Foundations
Integers
Learning Objectives
By the end of this section, you will be able to:
- Simplify expressions with absolute value
- Add and subtract integers
- Multiply and divide integers
- Simplify expressions with integers
- Evaluate variable expressions with integers
- Translate phrases to expressions with integers
- Use integers in applications
A more thorough introduction to the topics covered in this section can be found in the Elementary Algebra chapter, Foundations.
Simplify Expressions with Absolute Value
A negative numbers is a number less than 0. The negative numbers are to the left of zero on the number line. See (Figure).
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 and are the same distance from zero, each one is called the opposite of the other. The opposite of is and the opposite of is
The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.
(Figure) illustrates the definition.
We saw that numbers such as 3 and are opposites because they are the same distance from 0 on the number line. They are both three units from 0. The distance between 0 and any number on the number line is called the absolute value of that number.
The absolute value of a number is its distance from 0 on the number line.
The absolute value of a number is written as and for all numbers.
Absolute values are always greater than or equal to zero.
For example,
(Figure) 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 0 to 0 on the number line is zero units.
In the next example, we’ll order expressions with absolute values.
Fill in or for each of the following pairs of numbers:
ⓐⓑⓒⓓ
ⓐ
ⓑ
ⓒ
ⓓ
Fill in or for each of the following pairs of numbers:
ⓐⓑⓒⓓ
ⓐⓑⓒ
ⓓ
Fill in or for each of the following pairs of 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 like we do with parentheses.
Simplify:
Simplify:
16
Simplify:
9
Add and Subtract Integers
So far, we have only used the counting numbers and the whole numbers.
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
The whole numbers and their opposites are called the integers.
The integers are the numbers
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