# Integers

## Introduction to Integers


### Locate Positive and Negative Numbers on the Number Line

Do you live in a place that has very cold winters? Have you ever experienced a temperature below zero? If so, you are already familiar with negative numbers. A negative number is a number that is less than  Very cold temperatures are measured in degrees below zero and can be described by negative numbers. For example,  (read as “negative one degree Fahrenheit”) is  below  A minus sign is shown before a number to indicate that it is negative.  shows  which is  below 
Temperatures are not the only negative numbers. A bank overdraft is another example of a negative number. If a person writes a check for more than he has in his account, his balance will be negative.
Elevations can also be represented by negative numbers. The elevation at sea level is  Elevations above sea level are positive and elevations below sea level are negative. The elevation of the Dead Sea, which borders Israel and Jordan, is about  below sea level, so the elevation of the Dead Sea can be represented as  See .
Depths below the ocean surface are also described by negative numbers. A submarine, for example, might descend to a depth of  Its position would then be  as labeled in .
Both positive and negative numbers can be represented on a number line. Recall that the number line created in Add Whole Numbers started at  and showed the counting numbers increasing to the right as shown in . The counting numbers  on the number line are all positive. We could write a plus sign,  before a positive number such as  or  but it is customary to omit the plus sign and write only the number. If there is no sign, the number is assumed to be positive.
Now we need to extend the number line to include negative numbers. We mark several units to the left of zero, keeping the intervals the same width as those on the positive side. We label the marks with negative numbers, starting with  at the first mark to the left of  at the next mark, and so on. See .
The arrows at either end of the line indicate that the number line extends forever in each direction. There is no greatest positive number and there is no smallest negative number. 

### Order Positive and Negative Numbers

We can use the number line to compare and order positive and negative numbers. Going from left to right, numbers increase in value. Going from right to left, numbers decrease in value. See .
Just as we did with positive numbers, we can use inequality symbols to show the ordering of positive and negative numbers. Remember that we use the notation  (read  is less than ) when  is to the left of  on the number line. We write  (read  is greater than ) when  is to the right of  on the number line. This is shown for the numbers  and  in .
The numbers lines to follow show a few more examples.
ⓐ


 is to the right of  on the number line, so 
 is to the left of  on the number line, so 
ⓑ


 is to the left of  on the number line, so 
 is to the right of  on the number line, so 
ⓒ


 is to the right of  on the number line, so 
 is to the left of  on the number line, so 

### Find Opposites

On the number line, the negative numbers are a mirror image of the positive numbers with zero in the middle. Because the numbers  and  are the same distance from zero, they are called opposites. The opposite of  is  and the opposite of  is  as shown in (a). Similarly,  and  are opposites as shown in (b).

### Opposite Notation

Just as the same word in English can have different meanings, the same symbol in algebra can have different meanings. The specific meaning becomes clear by looking at how it is used. You have seen the symbol  in three different ways.


### Integers

The set of counting numbers, their opposites, and  is the set of integers.
We must be very careful with the signs when evaluating the opposite of a variable.

### Simplify Expressions with Absolute Value

We saw that numbers such as  and  are opposites because they are the same distance from  on the number line. They are both five units from  The distance between  and any number on the number line is called the absolute value of that number. Because distance is never negative, the absolute value of any number is never negative.
The symbol for absolute value is two vertical lines on either side of a number. So the absolute value of  is written as  and the absolute value of  is written as  as shown in .
We treat absolute value bars just like we treat parentheses in the order of operations. We simplify the expression inside first.
Absolute value bars act like grouping symbols. First simplify inside the absolute value bars as much as possible. Then take the absolute value of the resulting number, and continue with any operations outside the absolute value symbols.

### Translate Word Phrases into Expressions with Integers

Now we can translate word phrases into expressions with integers. Look for words that indicate a negative sign. For example, the word negative in “negative twenty” indicates  So does the word opposite in “the opposite of 
As we saw at the start of this section, negative numbers are needed to describe many real-world situations. We’ll look at some more applications of negative numbers in the next example.

### Key Concepts

1. Opposite Notation


2. Absolute Value Notation




### Practice Makes Perfect

Locate Positive and Negative Numbers on the Number Line 
For the following exercises, draw a number line and locate and label the given points on that number line.
Order Positive and Negative Numbers on the Number Line 
In the following exercises, order each of the following pairs of numbers, using  or 
Find Opposites 
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 each absolute value expression.
In the following exercises, evaluate each absolute value expression.
In the following exercises, fill in  to compare each expression.
In the following exercises, simplify each expression.
Translate Word Phrases into Expressions with Integers 
Translate each phrase into an expression with integers. Do not simplify.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.



ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.