 as four imaginary numbers, complex numbers. Take a look at this. What's the square root of four, right? Hopefully you can see this well enough. These pens are a little bit sharper, right? How do you take the square root of four? Well, when you want to take the square root of anything, it's called prime factorization. What we talked about in the first part here, right? A couple of days ago. You break this thing into things multiplied together to give it the top number, right? And square root means find two things that are identical to give it the multiply each other, right? It's like mech two, right? So square root means a pair inside the square root symbol can come out as a single, right? So this comes out as a two. So the answer to this is two. Now you have to think a little bit and decide if that's the only answer you have, right? Because what you're trying to do here with this number is break it down into numbers multiplied to give you that number, right? So here's another option for the square root of four. Do you think of two other numbers that multiply together to give you four that are identical? So two other identical numbers that multiply together to give you four. We have two times two gives us four, right? What are two other numbers that multiply together to give you four? Put equally right as four to the power of a half, I think. Yeah, indeed. Because the radical, as we talked about, right? That's a power. A radical is this number just goes in the denominator and the power, right? So if we write down three to the power of two over, I don't know, let's go three. Instead of going three, let's go 90. What do we want? We want a number that's actually, let's go 30. 30 to the power of two over three. You could rewrite that as cube root of 30 squared. Okay, that's what it really means. Chichou, I need to chill here for a bit. I just sent you an emotionally charged wrap via DM. I kind of got out of control with my energy. I need to calm down. Time to learn some math. Time to learn some math, baby mama. And depending on how big of a wrap that is, I might skim through it. I might read the whole thing, right? What square root two, by square root two, give root four, or would that give root two? Would say that again? Would square root two, I recorded it, it's just one verse, one verse, okay, awesome. So Snick Mc says, would square root two, by square root two, give root four, or would that give two? It would be both, right? So square root of two times the square root of two is equal to the square root of four, which is equal to, now it's not just two. Square root of four is not just two, because as I asked, what are two other identical numbers that multiply together to give you four? Well, negative two times negative two, okay? Negative two times negative two is four. So you could write this as negative two, negative two times negative two. Well, two identical things, that's what the root symbol means. This many identical things can come out as a single thing, right? So this would be negative two. So square root of four is really positive two and negative two, there's two answers to this, right? Should we do a quick review of radicals, roots? Let's do a quick little review of roots, okay, before we go any further. How's this stuff? I'm using new markers. I'm not sure if I really like these markers. They're hard to come off. We might revert back to the other markers. Let's check it out. Oh yeah, these are really hard to come out. Okay, this isn't, this isn't dry erase. This is a difficult dry erase. Chishol, you should look into the proof of numerology with regard to three, six, nice. Yeah, I've seen it in the past before, brother. For instance, yeah, yeah, I've seen it before in the past. It's cool stuff. There's a lot of patterns. And I rarely use any of these. I don't think there's anything left in this one. I think I got a little one here, do I? Let's check it out. Do we have a little one here? Oh, we do have a little guy here. It's rare I use these. I only got these things because sometimes you buy these things in packs and they come with it, right? I just cheaper buying it with this stunk. And then buying those things solo, right? The marker solo. Oh, look at this nastiness. Let's bring out a napkin. We're gonna do this with a napkin. The name of the wrap has elevated your brain and you're thinking 45 sets of me. Let's talk about radicals, let's talk about radicals. Yeah, definitely don't like these pens. Also, four plus five is nine. Oh, fun, fun. Let's see what we got. Let's see what we got. Let's see which color is gonna come up better. Let's talk about roots. Now think of radicals as this, right? Radicals are basically any number in the denominator in the exponents of radical, right? So if I write down 27 to the power of one over three. Okay. Then think of this as this three goes in the radical. So this is really the cube root of 27. Well, what's the cube root of 27? Cube root of 27, you break down 27, right? And thank you for the follows again. Apologies if I'm not catching those pens or proof they are. I need a new batch, I gotta go get another batch. So break down 27 again, so things multiply together to give you 27, right? So 27 becomes three times nine, three times three, right? So 27 is really three times three times three. Now this number for the radical up here, if there is no number, it means square root. It means two, right? So when there is no number, it means two. When they put a number, it means whatever the number is. And what that number means is if you find this many, of the same number inside, it can come back as one thing, right? So this thing says three things can come out as one thing. So three threes come out as a three, right? And there's nothing left in