The Imaginary Unit - Video Tutorials & Practice Problems
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Square Roots of Negative Numbers
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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 gonna 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 of a positive number like four, I know that the answer to this is just two because two times two is equal to four. 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. So I'm really just not sure what this could be, but the square root of negative one is actually just equal. 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 four, I can just factor this into the square root of negative one times four. And then further expand this using my rules for finding radicals into the square root of negative one times the square root of four. 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 four down. Now I know exactly how to take the square root of four. So I can just simplify this into I times two. Now, it looks a little funky the way that it's written right now. So we're actually gonna wanna write this as two times I with our whole number first and then our imaginary unit. So my solution here, the square root of negative four is equal to two I. 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 the square root of B. So any time I have the square root of a negative number, it will always be able to be simplified into I times the square root of that positive number. So let's look at a couple more examples of this. So 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 route 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 a square root of 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 um factor this into the square root of 16 times two. And we know using our rules of radicals that I could just pull a four out. And then I'm left with I times four root two. 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 eye and then my radical. So my solution here is actually going to be for I root two. 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 two, I, I route 17 and four, I route two. 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.
