Power of i

The Pattern of Powers of $i$

The imaginary unit $i$ is defined by $i^2 = -1$. From this single fact, every higher power of $i$ can be reduced to one of just four values: $1$, $i$, $-1$, or $-i$. The pattern repeats in a cycle of length 4.

Start with the smallest powers:

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

Once you reach $i^4 = 1$, the cycle resets. Multiply $i^4$ by $i$ to get $i^5$, and so on:

• $i^5 = i^4 \cdot i = 1 \cdot i = i$
• $i^6 = i^4 \cdot i^2 = 1 \cdot (-1) = -1$
• $i^7 = i^4 \cdot i^3 = 1 \cdot (-i) = -i$
• $i^8 = i^4 \cdot i^4 = 1 \cdot 1 = 1$

The pattern is clear: every fourth power brings you back to $1$.

Negative Powers of $i$

The same cyclic behaviour extends to negative exponents. Recall that $i^{-1}$ is the multiplicative inverse of $i$:

To write this in the standard $a+bi$ form, multiply numerator and denominator by $i$:

Now build the rest:

• $i^{-2} = \frac{1}{i^2} = \frac{1}{-1} = -1$
• $i^{-3} = \frac{1}{i^3} = \frac{1}{-i}$

Rationalise $i^{-3}$ by multiplying numerator and denominator by $i$:

• $i^{-4} = \frac{1}{i^4} = \frac{1}{1} = 1$

Notice that $i^{-1} = -i$, $i^{-2} = -1$, $i^{-3} = i$, $i^{-4} = 1$ — the same four values appear, just in a different order.

The General Formula for Any Integer $k$

The pattern for any integer $k$ (positive, negative, or zero) is captured by four cases based on the remainder when the exponent is divided by 4.

Here $k$ is any integer. The proof follows directly from the cycle:

• $i^{4k} = (i^4)^k = 1^k = 1$ …