Reciprocal of i

Theorem

$\dfrac 1 i = -i$

where $i$ denotes the imaginary unit.

Proof

\(\ds i^2\)

\(=\)

\(\ds -1\)

Definition of Imaginary Unit

\(\ds \leadsto \ \ \)

\(\ds \frac {i^2} i\)

\(=\)

\(\ds \frac {-1} i\)

\(\ds \leadsto \ \ \)

\(\ds i\)

\(=\)

\(\ds \frac {-1} i\)

\(\ds \leadsto \ \ \)

\(\ds -i\)

\(=\)

\(\ds \frac 1 i\)

$\blacksquare$