When we look at the special number $i$, we start to see an interesting pattern.

You might know that $i$ is defined as having the property that $i^2 = -1$. This simple idea helps us understand how $i$ behaves when we raise it to different powers.

Let’s check out some powers of $i$:

• $i^1 = i$
• $i^2 = -1$
• $i^3 = i \cdot i^2 = i \cdot -1 = -i$
• $i^4 = i \cdot i^3 = i \cdot -i = -i^2 = -(-1) = 1$

Now, if we keep going, we see that $i$ starts to repeat itself:

1. $i^5 = i^4 \cdot i = 1 \cdot i = i$
2. $i^6 = i^5 \cdot i = i \cdot i = i^2 = -1$
3. $i^7 = i^6 \cdot i = -1 \cdot i = -i$
4. $i^8 = i^7 \cdot i = -i \cdot i = -i^2 = 1$

These results show us that the powers of $i$ repeat every four steps. We can break it down like this:

• If the exponent (the number you raise $i$ to) is something like $4n$ (where $n$ is a whole number), then $i^{4n} = 1$.
• If it's $4n + 1$, then $i^{4n+1} = i$.
• If it's $4n + 2$, then $i^{4n + 2} = -1$.
• If it's $4n + 3$, then $i^{4n + 3} = -i$.

This repeating pattern not only makes it easier to work with $i$, but it also shows the cool symmetry in complex numbers.

So, the next time you deal with a higher power of $i$, remember this simple cycle to help you calculate more easily!