there. So the answer is three. So the cube root of 27, or 27 to the power of one third is three. Let's do a more complicated one. Let's go 32 to the power of two over five, right? Well, 32 to the power of two over five says this. Take the fifth root of 32 and then square it. That's what it means. And usually you do the denominator first. You take the radical first because it makes the number smaller. So the smaller the number, the easier you can deal with it, right? So fifth root of 32, well, 32 is four times eight. Four is two times two. Eight is two times two times two. Well, if you're looking for five of a kind, because that's what the fifth root says, if you're looking for five of a kind, here's five twos, right? Five twos merge into one two. Five of a kind becomes one. So this is two squared, which is equal to four. Okay, let's do one where it doesn't work out to an integer, right? What if you had, let's do this. Let's go 32 to the power of one third, right? So again, 32, but this time we're looking for three of a kind. So this goes in here becomes the cube root of 32. And 32 we already know is five twos, right? One, two, three, four, five. Two, two, two, two, two. There's five twos. Well, cube root means three of a kind. Three of a kind can come out as one thing. So three twos come out as a single two, right? And then what do you have left on the inside? You got two times two. Can you bring those out? No, the cube root says they need to be three of a kind for them to be able to come out as a single thing. So whatever that can't come out is still in there. So two times two is four. So this, 32 to the power of one third is two and the cube root of four. Does that make sense? I hope so. That's basically radicals, right? Now let's look at how the imaginary number comes into play here. Easier to take off, much easier to remove. Now take a look at this. So what if we had, what if we had as before square root of four, square root of four or four to a power of one half. Let's lay it all out from beginning, right? Just link everything up. So what if you have four to the power of a half? This means the square root of four because the two comes in the radical. But if it's a two, you don't need to write it. By definition, the square root just means two, right? So two numbers I multiply to give you four are two times two or negative two times negative two, right? Two times two is four, negative two times negative two is four. So we have two possible answers, plus or minus two. Now what if you had this? Negative four to the power of a half. So the whole thing to the power of a half. Keep this in mind that negative four to a power of a half, because this negative sign is not being taken to a power of a half, the way you write this is negative square root of four, which is gonna be negative plus and minus two, which is really gonna be minus plus two, which is the same thing in plus and minus two. It doesn't make a difference, right? So the negative sign in front here really doesn't change the game at all, because plus and minus two is the same thing as negative minus two. So you never really write down negative minus two, so you write down plus and minus two, okay? What about this guy? This guy says this, what's the square root of negative four? What's the square root of negative four? So what are two numbers I multiply to give you negative four? You can have two times negative two. Chicho, my mood on a scale of 10 is always five, plus and minus five. Hopefully you hit the range in between a little bit too. So it means you're either zero to 10. No, no, plus or minus five would be, you hit everything in the middle, so zero to 10, which is great, right? You're not only, you're part of the, I guess the integers would work as well, but the real numbers set, and everything in between. So two numbers that give you negative four are gonna be two times negative two, but that doesn't help us out because the square root says if two things are the same, you can bring them out, right, that doesn't work, or negative two times two, right? Which is the same thing as two times negative two. So if that doesn't work out, we can't bring it out. So instead of two numbers multiplied to give you four, or negative four, what do we look at three numbers that multiply to give you negative four? What are three numbers that could multiply to give you negative four? Hmm, well, you could have two times negative two times negative one. Well, that gives you positive, that still doesn't work. We need it to be negative. So how about negative, right? So you can have negative two times negative two, is positive four times negative one, is negative four, cool? Well, square root function is just defined to be the positive square root, mainly because it's nicer and lets it be a function. It's only system, system veil. They only say positive up to a certain grade. After a certain grade, you have to look at it as plus and minus. You have to look at both options, right? I wish they taught that earlier, right? They always say, oh, we define it as being the positive, but it's only a positive in certain systems where the negative is not allowed. In other systems, if the negative is allowed, it could be negative. So what I tell my students is, I start teaching them this in grade nine and grade 10, because in grade 11, you have to look at the positive and negative, right? Why do you have to look at the positive and negative here? Let me write this down. Ooh, what a beautiful equation this is, or formula. X is equal to negative B plus and minus the square root of B squared minus four AC over two A. Can you appreciate why it has to be positive and negative now? Because when you're solving, and this is the quadratic formula, right? The quadratic formula