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 is the imaginary unit i?
• How to calculate the powers of i?
• Does i have negative powers?

And some more! Let's go!

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!

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 (2^nd 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. To discover more, visit Omni's complex number calculator.

Just like the real numbers, imaginary numbers have their squares, cubes, and other powers. We've built this powers of i calculator so that you can easily and effortlessly compute every power of the imaginary unit.

How to use the tool? Just input the power n that you need to evaluate and the result will appear immediately!

When we need to evaluate the powers of i by hand, we can use a particularly nice feature: the consecutive powers follow a repetitive cycle, which we can exploit to quickly evaluate the power of i that we need. To see what we mean by the cycle, look at the table of a few first powers of i:

Notice how, every four powers, the results repeat? Perhaps you can already see how we can use this repetitiveness to simplify powers of i.

For instance, let's calculate i¹²³. We only need to know the place of 123 in the cycle. As the cycle has length four, this boils down to computing the remainder of 123 divided by 4:

123 / 4 = 30 remainder 3

So, if we wanted to compute i¹²³ by going through all the intermediate powers, we would make 30 full cycles and then in the last cycle we would still make three steps:

i¹²¹ = i
i¹²² = -1
i¹²³ = -i

So, i¹²³ is the same as i³ — we only needed the remainder 3 in the exponent.

As in the case of real numbers, we can also ask about negative powers of i. They work in exactly the same cyclic manner as we saw above, just running backwards:

The four possible values of the powers of the imaginary unit are: i, -1, -i, and 1. They form a cycle, and you only need to know the remainder of n divided by 4 to quickly determine the n^th power of i.

i⁴² = -1. To arrive at this answer, we express that i⁴² = i⁴⁰ × i². Then, we can notice that i⁴⁰ = (i²)²⁰ = (-1)²⁰ = 1. Therefore, we have i⁴² = i⁴⁰ × i² = i² = -1, as claimed.

Yes, for instance i² = -1 and i⁴ = 1. In fact, whenever you raise i to the power n that returns an even number when taken modulo 4, then the result of the exponentiation iⁿ will be real. More precisely,
iⁿ = 1 if n (mod 4) = 0, and
iⁿ = -1 if n (mod 4) = 2.

To simplify the n^th power of i:

1. Recall that the values of powers of i repeat in a cycle of length 4, so n (mod 4) corresponds to the place of your power in this cycle.
2. Compute n (mod 4). In other words, find the remainder of n divided by 4.
3. Determine iⁿ as follows:
   • If n (mod 4) = 0, then iⁿ = 1;
   • If n (mod 4) = 1, then iⁿ = i;
   • If n (mod 4) = 2, then iⁿ = -1; and
   • If n (mod 4) = 3, then iⁿ = -i.