36.2: Points on the Number Line

Lesson

Let's plot positive and negative numbers on the number line.

Exercise \(\PageIndex{1}\): A Point on the Number Line

Which of the following numbers could be \(B\)?

Figure \(\PageIndex{1}\)

\(2.5\qquad\frac{2}{5}\qquad\frac{5}{2}\qquad\frac{25}{10}\qquad 2.49\)

Exercise \(\PageIndex{2}\): What's the Temperature?

1. Here are five thermometers. The first four thermometers show temperatures in Celsius. Write the temperatures in the blanks.

Figure \(\PageIndex{2}\): Five vertical thermometers, a through e. Thermometer a is shaded up to 1. Thermometer b is shaded up to 2. Thermometer c is shaded up to the halfway point between 3 and 4. Thermometer d is shaded to the halfway point between negative 1 and 0. Thermometer has tick marks labeled negative 20, blank, blank, negative 5, blank, blank, ten, blank, blank, 25.

The last thermometer is missing some numbers. Write them in the boxes.

2. Elena says that the thermometer shown here reads \(-2.5^{\circ}\text{C}\) because the line of the liquid is above \(-2^{\circ}\text{C}\). Jada says that it is \(-1.5^{\circ}\text{C}\). Do you agree with either one of them? Explain your reasoning.

Figure \(\PageIndex{3}\)

3. One morning, the temperature in Phoenix, Arizona, was \(8^{\circ}\text{C}\) and the temperature in Portland, Maine, was \(12^{\circ}\text{C}\) cooler. What was the temperature in Portland?

Exercise \(\PageIndex{3}\): Folded Number Lines

Your teacher will give you a sheet of tracing paper on which to draw a number line.

1. Follow the steps to make your own number line.
   • Use a straightedge or a ruler to draw a horizontal line. Mark the middle point of the line and label it 0.
   • To the right of 0, draw tick marks that are 1 centimeter apart. Label the tick marks 1, 2, 3. . . 10. This represents the positive side of your number line.
   • Fold your paper so that a vertical crease goes through 0 and the two sides of the number line match up perfectly.
   • Use the fold to help you trace the tick marks that you already drew onto the opposite side of the number line. Unfold and label the tick marks -1, -2, -3. . . -10. This represents the negative side of your number line.
2. Use your number line to answer these questions:
   1. Which number is the same distance away from zero as is the number 4?
   2. Which number is the same distance away from zero as is the number -7?
   3. Two numbers that are the same distance from zero on the number line are called opposites. Find another pair of opposites on the number line.
   4. Determine how far away the number 5 is from 0. Then, choose a positive number and a negative number that is each farther away from zero than is the number 5.
   5. Determine how far away the number -2 is from 0. Then, choose a positive number and a negative number that is each farther away from zero than is the number -2.
      Pause here so your teacher can review your work.
3. Here is a number line with some points labeled with letters. Determine the location of points \(P, X,\) and \(Y\).

Figure \(\PageIndex{4}\)

If you get stuck, trace the number line and points onto a sheet of tracing paper, fold it so that a vertical crease goes through 0, and use the folded number line to help you find the unknown values.

Are you ready for more?

​At noon, the temperatures in Portland, Maine, and Phoenix, Arizona, had opposite values. The temperature in Portland was \(18^{\circ}\text{C}\) lower than in Phoenix. What was the temperature in each city? Explain your reasoning.

Summary

Here is a number line labeled with positive and negative numbers. The number 4 is positive, so its location is 4 units to the right of 0 on the number line. The number -1.1 is negative, so its location is 1.1 units to the left of 0 on the number line.

Figure \(\PageIndex{5}\)

We say that the opposite of 8.3 is -8.3, and that the opposite of \(\frac{-3}{2}\) is \(\frac{3}{2}\). Any pair of numbers that are equally far from 0 are called opposites.

Points \(A\) and \(B\) are opposites because they are both 2.5 units away from 0, even though \(A\) is to the left of 0 and \(B\) is to the right of 0.

Figure \(\PageIndex{6}\)

A positive number has a negative number for its opposite. A negative number has a positive number for its opposite. The opposite of 0 is itself.

You have worked with positive numbers for many years. All of the positive numbers you have seen—whole and non-whole numbers—can be thought of as fractions and can be located on a the number line.

To locate a non-whole number on a number line, we can divide the distance between two whole numbers into fractional parts and then count the number of parts. For example, 2.7 can be written as \(2\frac{7}{10}\). The segment between 2 and 3 can be partitioned into 10 equal parts or 10 tenths. From 2, we can count 7 of the tenths to locate 2.7 on the number line.

All of the fractions and their opposites are what we call rational numbers. For example, \(4, -1.1, 8.3, -8.3, \frac{-3}{2},\) and \(\frac{3}{2}\) are all rational numbers.

Glossary Entries

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: Opposite

Two numbers are opposites if they are the same distance from 0 and on different sides of the number line.

For example, 4 is the opposite of -4, and -4 is the opposite of 4. They are both the same distance from 0. One is negative, and the other is positive.

Figure \(\PageIndex{8}\)

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 \(\PageIndex{9}\)

Definition: Rational Number

A rational number is a fraction or the opposite of a fraction.

For example, 8 and -8 are rational numbers because they can be written as \(\frac{8}{1}\) and \(-\frac{8}{1}\).

Also, 0.75 and -0.75 are rational numbers because they can be written as \(\frac{75}{100}\) and \(-\frac{75}{100}\).

Practice

Exercise \(\PageIndex{4}\)

For each number, name its opposite.

1. \(-5\)
2. \(28\)
3. \(-10.4\)

1. \(0.875\)
2. \(0\)
3. \(-8,003\)

Exercise \(\PageIndex{5}\)

Plot the numbers \(-1.5, \frac{3}{2}, -\frac{3}{2},\) and \(-\frac{4}{3}\) on the number line. Label each point with its numeric value.

Figure \(\PageIndex{10}\)

Exercise \(\PageIndex{6}\)

Plot these points on a number line.

• -1.5
• the opposite of -2
• the opposite of 0.5
• -2

Exercise \(\PageIndex{7}\)

1. Represent each of these temperatures in degrees Fahrenheit with a positive or negative number.
   • \(5\) degrees above zero
   • \(3\) degrees below zero
   • \(6\) degrees above zero
   • \(2\frac{3}{4}\) degrees below zero
2. Order the temperatures above from the coldest to the warmest.

(From Unit 7.1.1)

Exercise \(\PageIndex{8}\)

Solve each equation.

1. \(8x=\frac{2}{3}\)
2. \(1\frac{1}{2}=2x\)
3. \(5x=\frac{2}{7}\)
4. \(\frac{1}{4}x=5\)
5. \(\frac{1}{5}=\frac{2}{3}x\)

(From Unit 6.1.5)

Exercise \(\PageIndex{9}\)

Write the solution to each equation as a fraction and as a decimal.

1. \(2x=3\)
2. \(5y=3\)
3. \(0.3z=0.009\)

(From Unit 6.1.5)

Exercise \(\PageIndex{10}\)

There are 15.24 centimeters in 6 inches.

1. How many centimeters are in 1 foot?
2. How many centimeters are in 1 yard?

(From Unit 3.2.3)