Question

Find the sum of the 5th roots of unity.

Answer 1

Answer on Question #78435 – Math – Complex Analysis

Question

Find the sum of the 5th roots of unity.

Solution

$$1 = e ^ {2 \pi i}.$$ $$\sqrt[5]{1} = \sqrt[5]{e ^ {2 \pi i}} = e ^ {\frac {2 \pi i}{n} m}, \quad m = 0, 1, 2, 3, 4.$$

Sum of the roots:

$$S = \sum_ {m = 0} ^ {4} e ^ {\frac {2 \pi i}{n} m} = \sum_ {m = 0} ^ {4} \left(e ^ {\frac {2 \pi i}{n}}\right) ^ {m} = \frac {1 - \left(e ^ {\frac {2 \pi i}{n}}\right) ^ {n}}{1 - e ^ {\frac {2 \pi i}{n}}} = \frac {1 - e ^ {2 \pi i}}{1 - e ^ {\frac {2 \pi i}{n}}} = 0.$$

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