1.4 Add and Subtract Integers
Learning Objectives
By the end of this section, you will be able to:
- Use negatives and opposites
- Simplify: expressions with absolute value
- Add integers
- 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 Figure 1.
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 Figure 2.
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 Figure 3 are called the integers. The integers are the numbers
EXAMPLE 1
Order each of the following pairs of numbers, using < or >: a) ___ b) ___ c) ___ d) ___.
It may be helpful to refer to the number line shown.
| a) 14 is to the right of 6 on the number line. | ___ | > |
| b) −1 is to the left of 9 on the number line. | ___ | |
| c) −1 is to the right of −4 on the number line. | ___ | |
| d) 2 is to the right of −20 on the number line. | ___ | > |
TRY IT 1.1
Order each of the following pairs of numbers, using < or > a) ___ b) ___ c) ___
d) ___.
Answer
a) > b) < c) > d) >
TRY IT 1.2
Order each of the following pairs of numbers, using < or > a) ___ b) ___ c) ___
d) ___.
Answer
a) < b) > c) < d) >
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
Opposite
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 4) illustrates the definition.
The opposite of 3 is .
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.
| Between two numbers, it indicates the operation of subtraction. We read as “10 minus 4.” |
|
| In front of a number, it indicates a negative number. We read −8 as “negati