In this video, we’re gonna look at how whole numbers can be written in fraction format, and also see how some fractions are equivalent to whole numbers.
First, let’s bring out a cake. It’s one whole cake, delicious! Now let’s get another cake, just the same. Now we’ve got two whole cakes even delicious-er! Okay. Delicious-er isn’t a real word, but this is a math video not an English one. Finally, for an even bigger dose of delicious-ity, let’s get a third cake. So that’s three cakes altogether. So if I cut each cake in half, I’ve still got three cakes, even though they’re in halves.
If we just think about the first cake, mm delicious, then we’ve got two halves, one half and another half make two halves, and two halves make one cake. So two over two, two-halves, is equivalent, or equal, to one. Now let’s look at the first two cakes together. And, as we said, that’s even delicious-er. So I’ve now got four halves. I’ve got one half, another half, another half, and another half. And four halves makes two whole cakes. And now let’s look at all three cakes together. So that’s a total delicious-ity. We’ve got one, two, three, four, five, six halves of a cake and that makes three whole cakes. So some fractions are equivalent to whole numbers. Two-halves is equivalent to one, four-halves is equivalent to two, and six-halves is equivalent to three.
Now let’s put the cakes away and represent the same numbers on a number line. So we’ve split each whole number up into two, or into halves. And each step along our number line here, we’re going up in one-half steps. So starting at zero, we’ve got zero halves. One step later, we’ve got one-half. Two steps later, we’ve got two-halves. Three steps later, we’ve got three-halves. Then we’ve got four-halves and five-halves and six-halves. So thinking back to the cakes for a moment, one cake took us from nothing and it had two halves. That took us all the way up to two-halves, which is equivalent to one. Two cakes had four halves in it, so four-halves is equivalent to two. And three cakes had six halves in it, so six-halves is equivalent to three.
Now we’re probably used to seeing numbers expressed like this on a number line. So zero, a half, one and a half, two, two and a half, three. But what we’ve just seen is that they can be expressed as zero halves, one-half, two-halves, three-halves, four-halves, five-halves, and six-halves. Some of the fractions are equivalent to whole numbers. Or, to put it the other way, whole numbers can also be expressed as equivalent fractions.
So now back to the cakes again, what if we cut them into four pieces each, rather than two. Now we’d have a whole load of quarters of cake. So thinking about one cake, that would be made up of four quarters. Two cakes would be made up of one, two, three, four, five, six, seven, eight quarters. And three cakes would be one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve quarters.
So we’ve now found some different fractions that those whole numbers are equivalent to. And remember, one, not only is equivalent to four-quarters or four-fourths, but it’s equal to two-halves. So two, before we said it was four-halves, it’s now also eight-quarters. And three, we said originally was six-halves, it’s also twelve-quarters.
And just quickly back to the number line again, if we’re splitting each unit into four, into quarters, we can now count along the top here. So zero quarters, one-quarter, two-quarters, three, four, and so on. And the first whole cake represented four-quarters, so that we can see that four-quarters is equivalent to one. When we had two whole cakes, we had eight-quarters, so eight-quarters is equivalent to two. And when we had three whole cakes, that gave us twelve of these quarters of cake, and that means that twelve-quarters is equivalent to three.
So we’ve been looking at fractions that turn out to be equivalent to whole numbers. But before we go, let’s just think about how we can express whole numbers as fractions. Well we can divide the space between zero and one on the number line into equal-sized chunks. For example, halves, thirds, and quarters. That means we can represent a half, or a third, or a quarter, or any other fraction on our number line. But what if we divided the space between zero and one into just one chunk? We could use the same language and see that one is equivalent to one-oneth. So we’ve divided it into one section and that starts off at zero-oneths and ends up at one-oneth. Now this distance here, one whole, is equivalent to one-oneth. Yes, that’s slightly weird terminology. We’d probably just say one over one usually.
Now when we divided up the space into halves or thirds and so on, we were quite happily counting up those little notches, one-half, two-halves, three-halves, four-halves. So now we’re splitting up into ones. We’ve got zero-oneth, one-oneth, two-oneths, three-oneths and four-oneths. Now what we can see, is that that gives us equivalent fractions to our whole numbers. So zero over one is equivalent to zero. One over one is equivalent to one, as we just saw. Two over one is equivalent to two. Three over one is equivalent to three. And four over one is equivalent to four, and so on. In fact, any number can be expressed as itself over one. Fifty-seven, for example, is fifty-seven oneths or fifty-seven over one.
Let’s just summarise what we’ve learnt then. Some fractions turn out to be equivalent to whole numbers. So zero-halves is equivalent to zero, two-halves is equivalent to one, four-halves is equivalent to two, and six-halves is equivalent to three. For all these whole number cases, we can see that the numerator on top is a simple multiple of the denominator on the bottom. In this case, three times two equals six, that’s a whole number. Two times two equals four, and two is a whole number. One times two equals one, and one is a whole number. And zero times two is equal to zero, and zero is a whole number. And any whole number can be expressed as itself over one. So zero is zero over one, one is one over one, two is two over one, three is three over one, four is four over one, and so on.
And we also learnt that one cake is delicious, two cakes are delicious-er, but three cakes is total delicious-ity.