Powers of i Calculator
Welcome to the powers of i calculator, which can help you quickly determine an arbitrary power of the imaginary unit! Not sure what we're talking about? Scroll down and discover what the imaginary unit i is, how to calculate the powers of i, whether i has negative powers, and some more! Let's go!
What is the imaginary unit i?
The imaginary unit i is defined as a number that satisfies the quadratic equation x² + 1 = 0. In other words, we have the equality i² = −1. Sometimes we write, a bit informally, that i = √−1. Remember these properties, as they greatly help when we need to evaluate powers of i!
💡 In some contexts (most often in electrical engineering) the imaginary unit is denoted by j instead of i. In this field, i typically stands for electric current, and so j is used instead.
Imaginary numbers arise by multiplying the imaginary unit i by real numbers. That is, every imaginary number is of the form βi, where β is a real number. All imaginary numbers have the property that their square (2nd power) is a negative number. Taking sums of real and imaginary numbers, α + βi, we arrive at the set of complex numbers, which are immensely important in both math and science.
How do I calculate powers of i?
The key observation is that the powers of i repeat in a cycle of length four. Starting from i&sup0; and multiplying by i each time, we get:
| Power | Value | Why |
|---|---|---|
| i&sup0; | 1 | Any nonzero number to the 0th power is 1 |
| i¹ | i | The unit itself |
| i² | −1 | By definition of i |
| i³ | −i | i² · i = −1 · i |
| i⁴ | 1 | i² · i² = (−1)(−1) |
After i⁴ = 1 the sequence starts all over again: i⁵ = i, i⁶ = −1, and so on. So to find iⁿ for any whole number n, you only need the remainder of n divided by 4:
- Remainder 0 → iⁿ = 1
- Remainder 1 → iⁿ = i
- Remainder 2 → iⁿ = −1
- Remainder 3 → iⁿ = −i
iⁿ = i⁽ⁿ mod ⁴⁾
Example. To evaluate i²², divide 22 by 4: 22 = 4 × 5 + 2, so the remainder is 2. Therefore i²² = i² = −1.
Negative powers of i
Yes — i has negative powers too! Recall that a negative exponent means a reciprocal: i⁻¹ = 1 / i. Multiplying the top and bottom by i gives 1/i = i/i² = i/(−1) = −i. The same cycle-of-four rule still works: just keep adding 4 to the exponent until it becomes non-negative, since adding 4 never changes the value.
- i⁻¹ = −i
- i⁻² = −1
- i⁻³ = i
- i⁻⁴ = 1
Using the powers of i calculator
- Enter the exponent n — any whole number, positive, negative, or zero.
- Hit Calculate and the tool instantly returns iⁿ (one of 1, i, −1, or −i).
- You'll also see the remainder of n ÷ 4, the reduced power i⁽ⁿ mod ⁴⁾, a highlighted view of the four-step cycle, and a clear step-by-step solution.
FAQ
What is i²&sup5;? Because 25 = 4 × 6 + 1, the remainder is 1, so i²⁵ = i.
What is i⁰? i⁰ = 1, just like any nonzero base raised to the power zero.
Is i&sup4; equal to 1? Yes. i⁴ = (i²)² = (−1)² = 1, which is exactly why the powers repeat every four steps.