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Section 4 Rational Numbers \(\mathbb{Q} \)

Definition 4.1.
The rational numbers, \(\mathbb{Q} \text{,}\) are all the integers as well as all the multiplicative inverses of the integers. That is to say that we are going to include the numbers that we multiply each integer by to get to 1.

Often times the rationals are referred to as the fractions and repeating decimals, but let's start by exploring the above definition a bit more.

When we talk about a multiplicative inverse, we are looking for a number that, through multiplication can get us to 1. For example, consider \(2 \left(\frac{1}{2} \right) = 1 \text{.}\) Since by multiplying \(2 \) by \(\frac{1}{2} \) we get a product of \(1 \) we would say that \(\frac{1}{2} \) is the multiplicative inverse of \(2 \text{.}\)

For the rational number section we are going to focus on non-integer rational numbers. This will mean working with fractions. The goal of this section is to explore how the different operations work with fractions. After we have had a chance to explore these operations, we will take a look at how the number line is up to this point.

Subsection 4.1 The Rational Operations

Let's start by listing all the operations that we are going to consider in this section.

  1. Multiplication of fractions
  2. Addition of fractions
  3. Reducing fractions
  4. Division of fractions

Each of the above topics will be presented with how they are computed, a few examples and finally, some exercises to consider.

Subsection 4.2 Multiplication of fractions

Before we get started with multiplication of fractions, we should first introduce some vocabulary. The top and the bottom of the fraction have names:

\begin{equation*} \textrm{fraction} = \frac{\textrm{numerator}}{\textrm{denominator}} \end{equation*}

As shown above, the top of the fraction is called the numerator and the bottom of the fraction is called the denominator. The bar that separates them will be called the division bar. This bar acts as both an operation as well as well as a way to group the numbers (or other mathematical objects) involved.

Multiplication of fractions is done by multiplying the numerators, this will give us the resultant numerator, and the we multiply the denominators, this will give us the resultant denominator. So,

\begin{equation*} \frac{a}{b} \left(\frac{c}{d}\right) = \frac{a \times c} {b \times d} \end{equation*}

Let's take a look at an example.

Given \(\frac{2}{3} \left(\frac{4}{6}\right) \text{.}\) To find the product, we multiply the numerators, \(2(4) \) and then the denominators, \(3(6) \text{.}\) We can use the multiplication table to help us find these. So,

\begin{equation*} \frac{2}{3} \left(\frac{4}{6}\right) = \frac{2(4)}{3(6)} = \frac{8}{18} \end{equation*}

Now it should be noted that we can reduce the fractions before multiplying them, but we will save this conversation till after we talk about fraction reduction.

Subsection 4.3 Addition of fractions

In this next section we are going to look at the process of adding fractions. As the name "fraction" suggests, these numbers can be thought of as a part of a "whole". The idea is that the denominator of a fraction, represents the "whole" that we are working with. So for example \(\frac{1}{5} \) can be thought of as 1 part of 5 pieces. The challenge here is that when we add fractions together, we need to make sure that we are working with the same "whole" for both fractions.

Consider the following,

\begin{equation*} \frac{1}{5} + \frac{2}{7} \end{equation*}
In this mathematical expression, the two fractions are working with two different "wholes". The fraction on the left is in 5, while the fraction on the right is in 7. Now to be able to add these two values together, we need to get them to share how many pieces that they are "out of". We call this the common denominator.

To find this common denominator, we are going to take \(\frac{1}{7} \) of \(\frac{1}{5} \) and then we will take 7 of them. This can be done mathematically by multiplying \(\frac{1}{5}\) by \(\frac{1}{7} \) and then find the product of that and 7. Here is that numerically:

\begin{equation*} 7 \left( \frac{1}{7} \right) \left(\frac{1}{5} \right) = 7 \left( \frac{1}{35} \right) = \frac{7}{35} \end{equation*}

Now the key thing to note here is that on the far left we started with \(7 \left( \frac{1}{7} \right) \left(\frac{1}{5} \right) \text{.}\) But if we have \(\frac{7}{7} \) that is to say, "seven pieces each of which is one seventh of the whole". So, \(\frac{7}{7} = 1 \text{.}\) Giving us that,

\begin{equation*} \frac{1}{5} = \frac{7}{35} \end{equation*}

If we do a similar thing with \(\frac{2}{7} = \left( \frac{5}{5} \right) \left( \frac{2}{7} \right) = \frac{10}{35} \text{,}\) we can see now that they share a denominator (both out of 35). With this common subdivision, we can now add them together.

\begin{equation*} \frac{1}{5} + \frac{2}{7} = \frac{7}{35} + \frac{10}{35} = \frac{17}{35} \end{equation*}

Find \(\frac{4}{7} + \frac{3}{4} \text{.}\)

So, to add these fractions together we end to get them into a common subdivision (common denominator). To do that we will further subdivide each using multiplication, and include enough of these new subdivisions so that we retain the whole. Numerically that would look like this:

\begin{equation*} \frac{5}{7} + \frac{3}{4} = \left( \frac{4}{4} \right) \frac{5}{7} + \left( \frac{7}{7} \right) \frac{3}{4} = \frac{20}{28} + \frac{21}{28} = \frac{41}{28} \end{equation*}
We can see that once the two fractions had a common subdivision of 28, we were able to determine how many 28ths we would have in total.

