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Now plot points at 1 5 and 4 5 .

A number line is shown. It shows 0, 1 fifth, 2 fifths, 3 fifths, 4 fifths, and 1. There are red dots at 1 fifth and at 4 fifths.

The only mixed number    to plot is 3 1 3 . Between what two whole numbers is 3 1 3 ? Remember that a mixed number is a whole number plus a proper fraction , so 3 1 3 > 3 . Since it is greater than 3 , but not a whole unit greater, 3 1 3 is between 3 and 4 . We need to divide the portion of the number line between 3 and 4 into three equal pieces (thirds) and plot 3 1 3 at the first mark.

A number line is shown with whole number 0 through 5. Between 3 and 4, 3 and 1 third and 3 and 2 thirds are labeled. There is a red dot at 3 and 1 third.

Finally, look at the improper fractions 7 4 , 9 2 , and 8 3 . Locating these points will be easier if you change each of them to a mixed number.

7 4 = 1 3 4 , 9 2 = 4 1 2 , 8 3 = 2 2 3

Here is the number line with all the points plotted.

A number line is shown with whole numbers 0 through 6. Between 0 and 1, 1 fifth and 4 fifths are labeled and shown with red dots. Between 1 and 2, 7 fourths is labeled and shown with a red dot. Between 2 and 3, 8 thirds is labeled and shown with a red dot. Between 3 and 4, 3 and 1 third is labeled and shown with a red dot. Between 4 and 5, 9 halves is labeled and shown with a red dot.

Locate and label the following on a number line: 3 4 , 4 3 , 5 3 , 4 1 5 , and 7 2 .

Solution

Start by locating the proper fraction 3 4 . It is between 0 and 1 . To do this, divide the distance between 0 and 1 into four equal parts. Then plot 3 4 .

A number line is shown. It shows 0, 1 fourth, 2 fourths, 3 fourths, and 1. There is a red dot at 3 fourths.

Next, locate the mixed number 4 1 5 . It is between 4 and 5 on the number line. Divide the number line between 4 and 5 into five equal parts, and then plot 4 1 5 one-fifth of the way between 4 and 5 .

A number line is shown. It shows 4, 4 and 1 fifth, 4 and 2 fifths, 4 and 3 fifths, 4 and 4 fifths, and 5. There is a red dot at 4 and 1 fifth.

Now locate the improper fractions 4 3 and 5 3 .

It is easier to plot them if we convert them to mixed numbers first.

4 3 = 1 1 3 , 5 3 = 1 2 3

Divide the distance between 1 and 2 into thirds.

A number line is shown. It shows 1, 1 and 1 third, 1 and 2 thirds, and 2. Below 1 it says 3 thirds. Below 1 and 1 third it says 4 thirds. Below 1 and 2 thirds it says 5 thirds. Below 2 it says 6 thirds. There are red dots at 1 and 1 third and 1 and 2 thirds.

Next let us plot 7 2 . We write it as a mixed number, 7 2 = 3 1 2 . Plot it between 3 and 4 .

A number line is shown. It shows 3, 3 and 1 half, and 4. Below 3 it says 6 halves. Below 3 and 1 half it says 7 halves. Below 4 it says 8 halves. There is a red dot at 3 and 1 half.

The number line shows all the numbers located on the number line.

A number line is shown. It shows the whole numbers 0 through 5. Between any 2 numbers are 10 tick marks. Between 0 and 1, between the 7th and 8th tick mark, 3 fourths is labeled and shown with a red dot. Between 1 and 2, 4 thirds and 5 thirds are labeled and shown with red dots. Between 3 and 4, 7 halves is labeled and shown with a red dot. Between 4 and 5, 4 and 1 fifth is labeled and shown with a red dot.
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Locate and label the following on a number line: 1 3 , 5 4 , 7 4 , 2 3 5 , 9 2 .


A number line is shown. It shows the whole numbers 0 through 6. Between 0 and 1, 1 third is labeled and shown with a red dot. Between 1 and 2, 5 fourths and 7 fourths are labeled and shown with red dots. Between 2 and 3, 2 and 3 fifths is labeled and shown with a red dot. Between 4 and 5, 9 halves is labeled and shown with a red dot.

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Locate and label the following on a number line: 2 3 , 5 2 , 9 4 , 11 4 , 3 2 5 .


