Section 24.5 Going Deeper: Intervals on the Number Line
In this section, we introduced a couple geometric representation of the numbers. We are going to focus our attention on the number line picture in the context of thinking about inequalities. Suppose that we have the inequality \(x \gt a\text{.}\) Here's how we would represent that on the number line:
The circle around the indicates that we do not want to include in the interval. It represents a "hole" in the arrow at that point. If we wanted to graph \(x \geq a\text{,}\) then it would look like this:
We could draw similar diagrams for \(x \lt a\) and \(x \leq a\text{:}\)
The arrows on the end of the thickened lines indicates that it extends forever in the indicated direction. This introduces the concept of infinity, which is denoted \(\infty\text{.}\) We are going to have to leave this at an intuitive level, as infinity turns out to be an incredibly complicated and nuanced topic. The key fact is that infinity is not a number. It is not a part of the number line. For our purposes, it represents the idea of continuing along in the same direction indefinitely. We also have an infinity in both directions, where \(\infty\) (sometimes written \(+\infty\) for emphasis) is off to the right and \(-\infty\) is off to the left. The diagram below is an attempt to convey this idea:
There are times that we want to restrict our values on two sides. For example, if you want a number between 1 and 5 (not restricting yourself to integers), you're actually asking for the number to meet two conditions at the same time: (1) The number must be greater than 1; (2) The number must be less than 5. This is an example of a compound inequality.
Graphically, this is not too hard to think about, as the betweenness property is captured intuitively. Notice that we wrote this as a single inequality. Even though this looks like just one inequality, it's actually shorthand for two inequalities: \(1 \lt x\) and \(x \lt 5\text{.}\)
It's important that the direction of the inequality is consistent. If the inequalities get turned around, we treat it as a meaningless statement. For example, \(1 \lt x \gt 8\) is not interpreted as \(1 \lt x\) and \(x \gt 8\text{.}\) If we look at what that would mean, we can see that there's a bit of redundancy in that interpretation. We would be looking for values to the right of 1 that are also to the right of 8. But every number to the right of 8 is already to the right of 1, so the first part doesn't really add anything but confusion. So it leads to cleaner communication to declare that such combinations are not allowed.
Sets of these types are known as intervals. They play an important role in understanding and describing functions and other mathematical objects. We have seen that we can describe intervals using diagrams and inequalities, and it turns out that there is one more method that we use, which is known as interval notation. The value of interval notation is that it allows us to describe an interval using symbols, but without introducing a variable. When talking about a number between and it's not necessarily helpful for us to arbitrarily pick a symbol to represent that quantity. This is especially true for more complicated situations where we're already working with several other variables.
Interval notation is easiest to understand through to the number line diagrams that we've been drawing. There are just a couple ideas that we need. The first idea is that we always want to think about the diagram from left to right. And this is easy to remember because your notation will be written out from left to right. The second is that we use a round bracket to exclude the endpoint and a square bracket to include it. Here is the example we were working with before.
Here is the same example, except that we're including the point 5 in our set.
As you can see, there are four different combinations of symbols that we might have, depending on whether each of the endpoints are included or excluded. Here they are presented together:
What about intervals that go off to infinity? The same ideas hold. In particular, we always use round brackets around the side with infinity because infinity is not actually included in the interval. We also need to use the appropriate sign on the infinity, depending on which side we're considering.
We also have one more case where both sides go to infinity. This is basically saying that we want to include every real number. Fortunately, this case follows all the same ideas as before, so the interval notation should not be a surprise.
With these examples in mind, you should be able to represent any interval using three different representations: a number line diagram, an inequality (or compound inequality), and interval notation. You should also be able to move freely between the different forms. For example, if you're given a number line diagram, you should be able to translate that into an inequality as well as writing it in interval notation.
Interval notation is used very often as part of the larger language of mathematics. While the notation is easy to understand on its own, it can be difficult to understand how we use it in applications without placing it in a larger context. Those larger contexts require some ideas that go beyond what we've developed. We will use one example that involves describing the behavior of a function. It will hopefully be intuitive, but don't worry too much if it's not. Consider the following graph:
The function is increasing on the interval \((2,5)\text{.}\) In other words, the function is increasing when \(2 \lt x \lt 5\text{.}\)