5.01 Integers and the number line

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

$\left\{\dots,-3,-2,-1,0,1,2,3,\dots\right\}${…,−3,−2,−1,0,1,2,3,…}

We know that $1$1 is less than $2$2, that $2$2 is less than $3$3, and so on. We also know that $-11$−11 is closer to $-10$−10 than $4$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$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$1 unit.

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

 

Reading the number line

We can identify an integer on the number line by plotting a point at that integer. On the number line below we can see that the point is at the tick labelled with the integer $-3$−3. This means that the point on the number line is at $-3$−3.

However, not every number line has labels for all the ticks. On the number line below, the location of the point on the number line has not been labelled.

In order to find the missing label we need to find the distance between each tick. 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$0 and $3$3. This means that there is a distance of $3$3 units shared between three gaps, so each gap will be equal to $1$1 unit. Since the point is located one tick to the right of $3$3, the integer we are looking for is $1$1 unit greater than $3$3. So the location of the point is $4$4.

Let's try a similar problem.

 

Worked example

example 1

Where is the point plotted on the number line?

Think: We can find the integer at the point by finding the distance between each tick.

Do: We can see that there are two gaps between $4$4 and $8$8. This tells us that one gap is equal to a distance of $2$2 units. Since the integer at the point is one tick to the right of $4$4, the integer we are looking for is $2$2 units greater than $4$4.

What integer is two units greater than $4$4? The answer is $6$6.

Reflect: When the number line is missing labels, we want to find the distance between ticks in order to find the integer where the point is located. We can do this by comparing the number of gaps to the number of units between adjacent labels.

Notice that the point was plotted at the tick halfway between $4$4 and $8$8. This tells us that the integer represented by the point will be halfway between $4$4 and $8$8. This is another way to find that the point is plotted at $6$6.

 

Moving on the number line

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

We follow similar steps if the point at $1$1 is shifted $3$3 units to the left. In this case, the point will end up at $-2$−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$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$0 by $4$4 places to the right, we move the point by $4$4 units in the positive direction to get $4$4.

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

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

 

 

Opposite integers

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

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

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

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

 

The opposite of the 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$3 is $-3$−3 and the opposite of $-3$−3 is $3$3. How can we use these results to make sense of the opposite of the opposite of $3$3?

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

 

 

Worked example

example 2

Find the opposite of the opposite of $-8$−8.

Think: We can use the fact that the opposite of $-8$−8 is $8$8, and the opposite of $8$8 is $-8$−8.

Do:

The opposite of the opposite of $-8$−8	$=$=	The opposite of (the opposite of $-8$−8)
	$=$=	The opposite of $8$8
	$=$=	$-8$−8

So the opposite of the opposite of $-8$−8 is $-8$−8.

Reflect: Finding the opposite of an integer is like flipping over a coin. Just as it takes two flips of the coin to return it to its starting orientation, finding the opposite of the opposite of an integer returns us to the starting integer.

 

Directed numbers

When we use the number line to understand a real world situation, we need to decide two things: where shall we put $0$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$0, and the sign (either positive or negative) of the integer tells us the direction from $0$0.

 

Worked example

example 3

Let the location of a city be represented by the integer $0$0, and let a point $7$7 km to the east of the city be represented by the integer $7$7. What integer represents the point $4$4 km to the west of the city?

Think: We can use a number line to represent this information. Since west is the opposite of east, the negative integers will represent points to the west of the city.

Do: The number line below shows the city at $0$0 and the point $7$7 km to the east at $7$7. One unit on the number line represents a distance of $1$1 km in the real world.

The point $4$4 km to the west of the city will be represented by the integer $-4$−4 on the number line.

 

Comparing and ordering integers

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.

 

 

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

Here is the same information represented with inequality symbols.

$-4$−4	$<$<	$0$0
$0$0	$<$<	$3$3
$8$8	$>$>	$3$3

 

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, $-4<0<3<8$−4<0<3<8. Here are the integers written in ascending order:

$-4,0,3,8$−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 $-4<0$−4<0 to $0>-4$0>−4 and $0<3$0<3 to $3>0$3>0, we can arrange the integers like so, $8>3>0>-4$8>3>0>−4. Here are the integers written in descending order:

$8,3,0,-4$8,3,0,−4

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

 

 

Practice questions

Question 1

Question 2

Question 3