# How do you simplify imaginary numbers?

## Question:

How do you simplify imaginary numbers?

## Imaginary Numbers:

If you remember your power rules, you know that whenever you take the square of a number you end up with a positive number as a result. This rule is true for all real numbers, but what if you could take the square of a number and get a negative as a result? This result can actually happen in math, but only when the number we square is not a real number. When we square a number and end up with a negative as a result we call the number we squared an imaginary number.

Imaginary numbers are generally written as a negative number under a square root sign. For example, {eq}\sqrt{-4} {/eq} is an imaginary number. In order to make things easier we write all imaginary numbers in terms of the imaginary number i, where {eq}i = \sqrt{-1} {/eq}. We can do this because the square root of any negative number can be written as the square root of the positive of that number multiplied by the square root of -1. This means our example imaginary number {eq}\sqrt{-4} {/eq} would be written in terms of i as follows.

{eq}\sqrt{-4} = \sqrt{4} \times \sqrt{-1} = 2i {/eq}

Knowing that the imaginary number i is equal to the {eq}\sqrt{-1} {/eq} is vital to simplifying imaginary numbers. For starters, it means that every power of i can be simplified by multiplying {eq}\sqrt{-1} {/eq} by itself as many times as the power indicates. To see how this works, let's look at the first four powers of i.

{eq}1)\; i^{1} = \sqrt{-1} = i \\ 2)\;i^{2} = \sqrt{-1} \times \sqrt{-1} = -1 \\ 3)\; i^{3} = \sqrt{-1} \times \sqrt{-1} \times \sqrt{-1} = -1 \times \sqrt{-1} = -1 \times i = -i \\ 4)\; i^{4} = \sqrt{-1} \times \sqrt{-1} \times \sqrt{-1} \times \sqrt{-1} = -1 \times -1 = 1 {/eq}

If we went any further than simplifying {eq}i^4 {/eq} this way things would start to get messy to write. Luckily, there's no reason to go any further than this ever. After {eq}i^{4} {/eq} the pattern of these four results begins to repeat with {eq}i^{5} = i {/eq}, {eq}i^{6} = -1 {/eq}, and so forth. This repeating pattern allows the simplification of imaginary numbers to be relatively straight forward. To simplify the imaginary number i raised to any power, all you need to do is divide the power i is raised to by 4. After dividing by 4, you take i and raise it to the power of the remainder. The result of i raised to the remainder is equal to i raised to the original power before dividing by 4.

In order to make sure we understand all this, let's look at an example.

{eq}i^{31} {/eq}

First we divide 31 by 4. Doing this we find that we get an answer of 7 with a remainder of 3.

{eq}31 \div 4 = 7\;R3 {/eq}

Next, we set i to be raised to the power of the remainder, which in this case is 3.

{eq}i^3 {/eq}

From our first four examples we already know that {eq}i^3 = -i {/eq}. Therefore, {eq}i^{31} {/eq} also equals {eq}-i {/eq}. If we wanted to put this all together, we could write it as follows.

{eq}i^{31} = i^{3} = -i {/eq}

Finally, I would like to note that sometimes you will get a remainder of 0 when doing this process. The rule that anything raised to 0 equals 1 also applies to the imaginary number i. This means that when you get 0 as a remainder your answer will always be 1. Note that this is also the same result you get when you raise i to a power of 4.