4A Negative Quantities and Absolute Value

Introduction: Negative numbers are very useful in our world today. In the stock market, a loss of $2400 can be represented by a negative value. If you live in cold climates, you may have seen the temperature dip below zero. If the unemployment rate decreases over a given time period, this can be represented as a negative slope. If you ever visited Death Valley, you would know that its altitude is below sea level. There are many other uses for negative numbers. Many functions and formulas stem from a good understanding of negative numbers.

Let’s look at a few examples. Suppose the temperature is 8 degrees below zero. Is this a positive quantity or a negative quantity?

Temperatures above zero are positive, while temperatures below zero are negative. So a temperature of 8 degrees below zero is $-8$.

Suppose we lost $375 on our trip to Las Vegas. Is this a positive quantity or a negative quantity?

A financial gain is positive, while a loss is negative. So our loss of $375 in Las Vegas corresponds to $-375$.

Let’s look at an example. Suppose a mountain range is 8,400 feet above sea level. Is this a positive quantity or a negative quantity?

Above sea level is positive, while heights below sea level are negative. So a mountain at 8,400 feet above sea level corresponds to $+\mathrm{8,400}$ or just 8,400. (You do not have to write the positive sign, though in this section it is helpful.)

Another useful topic is absolute value. The absolute value of a number is its distance from zero. So $|-16|$ is asking how far $-16$ is from zero on the number line. Since $-16$ is 16 places from zero, we know that $|-16|=+16$. Similarly we see that $|+10|=+10$ since $+10$ is 10 places from zero. It is not surprising that an absolute value will always be equal to a nonnegative quantity.

Do the following examples with your instructor:

Practice Problems Section 4A

Describe the following as a negative or positive quantity.

1. The depth of a submarine is 347 feet below sea level.

   Check answer

   negative

2. The temperature in Juno Alaska is 13 degrees below zero.

   Check answer

   negative

3. An airplane is flying at 600 feet above sea level.

   Check answer

   positive

4. Mario lost $650 while gambling at a Casino.

   Check answer

   negative

5. The number of people in a small town has increased by 850.

   Check answer

   positive

6. A whale is swimming 120 feet below the surface of the ocean.

   Check answer

   negative

7. The temperature in Montana is 19 degrees below zero.

   Check answer

   negative

8. An airplane is flying at an altitude of 200 meters above sea level.

   Check answer

   positive

9. Niki won $73 while playing poker with her friends.

   Check answer

   positive

10. The number of people in a small town has decreased by 124.

    Check answer

    negative

11. A stock price decreased $328 per share.

    Check answer

    negative

12. Sierra lost $1,450 while gambling at a Casino.

    Check answer

    negative

13. The number of cases of the flu has increased by about 3500.

    Check answer

    positive

Find the following absolute values

14. $$|-7|$$

    Check answer

    7

15. $$|-12|$$

    Check answer

    12

16. $$|+17|$$

    Check answer

    17

17. $$|+18|$$

    Check answer

    18

18. $$|-23|$$

    Check answer

    23

19. $$|+16|$$

    Check answer

    16

20. $$|-8.5|$$

    Check answer

    8.5

21. $$|+7.44|$$

    Check answer

    7.44

22. $$|-2.9|$$

    Check answer

    2.9

23. $$|-19|$$

    Check answer

    19

24. $$|-32|$$

    Check answer

    32

25. $$|+49|$$

    Check answer

    49

26. $$|+5⁤\frac{1}{4}|$$

    Check answer

    $$5⁤\frac{1}{4}$$

27. $$|-7⁤\frac{2}{5}|$$

    Check answer

    $$7⁤\frac{2}{5}$$

28. $$|+9⁤\frac{1}{8}|$$

    Check answer

    $$9⁤\frac{1}{8}$$

29. $$|-5.7|$$

    Check answer

    5.7

30. $$|+2⁤\frac{3}{4}|$$

    Check answer

    $$2⁤\frac{3}{4}$$

31. $$|+5.913|$$

    Check answer

    5.913

32. $$-|-22|$$

    Check answer

    ^-22

33. $$-|+3.5|$$

    Check answer

    ^-3.5

34. $$-|-6.25|$$

    Check answer

    ^-6.25

Adapted from Preparing for Algebra and Statistics, Third Edition, by Matt Teachout. License: CC BY 4.0. Copyright and sources.