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 **x ^{2} + 1 = 0**. In other words, we have the equality

**i**. Sometimes we write, a bit informally, that

^{2}= -1**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 (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.

## 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 | i |
---|---|

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 **i ^{123}**. 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 **i ^{123}** 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:

**i ^{121} = i**

**i**

^{122}= -1**i**

^{123}= -iSo, **i ^{123}** is the same as

**i**— we only needed the remainder 3 in the exponent.

^{3}🔎 If you are familiar with the notion of modulo, you will like the following notation: **i ^{n} = i^{n (mod 4)}**. Of course,

**i**. If you haven't yet learned about modulo, you may want to take a look at our modulo calculator.

^{0}= 1## 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 | i |
---|---|

-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 **n ^{th}** power of

**i**.

### What is i to the power of 42?

**i ^{42} = -1**. To arrive at this answer, we express that

**i**. Then, we can notice that

^{42}= i^{40}× i^{2}**i**. Therefore, we have

^{40}= (i^{2})^{20}= (-1)^{20}= 1**i**, as claimed.

^{42}= i^{40}× i^{2}= i^{2}= -1### Can a power of i be real?

Yes, for instance **i ^{2} = -1** and

**i**. In fact, whenever you raise

^{4}= 1**i**to the power

**n**that returns an even number when taken modulo 4, then the result of the exponentiation

**i**will be real. More precisely,

^{n}**i**if

^{n}= 1**n (mod 4) = 0**, and

**i**if

^{n}= -1**n (mod 4) = 2**.

### How do I simplify powers of i?

To simplify the n^{th} 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
**i**as follows:^{n}- If
**n (mod 4) = 0**, then**i**;^{n}= 1 - If
**n (mod 4) = 1**, then**i**;^{n}= i - If
**n (mod 4) = 2**, then**i**; and^{n}= -1 - If
**n (mod 4) = 3**, then**i**.^{n}= -i

- If

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