Okay, welcome back everyone. We're going to start a few video series on the idea of inequalities. And this video it's just going to give you a short one. It's going to give you the basic idea. The whole point here is to introduce you to symbols like a < b, x > y, c is less than or equal to d, z is greater than or equal to w, and this really funny one down here, which we often say, e << f. The first four of these are really well-defined mathematical concepts, the other one is a bit fuzzy, but you'll see it throughout all data science, so we really need to name it here. Let's start from the top and draw a real number line. And suppose here's 0, here's the number 2, here's the number 3.1, and let's consider this statement. 2 < 3.1. This funny little pacman symbol here pointing that way is read, is less than. So here's a statement, let's think about what the statement means and see why it's true. This statement, 2 < 3.1 means, 2 is to the left. Of 3.1 on the real number line, r, which is in fact true, 2 is the left of 3.1. Let's say 11.78 < 3.1, say it's about there. Fact, 11.78 is also to the left of 3.1 on the number line. So, in general, when I write A less than B, I mean A, wherever it is on the real number line. And I want to put and pick a B to make that true. B has to be somewhere over here. So A has to be to the left of B. A common misconception is that when you say A < B, say that means literally A is smaller than B. That actually works when A and B are both positive. Somehow 2 is smaller than 3.1 by the plain English meaning, but I would not say by plain English meaning that -11.78 is smaller than 3.1. It's really just to the left of it. When we write something like 3.1 > 2, this just means 3.1 is to the right of 2 on the real number line. Which is true. In fact, A < B is true if and only if, so this is a fun little mathematical symbol for if and only if, B > A. Okay, so we've really figured out what less than means, and what greater than means. Here's this funny symbol, x << y. And you actually never see this in a proper math textbook, because it's not really a proper mathematical concept. But you'll see this all the time in data science. What this really means, this means, x is much, much less than y. So for example, 1 << 1,000,000, might be a reasonable thing. I'd say reasonably not true. There's no real way for a judge to judge whether this is true or false. Whereas a statement like 2 < 3.1, you can say is that true or false. It's more much, much less. It's kind of in the eye of the beholder, but we tend to agree what it means. Somehow, 1,000,000 towards one, and that's also very important. Okay, so we figured out what A < B means, let's think about less than or equal to. So this statement a, I'm going to make this funny symbol. So this read, less than or equal to. First just in terms of writing, the way you write this is you first write the less than sign and then you put one line under there. And sometimes you really should put two for less than or equal to, and every now and then you'll see that, that a is less than and then the equals sign of b. But almost always you see less than and one line under it. What does this mean? This means that a < b, or a = b. It's literally just a shorthand in terms of the real number line. If this is b here, a less than or equal to b, means either a is less than b, which means a is somewhere here to the left of b or a is plopped right on top of b, and a equals b. Okay, fine. So how do we check if a claim, like a is less than or equal to b is true, if someone gives us specific examples? Let's see. So suppose someone proposed to me that 2 is less than or equal to 3.1. Do we believe them? Well, that person's really saying that at least one of two claims is true. This means that, 2 < 3.1, or 2 = 3.1. The first of these claims is true. The second claim is false, but taken together the claim is true. That's the nice thing about or you only need to satisfy one requirement. Say on the other hand someone claims that 2 is less than or equal to 2. So that means that the person says either, 2 < 2, or 2 = 2. First one's false. Second one's true. Taken together, they're true. And let's take one where we lose. Suppose that someone wants to say that 2 is less than or equal to 0.8. This means that either, 2 < 0.8, or 2 = 0.8. First one of these is false, the second one is false, and so we don't win. It's false overall. Okay, that concludes the video on basic inequalities. We've learned what less than means, greater than, less than or equal to, and greater than or equal to, all of which are proper mathematical concepts. We've also learned what it means to say that A << B, which is not a proper mathematical concept, but it's an intuitive one people use a lot. And that concludes our video for today.