}{60}} + e^{\frac{37\pi i}{60}} + e^{\frac{47\pi i}{60}} + e^{\frac{57\pi i}{60}}$

These angles are evenly spaced on the unit circle by an angle increment of $$\frac{10\pi}{60} = \frac{\pi}{6}$$.

### Step 3: Utilize symmetry

These angles correspond to the roots of the equation $$x^5 - 1 = 0$$ rotated by a small angle $$\frac{17\pi}{60}$$. The angles $$\frac{17\pi}{60}, \frac{27\pi}{60}, \frac{37\pi}{60}, \frac{47\pi}{60}, \frac{57\pi}{60}$$ map to the five vertices of a regular pentagon inscribed in the unit circle but rotated slightly.

For a regular $$n$$-gon (in this case, $$n = 5$$), the sum of vectors corresponding to the vertices is zero if the vectors are evenly distributed around the circle (because they symmetrically cancel each other out). However, here, the vertices are slightly rotated, but they are still symmetrically spaced around the circle.

### Step 4: Apply properties of roots of unity

Given the symmetry and spacing:

$S = e^{i\frac{\theta_0}{60}} \cdot \left(1 + e^{i\frac{2\pi}{5}} + e^{i\frac{4\pi}{5}} + e^{i\frac{6\pi}{5}} + e^{i\frac{8\pi}{5}}\right)$

Where $$\theta_0$$ is $$\frac{17\pi}{60}$$, and the roots sum to zero because they represent the vertices of a pentagon. Thus, the sum

:

$S = 0$

### Conclusion:

The exponential form of the given complex number is:

$\boxed{0}$

Aug 16, 2024