Remember that we use the notation:

[latex]a < b[/latex] (read “a is less than b”) when [latex]a[/latex] is to the left of [latex]b[/latex] on the number line.

[latex]a > b[/latex] (read “a is greater than b”) when [latex]a[/latex] is to the right of [latex]b[/latex] on the number line.

Now we need to extend the number line which shows the whole numbers to include negative numbers, too. The numbers marked by points in Figure 1.3.1 are called the integers. The integers are the numbers  {…}−3,−2,−1,0,1,2,3{…}

All the marked numbers are called integers.

You may have noticed that, on the number line, the negative numbers are a mirror image of the positive numbers, with zero in the middle. Because the numbers 2 and −2 are the same distance from zero, they are called opposites. The opposite of 2 is −2 and the opposite of −2 is 2.

Sometimes in algebra, the same symbol has different meanings. Just like some words in English, the specific meaning becomes clear by looking at how it is used. You have seen the symbol “−” used in three different ways.

When evaluating the opposite of a variable, we must be very careful. Without knowing whether the variable represents a positive or negative number, we don’t know whether [latex]x[/latex] is positive or negative. We can see this in the below figure.

We saw that numbers such as 2 and −2 are opposites because they are the same distance from 0 on the number line. They are both two units from 0. The distance between 0 and any number on the number line is called the absolute value of that number.

• [latex]−5[/latex] is [latex]5[/latex] units away from [latex]0[/latex], so [latex]\left|−5\right|=5[/latex].
• 5 is 5 units away from 0, so [latex]\left|5\right|=5[/latex].

The figure below illustrates this idea. The integers 5 and −5 are 5 units away from 0.

[latex]\xleftarrow[{-5\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}]{\overbrace{\;5\;units\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}}0\xrightarrow[{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;5}]{\overbrace{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;5units}}[/latex]

The absolute value of a number is never negative (because distance cannot be negative). The only number with absolute value equal to zero is the number zero itself, because the distance from 0 to 0 on the number line is zero units.

We now add absolute value bars to our list of grouping symbols. When we use the order of operations, first we simplify inside the absolute value bars as much as possible, then we take the absolute value of the resulting number.

In the next example, we simplify the expressions inside absolute value bars first, just like we do with parentheses.

Add Integers

Most students are comfortable with the addition and subtraction facts for positive numbers. However doing addition or subtraction with both positive and negative numbers may be more challenging.

We will use two colour counters to model the addition and subtraction of negatives so that you can visualize the procedures instead of memorizing the rules.

We let one colour (blue) represent positivity. The other colour (red) will represent the negatives. If we have one positive counter and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero.

We will use the counters to show how to add the four addition facts using the numbers 5,−5 and 3,−3.

To add [latex]5+3[/latex], we realize that [latex]5+3[/latex] means the sum of 5 and 3.

We start with 5 positives.
And then we add 3 positives.
We now have 8 positives. The sum of 5 and 3 is 8.

Now we will add [latex]-5+(-3)[/latex]. Watch for similarities to the last example [latex]5+3=8[/latex].

To add [latex]-5+(-3)[/latex], we realize this means the sum of −5 and −3.

We start with 5 negatives.
And then we add 3 negatives.
We now have 8 negatives. The sum of −5 and −3 is −8.

In what ways were these first two examples similar?

• The first example adds 5 positives and 3 positives—both positives.
• The second example adds 5 negatives and 3 negatives—both negatives.

In each case we got 8—either 8 positives or 8 negatives.

When the signs were the same, the counters were all the same colour, and so we added them.

Example 8

Add:

1. [latex]1+4[/latex]
2. [latex]-1+(-4)[/latex] 

Solution

a.

1 positive plus 4 positives is 5 positives.

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b.

 1 negative plus 4 negatives is 5 negatives.

	−5 + 3 means the sum of −5 and 3.
We start with 5 negatives.
And then we add 3 positives.
We remove any neutral pairs.
We have 2 negatives left.
The sum of −5 and 3 is −2.	[latex]−5+3 =−2[/latex]

Notice that there were more negatives than positives, so the result was negative.

Let’s now add the last combination, [latex]5+(−3)[/latex].

	5 + (−3) means the sum of 5 and −3.
We start with 5 positives.
And then we add 3 negatives.
We remove any neutral pairs.
We have 2 positives left.
The sum of 5 and −3 is 2.	[latex]5+(−3)=2[/latex]

When we use counters to model addition of positive and negative integers, it is easy to see whether there are more positive or more negative counters. So we know whether the sum will be positive or negative.

Now that we have added small positive and negative integers with a model, we can visualize the model in our minds to simplify problems with any numbers.

