Relationships between sets of rational numbers

0:00- [Instructor] In this video, we are going to become familiar with different categories of numbers. So let's first think about the types of numbers you learned when you were very young, when you just learned to count things. You might say hey, there's one apple, or you might say oh, there are two bananas, or there are three trees, and what we are doing right here, these numbers like one, two, three, four, and obviously, I could keep going on forever, we call these the natural numbers.

0:31Natural numbers, and these are the numbers that most people think of when they think of numbers, but they're not the only types of numbers out there. For example, some of you might have been thinking well, what about the number zero? So if we wanna include the number zero, now we are talking not just about natural numbers, so let me throw the number zero in here, if we put the number zero in here but also include all of the natural numbers, so I'm gonna put a circle around everything like that, now we are talking about whole numbers.

1:09Whole numbers, like that. So once again, whole numbers are zero, one, two, three, four, five, six, on and on and on, while natural numbers is all of those without the zero. So I put zero into this larger circle, into this larger set if you will. Now you might say hey okay, well those are a lot of the numbers, but I have also learned about negative numbers, so what if I wanted to include those as well?

1:35Numbers like negative one, negative two, negative three, so on and so forth. Well, that group of numbers, if we include it with everything else we've said, so I'm gonna put a big circle around everything, so this is going to go from negative, we could go super negative but then negative three, negative two, negative one, zero, one, two, three, four, so both positive and negative numbers and a zero, now we are talking about the set of integers.

2:05Now some of you might already be be feeling a little bit tired, but you might realize there are even more numbers than that. There are numbers that aren't in these sets here, numbers like one half. Numbers like four sevenths. Numbers like negative 105 over eight.

2:31These don't exist in any of these. Essentially, any number that you can represent as a fraction, sometimes people would say as a fraction like M over N, sometimes people use a notation P over Q, but these are called the rational numbers. Rational numbers, and they include not just these numbers here, but they include everything that we have talked about before.

2:57So these are all the numbers that you can represent as a fraction. So it's obviously not just these three numbers, it's any number that you can represent as something over something. Now you might say well, what about zero? Well, zero could be represented as zero over five, that's still zero.

3:13Two could be represented as two over one. Three could be represented three over one or six over two. Negative three could be represented as negative nine over three. So these are all still rational numbers, but now rational numbers, we can even represent things that we couldn't represent as integers.

3:33Now, some of you might be wondering, is there anything beyond the rational numbers? Things that you can't represent as a fraction? M over N or as a fraction P over Q? And that we will save for a future day.