Imaginary Constant

Tags

math

Is a constant used to represent the square root of the negative number \[ i = \sqrt{-1} \]

\( i \) is usually substituted in place of \( \sqrt{-1} \) in algebraic equations because it's easier to manipulate and interact with.

You can do this using regular surd arithmetic, for example:

\begin{align*} x &= \sqrt{-36} \\

&= \sqrt{36}\sqrt{-1} \\\\
&= 6i

\end{align*}

Powers of \(i\)

The various powers of \( i \) form a cyclic pattern with repeat invocations cancelling out our inverting the sign of the constant.

\begin{align*} \ldots \
i^{-1} &= -i, & i_0 &= 1, & i_1 &= i, & i_2 &= -1 \
i_3 &= -i, & i_4 &= 1, & i_5 &= i, & i_6 &= -1 \
\ldots \
\end{align*}

See also fig:i-cycle.

\begin{figure}[ht]
  \centering
  \incfig{20210614005505}
  \label{fig:i-cycle}
  \caption{Visualisation of the cyclic nature of the powers of \( i \).}
\end{figure}