1.01 Integers and absolute value

Identifying and representing integers

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.

Locating integers on 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 mark labeled 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 marks. On the number line below, the location of the point on the number line has not been labeled.

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$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 mark 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

Question 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 mark.

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 mark 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 marks 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 mark 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.

 

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.

This applet lets you visualize the idea of opposites. Slide point $A$A and see its opposite (point $B$B) move.

Created with Geogebra

 

Zero pairs

The number line isn't the only way to represent integers. Another common way is using two different colored counters. One color for positives and one color for negatives. If we have the same number of positive and negative counters, we call this a zero pair as each counter is canceled out resulting in zero. For example, below we have $2$2 positive counters ($+2$+2) and $2$2 negative counters ($-2$−2), so we have a zero pair. This tells us that $+2+(-2)=0$+2+(−2)=0.

Exploration

The applet below allows us to use counters to explore integers. To use this applet drag over a positive (purple) counter and if desired, click on it to change it to a negative (red) counter. See the text at the top for the value of all the counters together.

Use the guiding questions below:

1. Drag over a desired of counters and make them all negative. How many positive counters do you need to make a zero pair?
2. Why do you think we use different colors for positive and negative?
3. How might zero pairs help us in mathematics?
4. Where might we see zero pairs in the real world?

Created with Geogebra Credit: Duane Habecker

 

Practical applications of integers

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 signed numbers that we identify with locations in the 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

Question 2

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.

 

Practice questions

Question 3

Question 4

Question 5

Comparing and ordering integers

We have now seen that an integer is a whole number or its opposite:

$...,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,...$...,−6,−5,−4,−3,−2,−1,0,1,2,3,4,5,6,...

Notice that zero ($0$0) is an integer but it is neither positive or negative.

 

Comparing the sizes of integers

In what 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.

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.

We've seen before that the further an integer is to the right on a number line, the larger the integer is.

 

Worked examples

Question 6

Arrange $0,7,-1$0,7,−1, and $6$6 in ascending order.

Think:

Let's remember the number line: 

$-1$−1 is the smallest number as it is furthest to the left on the number line. $7$7 is furthest to the right, so it is the biggest.

Do: The numbers in ascending order are $-1,0,6,7$−1,0,6,7.

 

question 7

Arrange $-15,-2,-7$−15,−2,−7 and $-5$−5 in descending order.

Think: $-2$−2 is furthest to the right, so it is the biggest number. $-15$−15 is the smallest number as it is furthest to the left.

Do: $-2,-5,-7,-15$−2,−5,−7,−15

 

Practice questions

Question 8

Question 9

Question 10

Absolute value of integers

Let's consider the following situation:

A scuba diver is diving at a depth of $-50$−50 feet. At the same time, a helicopter pilot is flying overhead at $30$30 feet above the surface. Which person is closer to sea level?

Although the scuba diver is at an altitude much lower than the helicopter, the helicopter pilot is closer to sea level.

When making this comparison, we are considering the absolute value of each measurement. The absolute value of a number is the distance from the number to zero on the number line.

Exploration

The applet below shows the absolute value, or distance from zero for different integers on the number line. Move the point left and right and consider the following questions:

1. What do you notice about the absolute value of a positive number?
2. What do you notice about the absolute value of a negative number?

We can see that the absolute value of a positive number is the number itself. However, the absolute value of a negative number is its opposite. This is because the distance is always a positive number. This applies to all numbers on the number line!

The mathematical symbol for absolute value is "$\text{| |}$| |". For example, we would read "$\left|-6\right|$|−6|" as "the absolute value of negative six."

 

Worked example

Question 11

Evaluate: Which of the following are smaller than $\left|-20\right|$|−20|?

A) $-15$−15    B) $\left|-30\right|$|−30|     C) $\left|-5\right|$|−5|     D) $21$21

Think: We need to evaluate each of these terms, then compare them to $\left|-20\right|$|−20|.

Do: Let's start by evaluating all the absolute values:

$\left|-20\right|=20$|−20|=20, $\left|-30\right|=30$|−30|=30 and $\left|-5\right|=5$|−5|=5

Which of the four possible answers are smaller than $20$20?

So $\left|-20\right|$|−20| is greater than A) $-15$−15 and C) $\left|-5\right|$|−5|

 

Practice questions

Question 12

Question 13

Question 14