About this video
This video shows how sets of numbers grow. The whole numbers (0, 1, 2, 3, ...) sit inside the integers, which add the negatives, and the integers sit inside the rational numbers, which are all numbers that can be written as a ratio or fraction.
You'll be able to
- tell whether a number is a whole number, an integer or a rational number.
- explain why every integer is also a rational number.
- give examples of rational numbers that are not integers.
Key words
- whole numbers
- The counting numbers starting from zero: 0, 1, 2, 3, 4 and so on.
- integers
- The whole numbers together with their negatives: ..., -3, -2, -1, 0, 1, 2, 3, ...
- rational numbers
- Numbers that can be written as a ratio or fraction of two integers, like 1/2, 3/7 or -3/1.
- subset
- A set that sits completely inside a bigger set, like the whole numbers inside the integers.
Transcript
- [Instructor] At a pretty early age, we learned to count things. We learned that well, there might be no things, there might be one or two or three, and we could keep counting higher and higher from there. But then we learn later on in our educations that those aren't the only numbers that exist or that mathematicians think about.
These are the whole numbers, but there are numbers beyond the whole numbers. And so let me put a little circle around the whole numbers, which is 0, 1, 2, 3, 4, going on and on and on. But if we want to think a little bit broader than that, we can think about, well, what if we were to take the negative of those numbers and add those to the whole numbers?
Well, then we're going to be dealing with the integers. The integers include the whole numbers, 0, 1, 2, 3, 4, 5, but it also includes the negative of one, negative of two, negative three. So let me include those here. We have -1, -2, -3, and so on and so forth.
The integers include all of this. So the whole numbers are a subset of the integers, which includes the whole numbers and the, you could view it as the negative versions of the whole numbers, but we could go even broader. You have learned that there's numbers that are not just the integers, but maybe also numbers that are between the integers, maybe all of the numbers that can be represented as ratios or could be represented as fractions.
And we call those rational numbers. Notice the word ratio is in there, rational numbers. So that includes integers, which of course, includes whole numbers, 'cause you could represent any of these as a ratio. For example, if you wanted to represent -3 as a ratio, you could represent that as -3/1.
You could also represent that as -6/2. We could keep going like that and you could do something similar for whole numbers, but there are definitely rational numbers that are not integers. For example, I could say, well, 1/2, or let's say 3/7 or, and I could just keep making up more and more rational numbers if I want, 91/19.
These are all rational numbers that are not integers. So as we can see in mathematics, we can keep expanding the the types of numbers we talk about. And it turns out, and I will just do a little bit of foreshadowing here, that there's things even beyond the rational numbers that you are going to learn about in your future mathematical careers.
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