allows you to solve for X when given a quadratic equation, right? So the plus and minus comes into play because if you have a quadratic formula, which is really a parabola, right? Something like this, this X value is really giving you the X intercepts. It's giving you this point and this point. And the way you get both those points is because you have plus and minus the, in this quadratic formula. And the plus and minus plays like this. This is the axis of symmetry for quadratic equation, right? Negative B over two A is your axis of symmetry and plus square root of B squared minus four AC over two A is this distance here and minus is this distance here. It gives you both directions, right? Important, the plus and minus, where are we? The plus and minus is important. They just don't tell people how important it is until later on, which is unfortunate, right? They should be teaching this in grade eight and nine and people should know it well in grade 10, okay? The plus and minus appears in the quadratic formula because you took a square root, for sure. If you didn't take a square root to be only positive, the quadratic formula would be wrong. To be only, but by definition, if you take the square root, you need the plus and minus. The plus and minus doesn't just come into play in the quadratic formula, it's part of what happens. It's like, it's the reality of the situation. When you take the even root of any number, you're always gonna get a plus and minus. It's there. You can negate the minus. You can eliminate the minus if you want, right? Really, you can eliminate it or dismiss it or say it doesn't apply in your system, but you have to appreciate that it's there and by applying the language of mathematics because this is just straight up syntax. That's what it means, right? But by applying the mathematics in the real world, you can decide to accept the negative or not accept the negative. Let me try to reword this. Reword it, yeah, yeah, yeah, yeah. That'd be cool. System there. Plus and minus appears. Let me read what you wrote before as well. Again, the plus and minus appears in the quadratic formula because you took a square root, yeah? If you didn't take square to be only positive, the quadratic formula would be wrong. If you didn't take the square root, the square, if you didn't take the square root, that's the square root. If you didn't take square root to be only positive. Yeah, if you only took the square root to be positive, then the quadratic formula would be wrong because you wouldn't get this other half of the quadratic function. You would only get that half, right? Let's see what the rewording is before we move on. I like math tangents. They're cool. Or let me finish this off and then we'll deal with that, right? While the rewording comes in. Now take a look at this thing. Here's three numbers that multiplied to give you negative four. Here's three more numbers that can multiply to give you negative four. Instead of having a negative two here, get rid of the negative two and go times negative one. So negative two times negative two is four times negative one is negative four. So this works out. Two times two is four times negative one is negative four. So this works out as well. While the square root symbol says hey, If you have two things that are identical, you can bring them out as a single. So here's two negative twos. They can come out as a negative two. Here's two positive twos. They can come out as a two. So again, we have plus and minus two. Plus and minus two. However, in both situations, we have a negative one still in the square root symbol. Square root of negative one. Now there aren't two numbers that multiply to give you a negative number. Negative times a negative is positive. So this is a special number that appears. And our definition, it is a definition that we've come up with. We say, you know what, let's simplify this instead of, because this appears in a lot of places. It comes up a lot, right? Square root of negative one. Electrodynamics, magnetic methods of water, electricity, you get the square root of negative one in play in the real world. Okay, when you do the mathematics, and when you do quadratics, you get the imaginary numbers coming up, stuff, right? So what we did, because this appears a lot, just like the number pi, right? This is pi, everyone knows what this is. This is 3.1415 dot, dot, dot, dot, right? We don't have to write it down by decimal places. We just come up with a symbol to represent pi. Well, we just come up with a symbol to represent the square root of negative one. We call it i. So by definition, i is equal to square root of negative one. So to simplify what we wrote, we can write it as plus and minus i, right? Plus and minus, plus and minus two i, right? So all we do, we just replace the square root of negative one by i, okay? And in mathematics, whenever you see i with a number or just i by itself, we define it to be the square root of negative one, okay? That's all. And then there's different ways you can look at this. You can look at it as the third dimensional plane that's graphing in and what not. Unfortunately, I used to know how to apply this. We did apply it, we graphed it and stuff like this. And it definitely comes into play in electromagnetics and stuff because we graph. We actually provide graphs of the complex number readings that we're taking because they provide a certain type of information. Give us more information about anomalies that we're looking at. But I haven't been teaching it for like 20 years now. They took it out of the curriculum so I haven't done geophysics for like 20 years. So I'm not going to apply this, okay? I hope that's clear regarding i. That's all i is. i is the square root of negative one.