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Number Sets 1. Created by Sal Khan and Monterey Institute for Technology and Education.
Video transcript
We are asked, what number set does the number 8 belong to? So this is actually a good review of the different sets of numbers that we often talk about. So the first set under consideration is the natural numbers. And these are essentially the counting numbers, and you're not counting 0. So just if you were actually to count objects, and you have at least one of them, we're talking about the natural numbers. So that would be 1, 2, 3, so on and so forth. So clearly, 8 is a natural number. You can count up to 8 here. You could count 8 objects. So 8 is a member of the natural numbers. The next one we should consider, let's consider the whole numbers right over here. And I should say natural numbers. So let's consider the whole numbers. The whole numbers are essentially the same thing as the natural numbers, but we're now going to include 0. So this is 0, 1, 2, 3, so on and so forth. So clearly, 8 is one of these as well. You could eventually increment your way to 8, like you're just counting all of the whole numbers. Another way to view this is the non-negative numbers. So 8 clearly belongs to this as well. So let's expand our set a little bit. Let's think about integers. Now these are all the numbers starting with, well you could keep counting down, all the way up to negative 3, negative 2, negative 1, 0, 1, 2, 3, and you could just keep going there. Now clearly, 8 is one of these as well. You can just keep counting to 8. In fact, let me just put our check box there. In general, you have your integers that contain both the positive and the negative numbers and 0, depending on whether you consider that positive or negative, or neither. So that's the integers, right over here. And then the whole numbers is a subset of the integers. So I'll draw it like this. The whole numbers are right over here, that is the whole numbers. We've now excluded all of the negative numbers. So these are all the non-negative numbers. All the non-negative integers, I should say. So these are the whole numbers. And then the natural numbers are a subset of that. It's essentially everything. So the only thing that's in whole numbers that's not in the natural numbers is just the number 0. So this whole area right here just corresponds to the number 0. So it really should be a bit of a point. So let me make it clear. This circle is the whole numbers, and then I have the natural numbers, which is a subset of that. Obviously this isn't drawn to scale. The natural numbers is a subset of that. 8 is a member of all of them. 8 is sitting right over here. So it's a member of the natural numbers, whole numbers, and the integers. Now let's keep expanding things. Let's talk about rational numbers. Now these are numbers that can be expressed in the form p over q, where both p and q are integers. So can 8 be expressed this way? Well, you can express 8 as 8 over 1. Or actually 16 over 2. Or you could just keep going, 32 over 4, you can express it as a bunch of p's over q's, where both the p and q are integers. So it's definitely a rational number. And in fact, all of these things over here are rational numbers. So let me draw. So this is all a subset of rational numbers. So 8 is definitely a member of that as well. Rational numbers, so let me put the check box over here. Now what about irrational numbers? Irrational numbers. Well, by definition, these are numbers that are not rational. These are numbers that cannot be expressed in this form, where p and q are integers. So if something is rational, it just cannot be irrational. So 8 is not a member of the irrational numbers. The irrational numbers are just a completely separate set over here. So I would draw it like this. This area right over here, this would be the irrational numbers. Irrational. Rational is not a subset of irrational, they are exclusive. You can't be in both sets. So that's irrational right over there. And then finally let's ask, is 8 a member of the real numbers? Now the real numbers are essentially all of these. It's combining both the rational and the irrational. So the real numbers is all of this right over here. And so 8 is clearly a member of the real. It's a member of the real, and within the real, you either can be rational or irrational, 8 it is rational. It's an integer. It's a whole number. And it is a natural number. So it's definitely a member of the reals. And just to give you might be saying, hey well, what is an irrational number then? Can't almost every number be represented like this? Or every number you can think of be represented like this? And an example of maybe the most famous example of an irrational number is pi. Pi is equal to 3.14159, and people devote their lives to memorizing the digits of pi. But what makes this irrational is you can't represent it as a ratio, or as a rational expression, of integers, the way you can for rational numbers. And this right here is non-repeating. And if it was repeating, you actually could express it as a ratio of integers, and we do that in other videos. It is non-repeating and non terminating, so you never run out digits to the right of the decimal point. So this would be an example of an irrational number. So pi would sit here in the irrationals. Anyway, hopefully you found that helpful.