Powers of $i$

In complex numbers, $i$ is defined as the imaginary unit, satisfying:

This means $i^2 = -1$, which gives rise to an interesting pattern when we compute higher powers of $i$.

The Pattern

Let's compute the first few powers of $i$:

After that, the powers repeat every 4 steps:

General Rule

The powers of $i$ cycle every 4. So to simplify $i^n$, we reduce the exponent modulo 4:

And since the pattern is:

For example, $i^{23}$ simplifies to $i^3 = -i$ because $23 \bmod 4 = 3$.

Let's Practice