Now that we can find the common subdivisions and add the fractions together, we should now address how to recapture the original division. This is sometimes possible, but more often, once the addition is finished, we can find a fraction that is reduced to a "base" division of the whole.

Subsection 4.4 Reducing fractions

In the last section, we explored adding fractions together and talked about the idea around what a fraction represents. In this next section we will talk a look at some different ways that we can represent the same fraction, and talk about how to determine the one to use.

Let's start with the one of the calculations that we performed in the last section.

\begin{equation*} \frac{1}{5} = \frac{7}{7} \frac{1}{5} = \frac{7}{35} \end{equation*}

The above shows us that \(\frac{1}{5} = \frac{7}{35} \text{.}\) One way to think about this is that if we cut a large pizza into 5 pieces,

Figure 4.7. A fifth.

and then we cut a second large pizza into 35 pieces.

Figure 4.8. A seven thirty-fifths.

One slice of the first pizza would be equal to 7 slices of the second pizza.

Figure 4.9. A fifth broken into sevenths.

So what we can see here is that \(\frac{7}{35} \) is the same as \(\frac{1}{5}\text{.}\) So we can reduce \(\frac{7}{35} \) to \(\frac{1}{5}\text{.}\)

The next question that we should ask here is, "how to we preform this reduction". The key is going to be finding a common factor. If we can find a number that divides the numerator as well as the denominator, this common value can be reduced from the entire fraction.

Let's take a look at anther fraction. This time instead of starting with the reduced fraction, let's start with the unreduced one. Consider, \(\frac{9}{27} \text{.}\)

Figure 4.10. Nine twenty-sevenths.

The plan here is to find a factor of both 9 and 27. Now it should be noted that there may be more than one. But we can start with one of them and continue to reduce if necessary. In this case 3 and 9 both divide the numerator and the denominator. For the purpose of the example, let's work with 3. So mathematical we will have,

\begin{equation*} \frac{9}{27} = \frac{3(3)}{3(9)} = \left( \frac{3}{3} \right) \frac{3}{9} = \frac{3}{9} \end{equation*}

Figure 4.11. Nine twenty-sevenths.
Figure 4.12. Three Ninths.

And we can take this one step further and reduce again by noting that 3 is a factor of 3 and 3 is a factor of 9.

\begin{equation*} \frac{3}{9} = \frac{3(1)}{3(3)} = \left( \frac{3}{3} \right) \frac{1}{3} = \frac{1}{3} \end{equation*}

Figure 4.13. Three Ninths.

So we can now see that, \(\frac{9}{27} = \frac{1}{3} \) and we say that we reduced\(\frac{9}{27} \) to \(\frac{1}{3} \text{.}\)

Subsection 4.5 Division of fractions

Moving forward we are going to take a look at the process of dividing fractions. For this we are going to look at the structure of the notation first, then take a look at some of examples of the process in motion. Let's start with the notation:

\begin{equation*} \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a(d)}{b(c)} \end{equation*}
Now one important thing to note is that any of the values in the above equation could be 1. In some cases, that won't impact how the notation looks, but if either \(b\) or \(d\) is 1, traditionally they would not appear in the notation and the notation will look a bit different. For example, if \(b = 1\text{,}\)
\begin{equation*} \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{\frac{c}{d}} \end{equation*}
and if \(d = 1\text{,}\)
\begin{equation*} \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{\frac{a}{b}}{c} \end{equation*}
In practice, we will need to watch for these situations, since to compute the division, we will need to reintroduce the 1 back into the notation.

Find the following, using the process outlined above.

Compute, \(\displaystyle \frac{\frac{3}{2}}{\frac{4}{7}}. \)

Solution
To find this using the above pattern we,
\begin{equation*} \frac{\frac{3}{2}}{\frac{4}{7}} = \frac{3(7)}{2(4)} = \frac{21}{8} \end{equation*}

Since the numerator and the denominator of the fraction that we ended with have no common factors, this fraction is reduced and we are done.

Subsection 4.6 What Else is There?

Before we move on, let's look back at where we started.

Figure 4.16. Tree of the number hierarchy .

So far we have talked about natural numbers and how to add them. We have talked about the integers and some of the operations that we can apply to them. Finally, we introduced the relational numbers and have explored what working with those numbers is like. The question that this poses for us next is, "what else is there?"