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Section 24.1 Visualizing Numbers

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.

We can visualize any part of the real line that we want using any sense of scale that we want. It does not always need to include zero. If the actual location is important, then you should try to use an equal spacing. But sometimes all we need is a symbolic representation of the locations. Regardless, it is very important that we always keep the order the same, especially with negative numbers. For a portion of the line that only has negative numbers, remember that going to the left makes the numbers more negative. Here are some examples:

  • A number line with zero and positive numbers using an increment of 10:

  • A number line with negative numbers using an increment of 50:

  • A representation of the relative locations of the numbers -200 and 1:

  • An approximation of the locations of the numbers -15, 0, and 100:

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.

Definition 24.1.1. Greater Than and Less Than.

For any two numbers \(a\) and \(b\text{,}\) we say that \(a\) is greater than \(b\) (written symbolically as \(a \gt b\)) if \(a\) is to the right of \(b\) on the number line. We say that \(a\) is less than \(b\) (written symbolically as \(a \lt b\)) if \(a\) is to the left of \(b\) on the number line.

Using the number line to visualize the locations of numbers is a natural approach to comparing numbers. Some students learn a slightly complex set of rules for comparing numbers:

  • If both numbers are positive, then the bigger number is greater than the smaller number.

  • If one number is positive and the other is negative, then the positive number is greater than the negative number.

  • If both numbers are negative, then the bigger number is less than the smaller number.

While this is accurate, it ends up causing confusion because it turns it into a practice of rule-following rather than developing an understanding.

Activity 24.1.1. Ordering Numbers.

Ordering numbers is a skill that simply requires some practice. The best intuition comes from starting at and thinking about the number of steps in which direction is required to reach a value. For example, starting from 0, to get to the number -45 you would have to move to the left, and on the way you'll pass -21. And to reach the number 37 you would start from 0 and move to the right, and when you get there, you won't have yet passed 68. This could be represented on a number line.

Try it!

Put the numbers 36, 11, -58, -3, and 132 in order on a number line.

Solution.

Activity 24.1.2. Comparing Numbers.

Once you are comfortable with locating numbers on the number line, then comparisons thinking about "to the left of" (for "less than") and "to the right of" (for "greater than") are straightforward. Based on the number line diagram above, we can immediately check the following comparisons:

\begin{equation*} -45 \lt 37 \qquad 68 \gt 0 \qquad -21 \gt -45 \qquad -21 \lt 68 \end{equation*}

Try it!

Write all mathematical sentences that compare the numbers -14, 10, and 31.

Solution.
\begin{equation*} \begin{array}{ccccc} -14 \lt 10 \amp \phantom{.} \hspace{1cm} \phantom{.} \amp -14 \lt 31 \amp \phantom{.} \hspace{1cm} \phantom{.} \amp 10 \lt 31 \\ 10 \gt -14 \amp \amp 31 \gt -14 \amp \amp 31 \gt 10 \end{array} \end{equation*}

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:

Activity 24.1.3. Representing Integers with Base-10 Blocks.

You should be able to go back and forth between a number and combinations of base-10 blocks in order to represent any value.

Try it!

Represent the number 52 using base-10 blocks.

Solution.