Table of contents
What is the imaginary unit i?How do I calculate powers of i?Negative powers of iFAQsWelcome 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!
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. To discover more, visit Omni's complex number calculator.
How do I calculate powers of i?
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:
n | in |
|---|---|
1 | i |
2 | -1 |
3 | -i |
4 | 1 |
and it repeats... | |
5 | i |
6 | -1 |
7 | -i |
8 | 1 |
and it repeats... | |
9 | i |
10 | -1 |
11 | -i |
12 | 1 |
and it repeats... | |
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.
🔎 If you are familiar with the notion of modulo, you will like the following notation: in = in (mod 4). Of course, i0 = 1. If you haven't yet learned about modulo, you may want to take a look at our modulo calculator.
Negative powers of i
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:
n | in |
|---|---|
-1 | -i |
-2 | -1 |
-3 | i |
-4 | 1 |
and it repeats... | |
-5 | -i |
-6 | -1 |
-7 | i |
-8 | 1 |
and it repeats... | |
-9 | -i |
-10 | -1 |
-11 | i |
-12 | 1 |
and it repeats... | |
What are the four powers of i?
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 nth power of i.
What is i to the power of 42?
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.
Can a power of i be real?
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.
How do I simplify powers of i?
To simplify the nth power of i:
- 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.
- Compute n (mod 4). In other words, find the remainder of n divided by 4.
- 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.
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