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!
We want to find the exponential form of the complex number:
e17πi60+e27πi60+e37πi60+e47πi60+e57πi60
### Step 1: Recognize the terms as roots of unity
The terms e17πi60,e27πi60,e37πi60,e47πi60,e57πi60 are all complex numbers in exponential form. Notice that these exponents can be expressed as:
17πi60,27πi60,37πi60,47πi60,57πi60
These correspond to angles θ=17π60,27π60,37π60,47π60,57π60.
Since eiθ 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=e17πi60+e27πi60+e37πi60+e47πi60+e57πi60
These angles are evenly spaced on the unit circle by an angle increment of 10π60=π6.
### Step 3: Utilize symmetry
These angles correspond to the roots of the equation x5−1=0 rotated by a small angle 17π60. The angles 17π60,27π60,37π60,47π60,57π60 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=eiθ060⋅(1+ei2π5+ei4π5+ei6π5+ei8π5)
Where θ0 is 17π60, 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:
0