Sherpa In this GCSE Maths video, we look at whole numbers. There are many different types of numbers, including integers, rational numbers, square numbers, surds, prime and factors, plus many more. Numbers can be written in words. Both positive and negative numbers can be added, subtracted, multiplied and divided using rules. These rules must be applied in a specific order.

Okay, we're looking at whole numbers and first, we are going to look at what are they, what do they all mean, and second of all, we're going to look at the properties of our numbers. So there are five main properties. We're going to go through each one briefly and look at what it means. The whole numbers in itself, it's not really a standardised topic. It's not like you're going to get a whole numbers question in your exam paper. It's more about a general understanding of how numbers work and it's really applicable to many different topics, but it's kind of a behind-the-scenes thing that's kind of pervasive throughout many different topics.

So first of all, what are whole numbers? Well, first, to find what whole numbers are, it probably makes sense to just look at the three main categories of numbers because they're all slightly different. So imagine I've got a number line that goes from kind of minus infinity down here to plus infinity up here and we're just looking at a small section of this number line. Well, first of all, let's look at counting numbers. This is the most restrictive version of numbers.

So counting numbers is any number that is one or above that has no decimal point in it. So for example, 1, 2, 3, 4, 5, 6, they're all counting numbers. 17 is a counting number. 17.1 is not a counting number because it's got a decimal point in it. So these are our counting numbers.

Our second category is very similar to the counting numbers, but these are what we're looking at today. These are whole numbers. So whole numbers are the exact same as counting numbers, but it includes zero. So again, 15 is a whole number. 15.1 is not a whole number because it contains the decimal point.

And our final category would be integers. So integers basically anything between minus infinity and plus infinity, anything that doesn't have a decimal point is an integer. So for example, -4 is an integer, but it's not a whole number and it's not a counting number. Right, let's look at the properties of whole numbers. Now it's the properties of these numbers that go from zero all the way up to infinity.

So the first property is closure. And again, you don't need to memorise the names of each property. It's more just about getting a general understanding of how we can treat numbers and how we can apply these rules to various questions that we may tackle in the future. So the first rule is closure, and this is applicable to adding and multplying numbers. You'll see why we've made this caveat in a second.

And what closure says, it's very obvious. It says the whole number plus the whole number gives us a whole number. For example, 5 plus 7 gives us 12, right? So all these numbers are whole numbers. And likewise, we can use it for multiplying as well.

So we can say whole number times a whole number will always give us a whole number. So 5 times 7 will always give us a whole number as well, 35. So these are just a couple of examples. The reason why we specify that this is applicable to adding and multiplying is because it doesn't necessarily work all the time with minusing and dividing. So, in this case, we tried to divide a whole number by whole number.

It won't necessarily give us a whole number. For example, 5 divided by 7 gives us 0.71. And likewise, a whole number minus a whole number also won't necessarily give us a whole number. So 5 minus 7 gives us -2. So -2 is an integer, but it's not a whole number, it's not 0 or above.

And this caveat of adding and multiplying is going to apply to the next two as well. Right, let's look at the second property commutative. So if we say that the numbers are commutative, all we're saying is that A plus B is the exact same as B plus A. For example, in colour. For example, if we do 5 plus 7, we know 5 plus 7 is going to give us 12.

And likewise if we do 7 plus 5, again we get 12. That's what that rule is saying. And it's also applicable to multiplying. So for example, any whole number times a whole number is the exact same as the second whole number times the first whole number. For example, 5 times 7.

Well, thankfully we know from our five times tables that we get 35. And again we could say this is the exact same as 7 times 5. And so again, 7 times 5 gives us 35. That's commutative. Let's look at the third property that is they are associative.

So again, this is applicable to just adding or multiplying. What this rule says is basically A plus B plus C is the exact same as A plus B plus C. And we can look at a simple example and we can think of this as like 2 plus 7 is going to give us 9 and we can add 5 onto 9 and that obviously gives us 14. Or we can think of this as 5 plus 2. Well, 5 plus 2 gives us 7 and again 7 plus 7 gives us 14 and likewise, it's applicable to multiplying as well.

So A times B times C is the exact same as A times B times C.

Another example, we could do 2 times 7. We're following BIDMAS here, right? We're doing the brackets first. So we did 2 times 7 and that gives us 14 and we times 14 by 5, 14 times 5 bracket there. 14 times 5 is going to give us 70.

Or we can think of this as 5 times 2 times 7. So 5 times 2 is going to give us 10 and then 10 times 7 is going to give us again 70. So both of those are obviously the same number. Our fourth property, whole numbers are Distributive. And what this says is it basically says that we're allowed to expand brackets.

And you probably applied this without even knowing that you apply the Distributive rule many times before. So whenever you expand brackets, we're applying this rule. So in hard algebra, this basically says A times B plus C is the exact same as A times B plus A times C. And as an example, again, following BIDMAS with this, we would do the brackets first. So we would do 2 plus 7, that would give us 9.

We would do 5 times 9, and that's going to give us 45. Or we can expand the brackets and we can do 5 times 2 to get 10 and we can do 5 times 7 to get 35. And we can add these two together. And that's going to give us, again, the exact same number, 45. So Distributive is just expanding brackets really.

Our final property, the identity property. So what this says is two different things. One of them says that if we start with the whole number and we add 0 to it, we end up with the whole number. And that makes sense, right? Because adding 0 to anything, we just end up with whatever we started with.

So in this example, if we start off with a whole number, for example 5, we ADD 0 to it, nothing changes, but still left with our whole number of 5. And also, you can think of this as multiplying as well. So if we start off with the whole number, we times it by 1. Obviously, anything times 1 is just itself. So if we start off with the whole number, we're going to end up with a whole number when we multiply by one.

So in this example, we can start off with 5 times it by 1. We don't change anything, we just keep our original number and we're left with 5. So you may be thinking, what's the point in all this? Well, we could think of an example completely not really related to whole numbers. To show you why it's useful to understand whole numbers, we could get a powers question.

For example, you could have, say, A7 times M3 times A2. Well, first of all, we can use our permutative rule that says A times B is the exact same as B times A. I could think of this as M3 times A2, or I could think of this as A2 times M3. I can completely flip the order in which I do the multiplication in five brackets. So that would be applying our commutative rule.

And also, I know from my associative rule that A times B times C is the exact same as A times B times C. I could think of this as A times B times C, which we've got here. Or I can rearrange this a little bit and put my brackets around the first two terms. So I can think of this as A7 times A2 times M3. And all of a sudden, we've made it much nicer for ourselves.

We know that from our power rules, which we may have another lesson. A7 times A2 is going to give us A9, and we can end up with A9 times M3, or just A9M3. So that's how we can use the properties of our whole numbers as kind of a tool, seemingly me completely unrelated questions. Right, that is a brief summary of whole numbers.

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