When you need to add numbers such as [latex]37+(-53)[/latex], you really don’t want to have to count out 37 blue counters and 53 red counters. With the model in your mind, can you visualize what you would do to solve the problem?

Picture 37 blue counters with 53 red counters lined up underneath. Since there would be more red (negative) counters than blue (positive) counters, the sum would be negative. How many more red counters would there be? Because [latex]53-37=16[/latex], there are 16 more red counters.

Therefore, the sum of [latex]37+(-53)[/latex] is −16.

Let’s try another one. We’ll add [latex]-74+(−27)[/latex]. Again, imagine 74 red counters and 27 more red counters, so we’d have 101 red counters. This means the sum is −101.

Let’s look again at the results of adding the different combinations of 5,−5 and 3,−3.

Addition of Positive and Negative Integers

When the signs are different, some of the counters would make neutral pairs, so subtract to see how many are left.

Visualize the model as you simplify the expressions in the following examples.

The techniques used up to now extend to more complicated problems, like the ones we’ve seen before. Remember to follow the order of operations!

We will continue to use counters to model subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.

Perhaps when you were younger, you read “5−3” as “5 take away 3”. When you use counters, you can think of subtraction the same way!

We will model the four subtraction facts using the numbers 5 and 3.

[latex]5-3[/latex]        [latex]-5-(-3)[/latex]         [latex]-5-3[/latex]         [latex]5-(-3)[/latex]

To subtract 5-3, we restate the problem as “5 takes away 3”.

We start with 5 positives.
We ‘take away’ 3 positives.
We have 2 positives left.	The difference of 5 and 3 is 2.

Now we will subtract [latex]-5-(-3)[/latex]. Watch for similarities to the last example [latex]5-3=2[/latex].

To subtract [latex]-5-(-3)[/latex], we restate this as “−5 take away −3”.

We start with 5 negatives.
We ‘take away’ 3 negatives.
We have 2 negatives left.	The difference of −5 and −3 is −2.

Notice that these two examples are much alike: The first example, we subtract 3 positives from 5 positives and end up with 2 positives.

In the second example, we subtract 3 negatives from 5 negatives and end up with 2 negatives.

Each example used counters of only one colour, and the “take away” model of subtraction was easy to apply.

• To subtract [latex]-5-3[/latex], we restate it as −5 take away 3.

We start with 5 negatives. We need to take away 3 positives, but we do not have any positives to take away.

Remember, a neutral pair has value zero. If we add 0 to 5 its value is still 5. We add neutral pairs to the 5 negatives until we get 3 positives to take away.

We start with 5 negatives.
We now add the neutrals needed to get 3 positives.
We remove the 3 positives.
We are left with 8 negatives.
The difference of −5 and 3 is −8.	[latex]−5− 3=−8[/latex]

And now, the fourth case, [latex]5-(-3)[/latex]. We start with 5 positives. We need to take away 3 negatives, but there are no negatives to take away. So we add neutral pairs until we have 3 negatives to take away.

We start with 5 positives.
We now add the needed neutrals pairs.
We remove the 3 negatives.
We are left with 8 positives.
The difference of 5 and −3 is 8.	[latex]5-(-3) = 8[/latex]

Example 13

Subtract:

1. [latex]-3-1[/latex]
2. [latex]3-(-1)[/latex].

Solution

a.

Step 1: Take 1 positive from the one added neutral pair.

[latex]-3-1[/latex]

[latex]-4[/latex]

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b.

Step 1: Take 1 negative from the one added neutral pair.

[latex]3-(-1)[/latex]

[latex]4[/latex]

Have you noticed that subtraction of signed numbers can be done by adding the opposite? In example 1.3.13 (above),  [latex]-3-1[/latex] is the same as [latex]-3+(-1)[/latex] and [latex]3-(-1)[/latex] is the same as [latex]3+1[/latex]. You will often see this idea, the subtraction property, written as follows:

Of course, when you have a subtraction problem that has only positive numbers, like [latex]6-4[/latex], you just do the subtraction. You already knew how to subtract [latex]6-4[/latex] long ago. But knowing that [latex]6-4[/latex] gives the same answer as [latex]6+(-4)[/latex] helps when you are subtracting negative numbers. Make sure that you understand how [latex]6-4[/latex] and [latex]6+(-4)[/latex] give the same results!

Look at what happens when we subtract a negative.

Subtracting a negative number is like adding a positive!

You will often see this written as [latex]a-b=a+b[/latex].

Does that work for other numbers, too? Let’s do the following example and see.

What happens when there are more than three integers? We just use the order of operations as usual.

Be careful to get [latex]a[/latex] and [latex]b[/latex] in the right order!