# Foundations

## Add and Subtract Integers


### Use Negatives and Opposites

Our work so far has only included the counting numbers and the whole numbers. But if you have ever experienced a temperature below zero or accidentally overdrawn your checking account, you are already familiar with negative numbers. Negative numbers are numbers less than  The negative numbers are to the left of zero on the number line. See .
The arrows on the ends of the number line indicate that the numbers keep going forever. There is no biggest positive number, and there is no smallest negative number.
Is zero a positive or a negative number? Numbers larger than zero are positive, and numbers smaller than zero are negative. Zero is neither positive nor negative.
Consider how numbers are ordered on the number line. Going from left to right, the numbers increase in value. Going from right to left, the numbers decrease in value. See .
Remember that we use the notation:
a < b (read “a is less than b”) when a is to the left of b on the number line.
a > b (read “a is greater than b”) when a is to the right of b on the number line.
Now we need to extend the number line which showed the whole numbers to include negative numbers, too. The numbers marked by points in  are called the integers. The integers are the numbers 
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  are the same distance from zero, they are called opposites. The opposite of 2 is  and the opposite of  is 2.
 illustrates the definition.
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.
Our work with opposites gives us a way to define the integers.The whole numbers and their opposites are called the integers. The integers are the numbers 
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  is positive or negative. We can see this in .

### Simplify: Expressions with Absolute Value

We saw that numbers such as  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.
For example,
1.  units away from  so 
2.  units away from  so 
 illustrates this idea.
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  on the number line is zero units.
Mathematicians say it more precisely, “absolute values are always non-negative.” Non-negative means greater than or equal to zero.
In the next example, we’ll order expressions with absolute values. Remember, positive numbers are always greater than negative numbers!
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 as we do with parentheses.

### Add Integers

Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more challenging.
We will use two color counters to model addition and subtraction of negatives so that you can visualize the procedures instead of memorizing the rules.
We let one color (blue) represent positive. The other color (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  and 
To add  we realize that  means the sum of 5 and 3.
Now we will add  Watch for similarities to the last example 
To add  we realize this means the sum of 
In what ways were these first two examples similar?
1. The first example adds 5 positives and 3 positives—both positives.
2. 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 color, and so we added them.
So what happens when the signs are different? Let’s add  We realize this means the sum of  and 3. When the counters were the same color, we put them in a row. When the counters are a different color, we line them up under each other.
Notice that there were more negatives than positives, so the result was negative.
Let’s now add the last combination, 
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  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  there are 16 more red counters.
Therefore, the sum of  is 
Let’s try another one. We’ll add  Again, imagine 74 red counters and 27 more red counters, so we’d have 101 red counters. This means the sum is 
Let’s look again at the results of adding the different combinations of  and 
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!

### Subtract Integers

We will continue to use counters to model the subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.
Perhaps when you were younger, you read  as  take away  When you use counters, you can think of subtraction the same way!
We will model the four subtraction facts using the numbers  and 
To subtract  we restate the problem as  take away 
Now we will subtract  Watch for similarities to the last example 
To subtract  we restate this as  take away 
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 color, and the “take away” model of subtraction was easy to apply.
What happens when we have to subtract one positive and one negative number? We’ll need to use both blue and red counters as well as some neutral pairs. Adding a neutral pair does not change the value. It is like changing quarters to nickels—the value is the same, but it looks different.
1. To subtract  we restate it as  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.
And now, the fourth case,  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.
Have you noticed that subtraction of signed numbers can be done by adding the opposite? In ,  is the same as  and  is the same as  You will often see this idea, the subtraction property, written as follows:
Look at these two examples.
Of course, when you have a subtraction problem that has only positive numbers, like  you just do the subtraction. You already knew how to subtract  long ago. But knowing that  gives the same answer as  helps when you are subtracting negative numbers. Make sure that you understand how  and  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 
Does that work for other numbers, too? Let’s do the following example and see.
Let’s look again at the results of subtracting the different combinations of  and 
What happens when there are more than three integers? We just use the order of operations as usual.

### Key Concepts

1. Addition of Positive and Negative Integers

2. Property of Absolute Value:  for all numbers. Absolute values are always greater than or equal to zero!
3. Subtraction of Integers

4. Subtraction Property: Subtracting a number is the same as adding its opposite.

### Practice Makes Perfect

Use Negatives and Opposites of Integers
In the following exercises, order each of the following pairs of numbers, using < or >.
In the following exercises, find the opposite of each number.
In the following exercises, simplify.
In the following exercises, evaluate.
Simplify Expressions with Absolute Value
In the following exercises, simplify.
In the following exercises, fill in <, >, or  for each of the following pairs of numbers.
In the following exercises, simplify.
In the following exercises, evaluate.
Add Integers
In the following exercises, simplify each expression.
Subtract Integers
In the following exercises, simplify.
In the following exercises, simplify each expression.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?