Hi, today we want to talk about real numbers and interval notation. We'll go through real numbers quickly. We work with several groups of numbers in this class-- natural numbers. Natural numbers usually denoted by a letter n. Start at one and you count up by one. We also talk about integers, which are-- start with the natural numbers, but I include the negative versions of those. And I also includes 0 in this set. Rational numbers-- rational numbers you usually think of as fractions, or anything that can be written as a fraction of integers. So fractions involving integers, things like 2/3, 0.333 repeating, decimals. Also terminating decimals can be written as fractions. So 0.5, we recognize that as being equal to 1/2. Those are all rational numbers. Irrational numbers, these are anything not rational. So examples would be the square root of 2 cannot be written as a fraction. Also pi is a famous example of an irrational number. It's a non-terminating, non-repeating decimal. And in this big group, that we talked about the numbers being real, are rational and irrational numbers. So in the graphical format, it looks something like this. The smallest set is the set of natural numbers. Whole numbers is sometimes a group we talked about that includes, that's just natural numbers plus 0. Then integers includes all of those. And rational numbers here. Rational numbers and irrational numbers are split from each other. They don't overlap. And together they make up the real number system. Now we want to talk about interval notation, which is a shorthand notation that we use in mathematics to talk about intervals on the real line. So I think of this, I want to convert a less than or equal to x, less than or equal to b, into something that just involves the two endpoints. So on the real number line-- and this is what we mean by an interval. There's a. There's b. And if I were to draw this, this includes a to b and this entire group of numbers between those. That is called an interval on the real line. And so the way interval notation works, is it involves the left hand point, a. It involves the right endpoint, b. And then these two brackets on the end tell me whether or not I am including this. I have a closed circle here. That means a bracket. And a closed circle here. That also means a bracket. But these involve the left and endpoints-- left endpoint, right endpoint here. So some more examples of things that we can write in the shorthand-- let me write down the one that we just did-- a less than or equal to x, less than or equal to b is written like this. If I don't have equality here, I still have the same endpoints. I still include the same interval. But when I don't want to include endpoints, I use these open parentheses. Parentheses for not including endpoints. What I mean by not including the endpoints is, this doesn't have a less than or equal to. I'm missing that here. It's just less than, which means x cannot be equal to a. So when this is just less than instead of less than or equal to, you get this open parentheses. These can also be mixed. a, b, I include a as an endpoint, but not b. And similarly here. Now these four examples that I've written down are examples when a and b are finite. When they're actual hardcore numbers. There are examples of inequalities that I can write that lead to infinite endpoints. For example, a less than or equal to x. If I draw that on the real number line, it looks something like this. a is included as an endpoint. And I want a is less than x. So x is bigger than a. x is anything to the right of a on a real number line. So when I think about this in terms of left and right endpoints, my left endpoint is a. And my right endpoint goes all the way out. And I want to keep going until I get out to this idea of infinity. Now infinity isn't a real number. So I will always use parentheses here. And here a is included in the solution. So I use this since I have a less than or equal to x. I use close brackets. I just want to highlight, always use parentheses with infinity, plus or minus infinity. So other examples that look like this could be a less than x. x less than or equal to b. That's everything to the left of b. So I get minus infinity to b here. And again I use open parentheses next to minus infinity. So these are the 8 examples of interval notation that we'll see in this class. Let's do a couple concrete examples with numbers. Write negative 2 is less than x is less than or equal to 0 in interval notation. When I shade this in, negative 2-- it's less than, not less than or equal to. So I have an open circle there. And it's less than or equal to 0. So I have a closed circle here. I get everything in between. My left endpoint is negative 2. 0. And this should be that same inequality statement in interval notation right here. This is the solution. x is greater than 3. Looks something like this. One of my endpoints is the number 3. This should be an open circle because it's greater than, not greater than or equal to. And I get everything to the right of 3. My left endpoint is 3. And this should go all the way out to infinity. Don't include infinity ever. And there's greater than, not an equal to sign here. So this should be the solution in interval notation.