The Imaginary Unit - Online Tutor, Practice Problems & Exam Prep

To evaluate the square roots of negative numbers, we use the imaginary unit, denoted as $I^2=-1$. For example, $\sqrt{-4}$ simplifies to $2I$. Generally, $\sqrt{-b}$ can be expressed as $I\sqrt{b}$, where $b$ is a positive number. This concept is essential for understanding imaginary numbers in mathematics.

Square Roots of Negative Numbers

Hey, everyone. In the last chapter, we said that the square roots of positive numbers are real, but the square roots of negative numbers are not real, and we just left it at that. But you're actually now going to be asked to actually evaluate the square roots of negative numbers and you can't just say that they're not real and move on. So I'm going to show you exactly how you can take the square root of a negative number using something created especially for this purpose called the imaginary unit. So let's get started.

Now, if I were asked to take the square root of a positive number like 4, I know that the answer to this is just 2 because 2 times 2 is equal to 4. But if I were asked to take the square root of negative one, there's nothing that I can think of that multiplies by itself to give me negative one. I'm really just not sure what this could be. But the square root of negative one is actually just equal to i. Now, i is something that mathematicians came up with, especially to solve this problem. And i is literally just equal to the square root of negative one, and it is referred to as the imaginary unit.

Now, the imaginary unit is going to be super useful for us throughout this course and it has a ton of different uses. So let's look at how we would take the square root of a negative number using the imaginary unit. So if I'm asked to simplify a square root, I can simply factor in order to separate the negative out. So if I'm asked to take the square root of negative 4, I can just factor this into the square root of negative one times 4 and then further expand this using my rules for simplifying radicals into the square root of negative one times the square root of 4. Now we literally just said that the square root of negative one is equal to i, so I can simply replace that square root of negative one with an i and then pull that square root of 4 down. Now I know exactly how to take the square root of 4, so I can just simplify this into i \times 2.

Now it looks a little funky the way that it's written right now, so we're actually going to want to write this as 2 \times i with our whole number first and then our imaginary unit. So my solution here, the square root of negative 4, is equal to 2i. Now this works for the square root of any negative number. So if I have the square root of some negative number, b is just equal to any old number, I can further expand this or factor it into being the square root of negative one times that number, so times b, and then separate it again into the square root of negative one times the square root of that number. Now we know that the square root of negative one is just i, so this will just simplify into i \times \sqrt{b}. So anytime I have the square root of a negative number, it will always be able to be simplified into i \times \sqrt{b}.

Let's look at a couple more examples of this. Looking at my first example, I have the square root of 17, or a square root of negative 17. Now, we just said that the square root of any negative number, we can always simplify into i and then the square root of the positive number. Now I actually can't simplify the square root of 17 anymore, so this is actually just going to be my solution. Now I know that I just said that we want to write our i after a whole number, but when we have a radical, we actually want to write it first because if I were to write root 17 i, you can't really tell if that i is supposed to be under the radical or outside of it, so we don't want to write it like that so we don't confuse ourselves later. So my answer is just i \times \sqrt{17}.

Let's take a look at our next example. So here I have the square root of negative 32. Now, of course, the square root of a negative number, I can just write as i times the square root of a positive number. So further simplifying this, I can actually factor this into the square root of 16 times 2. And we know using our rules of radicals that I could just pull out a 4, and then I'm left with i \times 4 \sqrt{2}. Now when we write answers that have both a whole number and a radical with our imaginary unit, I actually want to write this as my whole number first, then my i, and then my radical. So my solution here is actually going to be 4i \sqrt{2}. So whenever we're writing answers that have all of these different things going on, we're always going to write our whole number first, then our imaginary unit, and then our radical. Now, looking at all of the solutions that I have for each of my examples, I had 2i, i\sqrt{17}, and 4i\sqrt{2}. Since all of these solutions include the imaginary unit along with some other numbers, these are actually all called imaginary numbers. That's all for this one. Let me know if you have questions.

Simplify the given square root. $\sqrt{-75}$

• A. $25i\sqrt3$
• B. $5i\sqrt3$
• C. $3i\sqrt5$
• D. $75i$

Here’s what students ask on this topic:

What is the imaginary unit and how is it used to evaluate the square roots of negative numbers?

The imaginary unit, denoted as $i$, is defined as the square root of -1. It is used to evaluate the square roots of negative numbers by allowing us to express these roots in terms of $i$. For example, the square root of -4 can be simplified as follows: $\sqrt{-4}=\sqrt{-1}*\sqrt{4}=i*2=2i$. Generally, the square root of any negative number -b can be expressed as $i\sqrt{b}$, where b is a positive number.

How do you simplify the square root of a negative number using the imaginary unit?

To simplify the square root of a negative number using the imaginary unit, follow these steps: First, factor the negative number into the product of -1 and a positive number. Then, separate the square root into the product of the square root of -1 and the square root of the positive number. Replace the square root of -1 with the imaginary unit $i$. For example, to simplify $\sqrt{-9}$: $\sqrt{-9}=\sqrt{-1}*\sqrt{9}=i*3=3i$.

What are imaginary numbers and how are they written?

Imaginary numbers are numbers that include the imaginary unit $i$, which is defined as the square root of -1. They are written in the form $ai$, where $a$ is a real number. For example, $2i$ and $4i$ are imaginary numbers. When writing solutions that include both a whole number and a radical with the imaginary unit, the standard format is to write the whole number first, followed by the imaginary unit, and then the radical. For example, $4i\sqrt{2}$.

How do you handle the square root of a negative number that includes a radical?

When handling the square root of a negative number that includes a radical, you first express the negative number as the product of -1 and a positive number. Then, separate the square root into the product of the square root of -1 and the square root of the positive number. Replace the square root of -1 with the imaginary unit $i$. For example, to simplify $\sqrt{-32}$: $\sqrt{-32}=\sqrt{-1}*\sqrt{32}=i*\sqrt{16*2}=i*4*\sqrt{2}=4i\sqrt{2}$.

Why is the imaginary unit important in mathematics?

The imaginary unit $i$ is important in mathematics because it allows us to extend the real number system to include solutions to equations that do not have real solutions. For example, the equation $x^2+\; 1\; =\; 0$ has no real solutions because the square of any real number is non-negative. However, using the imaginary unit, we can solve this equation as $x=\pm i$. This extension leads to the field of complex numbers, which has applications in engineering, physics, and other sciences.