 So in this lecture, we've talked a lot about complex numbers. Now let's kind of focus on square roots in general, because when I see these complex numbers with the number i, we talk about the square root of negative 1, but what about the square roots of other negative numbers? Because, for example, if we have some positive number capital N right here, how does one take the square root of negative N? Well, there's always two square roots. So let's first talk about the principal square root for which this, by this, we'll denote the square root of negative N right here. When you take the square root of a negative number, its principal square root we'll just call this i times the square root of N. Or if you want to, you can put the i in the back. Sometimes you get a little bit confused if you draw your little radical sign too far. So generally speaking, I would put it in the front, but when you take the square root of negative N, this just means i times the square root of N. And since I recall is the square root of negative 1, the idea here is the square root of negative N, you can factor it as the square root of negative 1 and the square root of N giving us i square root of N. So in order for the square root to be multiplicative, even for negative numbers, we define it the way we do. And notice, of course, that when you square this number, you're gonna end up with an i squared times the square root of N squared, where i squared is negative 1, and then the square root of N, or as in the positive number, the square root of N is already a real number, you get N, and thus you end up with a negative N. So this number is a square root of negative N. It's the principal square root. There's always two square roots, so you're gonna get this one right here, but then if you take the negative square root, that's the other one right there. The principal one is what we refer to as the positive one. So as we're trying to calculate these square roots, we get things like the following. What is the square root of negative 1? Well, that would just be i, as we were referring to it earlier, nothing fancy there. But then if we come down to here like the square root of negative 4, well, this would mean i times the square root of 4, which the square root of 4 is 2, 2 times 2 is 4, and so the square root of negative 4 is what we refer to as 2i. Then if you look at, for example, the square root of negative 8, well, the first thing to do when you take the negative square root is always pull out an i, and then we have to calculate the square root of 8, which the square root of 8 itself is not a perfect square, but it is 4 times 2, 4 is a perfect square, and so we can take the square root of 4, which is 2, that does leave a square root of 2 behind, and I would probably write this as 2i times the square root of 2, and so that's how I like to write these things. I always, if there's any whole number coefficients or even decimal coefficients, put those in front. If there's an i, then put it between the coefficient and the radical, then anything else that can't be simplified with the radical without a numerical approximation, you'll leave that inside of the square root. So 2i square root of 2.