A number line is shown. It shows the whole numbers 0 through 6. Between 0 and 1, 2 thirds is labeled and shown with a red dot. Between 2 and 3, 9 fourths, 5 halves, and 11 fourths are labeled and shown with red dots. Between 3 and 4, 3 and 2 fifths is labeled and shown with a red dot.

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In Introduction to Integers , we defined the opposite of a number. It is the number that is the same distance from zero on the number line but on the opposite side of zero. We saw, for example, that the opposite of 7 is −7 and the opposite of −7 is 7 .

A number line is shown. It shows the numbers negative 7, 0 and 7. There are red dots at negative 7 and 7. The space between negative 7 and 0 is labeled as 7 units. The space between 0 and 7 is labeled as 7 units.

Fractions have opposites, too. The opposite of 3 4 is 3 4 . It is the same distance from 0 on the number line, but on the opposite side of 0 .

A number line is shown. It shows the numbers negative 1, negative 3 fourths, 0, 3 fourths, and 1. There are red dots at negative 3 fourths and 3 fourths. The space between negative 3 fourths and 0 is labeled as 3 fourths of a unit. The space between 0 and 3 fourths is labeled as 3 fourths of a unit.

Thinking of negative fractions as the opposite of positive fractions will help us locate them on the number line. To locate 15 8 on the number line, first think of where 15 8 is located. It is an improper fraction , so we first convert it to the mixed number 1 7 8 and see that it will be between 1 and 2 on the number line. So its opposite, 15 8 , will be between −1 and −2 on the number line.

A number line is shown. It shows the numbers negative 2, negative 1, 0, 1, and 2. Between negative 2 and negative 1, negative 1 and 7 eighths is labeled and marked with a red dot. The distance between negative 1 and 7 eighths and 0 is marked as 15 eighths units. Between 1 and 2, 1 and 7 eighths is labeled and marked with a red dot. The distance between 0 and 1 and 7 eighths is marked as 15 eighths units.

Locate and label the following on the number line: 1 4 , 1 4 , 1 1 3 , −1 1 3 , 5 2 , and 5 2 .

Solution

Draw a number line. Mark 0 in the middle and then mark several units to the left and right.

To locate 1 4 , divide the interval between 0 and 1 into four equal parts. Each part represents one-quarter of the distance. So plot 1 4 at the first mark.

A number line is shown. It shows the numbers negative 4, negative 3, negative 2, negative 1, 0, 1, 2, 3, and 4. There are 4 tick marks between negative 1 and 0. There are 4 tick marks between 0 and 1. The first tick mark between 0 and 1 is labeled as 1 fourth and marked with a red dot.

To locate 1 4 , divide the interval between 0 and −1 into four equal parts. Plot 1 4 at the first mark to the left of 0 .

A number line is shown. It shows the numbers negative 4, negative 3, negative 2, negative 1, 0, 1, 2, 3, and 4. There are 4 tick marks between negative 1 and 0. There are 4 tick marks between 0 and 1. The first tick mark between 0 and 1 is labeled as 1 fourth and marked with a red dot. The first tick mark between 0 and negative 1 is labeled as negative 1 fourth and marked with a red dot.

Since 1 1 3 is between 1 and 2 , divide the interval between 1 and 2 into three equal parts. Plot 1 1 3 at the first mark to the right of 1 . Then since −1 1 3 is the opposite of 1 1 3 it is between −1 and −2 . Divide the interval between −1 and −2 into three equal parts. Plot −1 1 3 at the first mark to the left of −1 .

A number line is shown. The integers from negative 2 to 2 are labeled. Between negative 2 and negative 1, negative 1 and 1 third is labeled and marked with a red dot. Between 1 and 2, 1 and 1 third is labeled and marked with a red dot.

To locate 5 2 and 5 2 , it may be helpful to rewrite them as the mixed numbers 2 1 2 and −2 1 2 .

Since 2 1 2 is between 2 and 3 , divide the interval between 2 and 3 into two equal parts. Plot 5 2 at the mark. Then since −2 1 2 is between −2 and −3 , divide the interval between −2 and −3 into two equal parts. Plot 5 2 at the mark.

A number line is shown. The integers from negative 4 to 4 are labeled. Between negative 3 and negative 2, negative 5 halves is labeled and marked with a red dot. Between 2 and 3, 5 halves is labeled and marked with a red dot.
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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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