7.01 Integers and the number line

The integers are made up of the positive and negative whole numbers, as well as the number 0. Here is a partial list of the set of integers, which continues up toward +\infty and down toward -\infty.

\left\{\ldots ,-3,\,-2,\,-1,\,0,\,1,\,2,\,3,\,\ldots\right\}

We know that 1 is less than 2, that 2 is less than 3, and so on. We also know that -11 is closer to -10 than 4. All of this information can be represented visually using a number line, shown below. The arrows at each end indicate that the line extends infinitely in the positive and negative directions.

This number line uses a few rules to help us keep track of which integers are greater or lesser and by how much. On the number line above, the values of the integers are increasing from left to right, with 0 separating the positive integers and the negative integers. On this number line, each tick has been labelled and we can see that the gap between ticks is 1 unit.

Looking at the number line, we can also see that the integer 2 is three ticks to the left of -1. This tells us that 2 is greater than -1 by 3 units.

However, not every number line has labels for all the marks. In order to find the missing label we need to find the distance between each mark. We can do this by comparing the number of gaps between the existing labels with the distance between the integer labels.

In this case, we can see that there are three gaps between the labels of 0 and 3. This means that there is a distance of 3 units shared between three gaps, so each gap will be equal to 1 unit. Since the point is located one mark to the right of 3, the integer we are looking for is 1 unit greater than 3. So the location of the point is 4.

We can also use the number line to find the location of a point after it has been moved.

For example, if a point located at 1 is then shifted 3 units to the right, it will end up at 4. We can see this by moving the point on the number line.

We follow similar steps if the point at 1 is shifted 3 units to the left. In this case, the point will end up at -2.

On this number line, shifting the point to the left will decrease the integer value, and shifting the point to the right will increase the integer value.

It should be noted that if we shift a point at 0 by some number of units, the integer we end up at will always represent the number of units and the direction in which we shifted the point.

If we shift the point at 0 by 4 places to the right, we move the point by 4 units in the positive direction to get 4.

If we shift the point at 0 by 4 places to the left, we move the point by 4 units in the negative direction to get -4.

Notice that we referred to the directions as positive and negative. We can do this because the integers to the right of 0 are positive and the integers to the left of 0 are negative on this number line.

Examples

When looking for the opposite of a meaning we usually try to reverse it. For example, the opposite of left is right because we can reverse moving to the left by moving to the right. When trying to find opposites on a number line, we can use the same approach.

Consider the integer 3. On this number line, the integer 3 represents "the location 3 units to the right of 0", shown in green. The opposite of this would involve reversing the direction. In other words, the opposite would be "the location 3 units to the left of 0", shown in blue.

This example shows that the opposite of the integer 3 is the integer -3.

We can use the same method to find the opposite of a negative integer. Consider the integer -3. This number represents "the location 3 units to the left of 0", shown in green, so its opposite will be "the location 3 units to the right of 0", shown in blue.

Two integers are opposite if their locations on the number line are the same distance from 0, but on different sides of 0.

What about 0 itself? We can think about the opposite of 0 as being the number -0. But since -0 is the same as 0, the opposite of 0 is again 0. That is, the integer 0 is its own opposite.

Every integer has an opposite, which is also an integer. This means that the opposite of an integer also has an opposite, which we can call the opposite of the opposite of that initial integer.

Look back at the examples in the previous section, where we found that the opposite of 3 is -3 and the opposite of -3 is 3. How can we use these results to make sense of the opposite of the opposite of 3?

In the statement "the opposite of the opposite of 3", we can replace "the opposite of 3" with -3 to get the equivalent statement "the opposite of -3". But we know that "the opposite of -3" is 3, so we can return to the original statement and say that the opposite of the opposite of 3 is 3.

When we use the number line to understand a real world situation, we need to decide two things: where shall we put 0, and what direction shall be positive? In this way, the integers on the number line become directed numbers that we identify with locations in real world. The size of the integer tells us the distance from 0, and the sign (either positive or negative) of the integer tells us the direction from 0.

Examples

In the examples we have seen so far the integers are increasing from left to right on the number line. This means that when we compare two integers, the integer further to the right is always greater and the integer further to the left will always be lesser.

Inequality symbols can be used to show the relative ordering of two integers on the number line.

The symbol < represents the phrase is less than. For example, -3 is less than 4 can be represented by -3 \,\lt\, 4.

The symbol > represents the phrase is greater than. For example, 4 is greater than -3 can be represented by 4 \,>\, -3.

We can use a number line to clearly see the relationship between different integers.

• Since the point at -4 is to the left of 0, we know that -4 is less than 0.
• Since the point at 0 is to the left of 3, we know that 0 is less than 3.
• Since the point at 8 is to the right of 3, we know that 8 is greater than 3.

Here is the same information represented with inequality symbols.

\begin{aligned} -4 & \lt 0 \\ 0 & \lt 3 \\ 8 & \gt 3 \end{aligned}

We can arrange these four integers in ascending order by writing them left to right in order from the least integer to the greatest integer. We can use the < symbol to arrange the integers like so, -4 \,<\, 0 \,<\, 3 \,<\,8. Here are the integers written in ascending order:

-4, 0, 3, 8

Now using the > symbol, we can arrange these same integers in descending order, written left to right from greatest to least. Rearranging -4 \,<\, 0 to 0 \,>\,-4 and 0 \,<\, 3 to 3 \,>\, 0, we can arrange the integers like so, 8 \,>\, 3 \,>\, 0 \,>\, -4. Here are the integers written in descending order:

8, 3, 0, -4

Notice that the descending order of the integers is the reverse of the ascending order.

There is more than one orientation that we can choose for a number line. Most often we will use a number line that is increasing from left to right. However, this is not the only type we can encounter.

For example, we could have a number line that decreases from left to right.

Which direction is positive: left or right? For any pair of integers, will the integer to the left or the right be greater?

Examples