5.1.1: Interpreting Negative Numbers

Lesson

Let's review what we know about signed numbers.

Exercise \(\PageIndex{1}\): Using the Thermometer

Here is a weather thermometer. Three of the numbers have been left off.

Figure \(\PageIndex{1}\): A vertical thermometer measured in degrees Celsius. There are 9 evenly spaced tick marks and starting from the bottom of the thermometer, negative 10 is on the first tick mark, zero on the third, 5 on the fourth, 15 on the sixth, 20 on the seventh, and 30 on the ninth. The second, fifth, and eighth tick marks each are labeled with a box. The thermometer is shaded starting from the bottom of the thermometer to halfway between the second and third tick marks.

1. What numbers go in the boxes?
2. What temperature does the thermometer show?

Exercise \(\PageIndex{2}\): Fractions of a Degree

1. What temperature is shown on each thermometer?
2. Which thermometer shows the highest temperature?
3. Which thermometer shows the lowest temperature?
4. Suppose the temperature outside is \(-4^{\circ}\text{C}\). Is that colder or warmer than the coldest temperature shown? How do you know?

Figure \(\PageIndex{2}\): Four vertical thermometers measured in degrees Celsius. There are 16 evenly spaced tick marks and starting from the bottom of the thermometer, negative 5 is on the first tick mark, zero on the sixth, 5 on the eleventh, and 10 on the sixteenth. The first thermometer is shaded starting from the bottom of the thermometer to the tenth tickmark. The second thermometer is shaded starting from the bottom of the thermometer to the third tickmark. The third thermometer is shaded starting from the bottom of the thermometer to between the eleventh and twelfth tickmark. The fourth thermometer is shaded starting from the bottom of the thermometer to between the fourth and fifth tickmark.

Exercise \(\PageIndex{3}\): Seagulls Soar, SHarks Swim

Here is a picture of some sea animals. The number line on the left shows the vertical position of each animal above or below sea level, in meters.

Figure \(\PageIndex{3}\): A graph. Vertical axis is labeled vertical position, in meters. 5 tick marks. Scale from negative 15 to 15, by 5s. Horizontal axis has no label or scale. Seagull is at 10. Dolphin is at 3. Penguin is at 0. Shark is at negative 3. Fish is at negative 7. Octopus is at negative 10.

1. How far above or below sea level is each animal? Measure to their eye level.
2. A mobula ray is 3 meters above the surface of the ocean. How does its vertical position compare to the height or depth of:
   The jumping dolphin?
   The flying seagull?
   The octopus?
3. An albatross is 5 meters above the surface of the ocean. How does its vertical position compare to the height or depth of:
   The jumping dolphin?
   The flying seagull?
   The octopus?
4. A clownfish is 2 meters below the surface of the ocean. How does its vertical position compare to the height or depth of:
   The jumping dolphin?
   The flying seagull?
   The octopus?
5. The vertical distance of a new dolphin from the dolphin in the picture is 3 meters. What is its distance from the surface of the ocean?

Are you ready for more?

The north pole is in the middle of the ocean. A person at sea level at the north pole would be 3,949 miles from the center of Earth. The sea floor below the north pole is at an elevation of approximately -2.7 miles. The elevation of the south pole is about 1.7 miles. How far is a person standing on the south pole from a submarine at the sea floor below the north pole?

Exercise \(\PageIndex{4}\): Card Sort: Rational Numbers

1. Your teacher will give your group a set of cards. Order the cards from least to greatest.
2. Pause here so your teacher can review your work. Then, your teacher will give you a second set of cards.
3. Add the new set of cards to the first set so that all of the cards are ordered from least to greatest.

Summary

We can use positive numbers and negative numbers to represent temperature and elevation.

When numbers represent temperatures, positive numbers indicate temperatures that are warmer than zero and negative numbers indicate temperatures that are colder than zero. This thermometer shows a temperature of -1 degree Celsius, which we write \(-1^{\circ}\text{C}\).

Figure \(\PageIndex{4}\): A thermometer positioned horizontally and with the bottom of the thermometer on the left, is labeled degrees Celsius. There are 4 evenly spaced large tick marks. Starting on the left, the large tick marks are labeled negative 5, 0, 5, and 10. Between each large tick mark, there are 4 evenly spaced small tick marks. The thermometer is shaded, starting from the left, up to negative 1.

When numbers represent elevations, positive numbers indicate positions above sea level and negative numbers indicate positions below sea level.

We can see the order of signed numbers on a number line.

Figure \(\PageIndex{5}\)

A number is always less than numbers to its right. So \(-7<-3\).

We use absolute value to describe how far a number is from 0. The numbers 15 and -15 are both 15 units from 0, so \(|15|=15\) and \(|-15|=15\). We call 15 and -15 opposites. They are on opposite sides of 0 on the number line, but the same distance from 0.

Glossary Entries

Definition: Absolute Value

The absolute value of a number is its distance from 0 on the number line.

Figure \(\PageIndex{6}\)

The absolute value of -7 is 7, because it is 7 units away from 0. The absolute value of 5 is 5, because it is 5 units away from 0.

Definition: Negative Number

A negative number is a number that is less than zero. On a horizontal number line, negative numbers are usually shown to the left of 0.

Figure \(\PageIndex{7}\)

Definition: Positive Number

A positive number is a number that is greater than zero. On a horizontal number line, positive numbers are usually shown to the right of 0.

Figure \(\PageIndex{8}\)

Practice

Exercise \(\PageIndex{5}\)

It was \(-5^{\circ}\text{C}\) in Copenhagen and \(-12^{\circ}\text{C}\) in Oslo. Which city was colder?

Exercise \(\PageIndex{6}\)

1. A fish is 12 meters below the surface of the ocean. What is its elevation?
2. A sea bird is 28 meters above the surface of the ocean. What is its elevation?
3. If the bird is directly above the fish, how far apart are they?

Exercise \(\PageIndex{7}\)

Compare using >, =, or <.

1. \(3\) _____ \(-3\)
2. \(12\) _____ \(24\)
3. \(-12\) _____ \(-24\)
4. \(5\) _____ \(-(-5)\)
5. \(7.2\) _____ \(7\)
6. \(-7.2\) _____ \(-7\)
7. \(-1.5\) _____ \(\frac{-3}{2}\)
8. \(\frac{-4}{5}\) _____ \(\frac{-5}{4}\)
9. \(\frac{-3}{5}\) _____ \(\frac{-6}{10}\)
10. \(\frac{-2}{3}\) _____ \(\frac{1}{3}\)

Exercise \(\PageIndex{8}\)

Han wants to buy a $30 ticket to a game, but the pre-order tickets are sold out. He knows there will be more tickets sold the day of the game, with a markup of 200%. How much should Han expect to pay for the ticket if he buys it the day of the game?

(From Unit 4.2.2)

Exercise \(\PageIndex{9}\)

A type of green paint is made by mixing 2 cups of yellow with 3.5 cups of blue.

1. Find a mixture that will make the same shade of green but a smaller amount.
2. Find a mixture that will make the same shade of green but a larger amount.
3. Find a mixture that will make a different shade of green that is bluer.
4. Find a mixture that will make a different shade of green that is more yellow.

(From Unit 2.1.1)