By using the definition of the complex multiplication, prove that $$i^2 = -1.$$ By induction, prove

Question

By using the definition of the complex multiplication, prove that

\[i^2 = -1.\]

By induction, prove the following for every non-negative integer $n$:

\[i^n = \begin{cases} 1 & \text{if}\hspace{0.5em} n \equiv 0 \pmod{4}; \\ i & \text{if}\hspace{0.5em} n \equiv 1 \pmod{4}; \\ -1 & \text{if}\hspace{0.5em} n \equiv 2 \pmod{4}; \\ -i & \text{if}\hspace{0.5em} n \equiv 3 \pmod{4}; \end{cases}\]

Accepted Answer

$i^2=-1$. For integers $n\ge0$ with $n\equiv0,1,2,3\pmod4$, respectively, $i^n=1,i,-1,-i$, proved by induction.

Iterated powers of \(i\)

Iterated powers of \(i\)

Solution:

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