Problem: The highest elevation in North America is Mt. McKinley, which is 20,320 feet above sea level. The lowest elevation is Death Valley, which is 282 feet below sea level. What is the distance from the top of Mt. McKinley to the bottom of Death Valley?
Solution: The distance from the top of Mt. McKinley to sea level is 20,320 feet and the distance from sea level to the bottom of Death Valley is 282 feet. The total distance is the sum of 20,320 and 282, which is 20,602 feet.
The problem above uses the notion of opposites: Above sea level is the opposite of below sea level. Here are some more examples of opposites:
top, bottom | increase, decrease | forward, backward | positive, negative
We could solve the problem above using integers. Integers are the set of whole numbers and their opposites. The number line is used to represent integers. This is shown below.
Definitions
- The number line goes on forever in both directions. This is indicated by the arrows.
- Whole numbers greater than zero are called positive integers. These numbers are to the right of zero on the number line.
- Whole numbers less than zero are called negative integers. These numbers are to the left of zero on the number line.
- The integer zero is neutral. It is neither positive nor negative.
- The sign of an integer is either positive (^{+}) or negative (^{-}), except zero, which has no sign.
- Two integers are opposites if they are each the same distance away from zero, but on opposite sides of the number line. One will have a positive sign, the other a negative sign. In the number line above, ^{+}3 and ^{-}3 are labeled as opposites.
Let's revisit the problem from the top of this page using integers to solve it.
Problem: The highest elevation in North America is Mt. McKinley, which is 20,320 feet above sea level. The lowest elevation is Death Valley, which is 282 feet below sea level. What is the distance from the top of Mt. McKinley to the bottom of Death Valley?
Solution: We can represent the elevation as an integers:
Elevation | Integer |
20,320 feet above sea level | ^{+}20,320 |
sea level | 0 |
282 feet below sea level | ^{-}282 |
The distance from the top of Mt. McKinley to the bottom of Death Valley is the same as the distance from ^{+}20,320 to ^{-}282 on the number line. We add the distance from ^{+}20,320 to 0, and the distance from 0 to ^{-}282, for a total of 20,602 feet.
Example 1: Write an integer to represent each situation:
10 degrees above zero | ^{+}10 | |
a loss of 16 dollars | ^{-}16 | |
a gain of 5 points | ^{+}5 | |
8 steps backward | ^{-}8 |
Example 2: Name the opposite of each integer.
^{-}12 | ^{+}12 | |
^{+}21 | ^{-}21 | |
^{-}17 | ^{+}17 | |
^{+}9 | ^{-}9 |
Example 3: Name 4 real life situations in which integers can be used.
Spending and earning money.
Rising and falling temperatures.
Stock market gains and losses.
Gaining and losing yards in a football game.
Note: A positive integer does not have to have a ^{+} sign in it. For example, ^{+}3 and 3 are interchangeable.
Summary: Integers are the set of whole numbers and their opposites. Whole numbers greater than zero are called positive integers. Whole numbers less than zero are called negative integers. The integer zero is neither positive nor negative, and has no sign. Two integers are opposites if they are each the same distance away from zero, but on opposite sides of the number line. Positive integers can be written with or without a sign.
Exercises
Directions: Read each question below. Click once in an ANSWER BOX and type in your answer; then click ENTER. Do not enter commas in your answers. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR. Use the + key to write a positive integer, and the - key to write a negative integer. Omit words and labels from your answers.
1. | Write an integer to represent the following situation: Earnings of 15 dollars |
2. | Write an integer to represent the following situation: A loss of 20 yards |
3. | What is the opposite of ^{-}231? |
4. | What is the opposite of ^{+}1096? |
5. | Solve the following problem using integers. Jenny has $2. She earns $5, spends $10, earns $4, then spends $3. How many dollars does she have or owe? (Enter an integer in the space below.) |