Foundational Mathematics for Corequisite Courses: Succeeding at College Mathematics

An important aspect of mathematical thinking is the ability to represent ideas in several different ways. Even something as simple as a number can have multiple representations, and those different representations have different applications. And it may seem simple, but even just representing numbers as a collection of dots or squares is a technique used in higher levels of mathematics to help discover patterns and prove mathematical relationships.

We’ve already used two different representations of numbers in this book. The first is the number line, and the second is the place value system. We’re going to spend some time exploring these ideas a little more closely to deepen our mathematical thinking.

The number line is the ordering of numbers in a straight line based on their values. Traditionally, this line is drawn horizontally, though there is no reason that this has to be the case. In fact, the coordinate plane is actually two number lines drawn together where one of them is drawn vertically.

We can think of the number of as being at the middle of the number line, with positive numbers to the right of it and negative numbers to the left of it.

As we move away from the numbers get bigger in size. The phrase "bigger in size" is important, because the words "bigger" and "smaller" on their own create confusion with negative numbers. The size of a number is often called its absolute value (or magnitude). Intuitively, is a "big" number by size, but it can also represent a large deficit instead of a large quantity. To avoid that confusion, we use the phrases "greater than" and "less than" when comparing numbers. These words take away the possibility of misinterpreting the comparison of two numbers.

The second representation of numbers that we’ve used is the place value system. This is the way that you’re already familiar with writing numbers, and we discussed it briefly when we looked at decimals. But there is another way of looking at these numbers that have an important generalization to other mathematical ideas.

In early elementary school, a common manipulative that’s used to help teach children numbers are known as base-10 blocks. These are basically just plastic or wooden pieces that come in three different shapes.

Furthermore, we have containers that the various pieces fit into. We have a tray that fits ten units and a tray that fits ten rods. We will represent these by uncolored boxes, and as pieces are put in, they will be colored in.

Notice that an empty unit cube tray looks a lot like a tens rod, and that a tens rod tray looks like a hundreds flat. This is because in practice, students would get to exchange their full tray of unit cubes for a rod (or the other way around), and this helps to reinforce arithmetic using the system. (We’ll see this again in a little bit.)

If we wanted to represent a number, we could simply pick the appropriate number of each piece. Here is an example:

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Algebra is a skill, which means it requires practice to become proficient. But it will take more than rote repetition to get there. Deliberate practice is the thoughtful repetition of a task. For each of these sections, you will be given a list of specific skills or ideas to focus on as you practice thinking through the problems.

The ability to see the same idea from multiple perspectives creates the opportunity to apply different approaches to solving problems. In this section, we saw two different ways to represent numbers.

The number line is a purely geometric framework. It turns out that the Greek mathematicians thought of numbers this way, but in an even more strict sense. If they wanted to compare the numbers and then they would (essentially) say that is longer than But how would this work with negative numbers? As it turns out, the Greeks never really bothered with negative numbers. To them, they didn’t exist because they only worked with counting numbers and lengths. Since the number line contains negative numbers, we will have an additional set of tools that the Greeks did not have when it comes to thinking about numbers and mathematical ideas.

The base-10 blocks gives us another geometric framework, but it ties more closely with our sense of how we represent numbers instead of giving us insights into the numbers themselves. It may seem obvious to us because we’ve worked with numbers this way our whole lives. But not every system of writing numbers uses a place value system. For example, Roman numerals are notoriously difficult for students to learn because it’s built around rules that are not always intuitive and sometimes feel arbitrary.

As you come to understand more mathematics, you can start to develop a more flexible mindset for looking at ideas. This can often lead to new insights and more interesting questions.

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 \(2 \lt x \lt 5\text{.}\)