Powers of i Calculator (iⁿ for any whole-number exponent, with steps)

Understanding the powers of i calculator — concept and principle

This calculator simplifies iⁿ, a power of the imaginary unit i, for any whole-number exponent, including negative ones. It shows the remainder of n ÷ 4, the working, the real and imaginary parts, the angle on the complex plane and the sum i¹ + i² + … + iⁿ. Use it for algebra, precalculus and complex-number homework.

How it is calculated

1. i is defined by i² = −1, so i¹ = i, i² = −1, i³ = −i and i⁴ = 1.
2. Divide n by 4: n = 4q + r with remainder r = 0, 1, 2 or 3 (also for negative n).
3. iⁿ = (i⁴)^q · i^r = i^r.
4. The angle on the complex plane is 90° × r.
5. Any four consecutive powers add up to 0, so the running sum depends only on r.

The principle

Because i⁴ = (i²)² = (−1)² = 1, every fourth power brings you back to 1, and the pattern i, −1, −i, 1 repeats forever. Geometrically, multiplying by i rotates a point on the complex plane by a quarter turn (90°) counter-clockwise. Four quarter turns make a full circle, which is why the cycle has length four. For negative exponents the same rule holds as long as the remainder is taken between 0 and 3: i⁻¹ = 1/i = −i, because −1 = 4·(−1) + 3.

Worked example

Default input: n = 35.

1. 35 ÷ 4 = 8 remainder 3, so 35 = 4 × 8 + 3.
2. i³⁵ = (i⁴)⁸ · i³ = 1 · i³.
3. i³ = i² · i = −i, so i³⁵ = −i.
4. Real part 0, imaginary part −1, angle 270°.
5. Sum i¹ + … + i³⁵: eight full cycles add to 0, leaving i + (−1) + (−i) = −1.

How to read the result

The result is always one of four values: 1, i, −1 or −i. The remainder tells you which, and the angle shows where the point sits on the unit circle: 0° (1), 90° (i), 180° (−1) or 270° (−i). If your answer is anything else, the reduction step went wrong.

Common mistakes

• Using the quotient instead of the remainder.
• Getting a negative remainder for negative exponents; it must be 0 to 3.
• Thinking i³ = i or i³ = −1; it is −i.
• Forgetting that i⁰ = 1.

Frequently asked questions

What is i to the power of 100?

100 is divisible by 4, so i¹⁰⁰ = 1.

What is i⁻¹?

1/i = −i, which matches the remainder rule since −1 leaves remainder 3.

Why is the sum of four consecutive powers zero?

i + (−1) + (−i) + 1 = 0, and every block of four is that pattern multiplied by a power of i.

Sources: OpenStax, College Algebra with Corequisite Support 2e (2021), §2.4 Complex Numbers