Section 31.5 Going Deeper: Remainders and Decimals
In this section, we mostly avoided remainders so that the focus would be on the primary concept of division. We're going to take a deeper dive into this topic to see how this idea relates back to one of the core concepts of fractions, and also take a deeper look at the decimals associated with those fractions.
It's often the case that division problems don't work out evenly. In elementary school, children learn that when you have leftover pieces, it's called a remainder. And in the long division process, they are often taught to write their answers with an R, so that a calculation like \(11 \div 3\) results in \(3 \, \text{R}2\text{.}\) What's interesting about this notation is that there's no other place in mathematics where we use it.
The main tool for working with remainders is the idea of parts of a whole, which we discussed in detail in Section 17.1. The basic idea is that the fraction means to take a whole unit (often represented by a circle) and cut it into pieces, and then the quantity is the amount you would have if you had of those pieces. breaking a whole unit into pieces allows us to make groupings and equal-sized distributions when the calculation doesn't work out evenly.
For example, in the calculation \(11 \div 3\text{,}\) if we were thinking about making groupings, we would have three full groups and two out of three pieces to make another group. The language of "two out of three pieces" is exactly mirroring the language we use with parts of a whole. Here's what looks like as a parts of a whole diagram:
This gives us the explicit image of having three full groups and two out of three pieces needed for another group. If we wanted to view this as equal distribution, we would have a diagram that looks more like this:
In both situations, the need for having parts of a whole comes up naturally.
In Section 22.2, there were a couple exercises on the worksheet that discussed the conversion of fractions to decimals. In those sections, we simply assumed that you had sufficient familiarity with decimals that you would be able to come up with the answers and they would make sense to you. But we didn't really talk about what was going on with those decimals.
Recall that decimals are really just fractions with powers of ten in the denominator. And so in some sense, decimal expansions are just another way of trying to express a division calculation. In most practical settings, it's far more likely that remainders will be expressed as decimals instead of fractions. There are many reasons for this, including the simple fact that we use computers and calculators and they generally give us decimal answers.
Some fractions have terminating decimal expansions. This means that at some point the decimal stops and gives us an exact answer. The simplest examples are the ones where we start with a power of ten in the denominator. For example, \(\frac{3}{10} = 0.3\) and \(\frac{51}{100} = 0.51\text{.}\) These are quite literally the direct translation of fractions to decimals.
In other cases, we may not initially have a power of ten in the denominator, but we can get there by finding a different representation of that fraction. Here are a couple examples of this:
In both of these situations, there's a way to cut up our parts of a whole diagram so that we have a power of te pieces. We can visualize this with but with it's very difficult to see all subdivisions so we didn't draw all of them. But you should get the general idea from the diagram.
Next, we have fractions for which we simply cannot use integers to get that representation. For example, with the fraction \(\frac{1}{3}\) there is no integer value \(x\) so that is equal to a power of ten:
We can see that no matter how far out we go, we'll never get an integer answer.
These values were written as mixed numbers for a reason. That reason is that it makes it clear that even though there is no integer value, there's a definite pattern that's developing. There is some number of 3s but then there's always one out of three pieces left over. And this happens over and over again. Because of this, we know that the decimal expansion for \(\frac{1}{3}\) is a repeating decimal.
The idea of a repeating decimal is that there is some pattern of numbers that repeats indefinitely. The pattern is not limited to a single digit, nor does it have to start immediately after the decimal. The feature is that the decimal eventually falls into a fixed pattern. There are two notations for repeating decimals. The implicit notation leaves it to the person reading it to identify the pattern. For example, here is the decimal expansion of
In this case, the pattern is pretty easy to spot. But there's also an explicit notation that specifically marks out the pattern. This is good for patterns that take longer to repeat:
We're going to spend some time exploring the nature of these repeating decimals by thinking about fractions. Can we explain why those decimals repeat? If we look at the column of values we had for the fraction we can see that keeps appearing over and over again. The process being shown above in equations is a bit easier to understand in pictures. We are trying to calculate \(\frac{1}{3} = 1 \div 3\text{.}\) We will represent by a bar:
We want to divide this into three equal pieces. However, decimals constrain us to having to break things into ten pieces. So we'll do that and do our best to create three equal groupings without breaking things up any further:
We get three groups and then one leftover piece. In order to try to create equal groupings, we're going to need to break this piece up. But again, the limitation of using decimals is that we can only break this up into ten pieces. But taking one item and breaking it into ten pieces is exactly what we just did, so we already know the result. We're going to end up with three more groupings, and then one leftover piece. And if we were to try to take that leftover piece and break it into ten again, it's just going to be the same thing forever.
This pattern of breaking things into ten, dividing, and looking at the remainder is what leads to repeating decimals. Notice that breaking into ten pieces is the same as multiplying the number of pieces by ten, so that we can interpret this process as multiplying by ten, dividing, and looking at the remainder. We can see that the size of the remainder is what controls the next step in the process.
Let's look at another example that has a little but more going on. Let's say we were trying to get a decimal for \(\frac{2}{11}\)
The of the mixed number is our first decimal value, and the 9 in the numerator is our remainder. We can then repeat this process to get the next decimal:
But notice that we have a remainder of 2, which is a number that we've already seen. In fact, we explicitly see the \(\frac{2}{11}\) that we started with in the calculation. So we can return back to that value to create the loop that gives us the repeating decimal expansion!
We're going to condense the notation a bit so that it takes up less space. Since each arrow represents multiplying by 10 and dividing by the denominator, we're just going to give the answers at each step. So the above diagram can be reduced to this one:
Let's take a look at an example with a longer pattern. Here is what we get when working with \(\frac{1}{7}\text{:}\)
In this case, we have a cycle that runs through every single remainder. This means that we have the full information about the decimal expansions when the denominator is 7. If we wanted the decimal expansion of \(\frac{2}{7}\) we start from the 2 and write down the integer parts of the mixed numbers in order: \(\frac{2}{7} = 0.\overline{285714}\text{.}\)
Different denominators will have different diagrams. Here is what the denominator of 6 looks like. Notice that we don't reduce the fractions (because that changes the value of the numerator and the denominator).
In this case, there are three separate pieces, two of which repeat (but don't repeat the entire chain) and one that terminates. So it's possible to get denominators that multiple possibilities.
Here are a few questions to consider. If you were to be given some other denominator (such as 12):
Can you predict how many fractions will terminate and how many won't?
Can you predict how many different cycles you will have?
Can you predict which cycles will have full loops and which ones will have partial loops?
You actually have all the tools you need to explore these questions. You may even be able to come up with some conjectures and have some explanations for why you think your ideas are correct. At this point, you may not have the tools to fully explain everything, but that's part of the learning process. As you get interested in new ideas, that's the starting point for developing new tools and new skills to try to find out more.