Complex Arithmetic/Examples/Sum of Powers of i from 0 to 7

Example of Complex Arithmetic

$1 + i + i^2 + i^3 + i^4 + i^5 + i^6 + i^7 = 0$

Proof

\(\ds \)

\(\)

\(\ds 1 + i + i^2 + i^3 + i^4 + i^5 + i^6 + i^7\)

\(\ds \)

\(=\)

\(\ds \dfrac {1 - i^8} {1 - i}\)

Sum of Geometric Sequence

\(\ds \)

\(=\)

\(\ds \dfrac {1 - \paren {i^4}^2} {1 - i}\)

simplifying

\(\ds \)

\(=\)

\(\ds \dfrac {1 - 1^2} {1 - i}\)

Powers of Imaginary Unit

\(\ds \)

\(=\)

\(\ds 0\)

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory: Example $3$.