6.04 Rational numbers and the number line

We are ready now to combine what we have learned about positive and negative numbers with our knowledge of fractions and decimals. Just like we saw with integers, we can order and compare fractions and perform addition and subtraction of fractions using the number line.

 

Number Lines with Fractions

On the number line below, each tick is labelled with a multiple of the fraction $\frac{1}{5}$15​. We can see that the point furthest to the left is plotted at the fraction $-\frac{3}{5}$−35​, and the point furthest to the right is plotted at the fraction $\frac{6}{5}$65​.

 

 

It is common to see number lines where only the integers are labelled, with ticks between each integer that represent a fraction of one whole. A point has been plotted on this next number line. Can you see which fraction the point lies on?

 

 

We can find the denominator of the fraction using the number of ticks between each integer. To go from $0$0 to $1$1 we need to move up $3$3 ticks, so each tick represents $\frac{1}{3}$13​. To find the numerator of the fraction we can count the ticks from $0$0 to the point, which gives $8$8. This means that the point is plotted at the fraction $\frac{8}{3}$83​.

Another way to identify the fraction is to see that the point is two thirds to the right of the integer $2$2. So it lies on the number $2+\frac{2}{3}$2+23​, which we can write as the mixed number $2\frac{2}{3}$223​.

 

Worked example 1

Where is the point plotted on the number line?

 

Think: The point lies in between two integers, in the negative direction from $0$0, so it will be a negative fraction. We can look at the spacing between each integer to determine the denominator of the fraction, and count the ticks from $0$0 to find the numerator.

Do: Each integer is separated by $7$7 ticks, so each tick represents $\frac{1}{7}$17​. If we start at $0$0 we will move $10$10 ticks to the left to get to the plotted point. This means that the point lies on the number $-\frac{10}{7}$−107​.

Reflect: We counted down by sevenths to get to the plotted point; $0,-\frac{1}{7},-\frac{2}{7},-\frac{3}{7},\dots,-\frac{9}{7},-\frac{10}{7}$0,−17​,−27​,−37​,…,−97​,−107​. If our point was very far from $0$0 this could take a long time. Alternatively, we could start counting from the nearest labelled integer, which in this case is $-1$−1. We can write $-1$−1 as $-\frac{7}{7}$−77​, and count down from here to the point like so: $-\frac{7}{7},-\frac{8}{7},-\frac{9}{7},-\frac{10}{7}$−77​,−87​,−97​,−107​.

 

Fractions with different denominators

If we look at any two integers, it is simple to see which is greater. But if we are given two fractions, it can be less obvious to see which is greater if the fractions do not have the same denominator.

The point on this number line lies on the fraction $-\frac{5}{6}$−56​.

 

If we wanted to plot another point at the fraction $-\frac{2}{3}$−23​, we need to first write $-\frac{2}{3}$−23​ as an equivalent fraction with a denominator of $6$6. Since $\frac{1}{3}$13​ is the same as $\frac{1\times2}{3\times2}=\frac{2}{6}$1×23×2​=26​, then $-\frac{2}{3}$−23​ is the same as $-\frac{2\times2}{3\times2}=-\frac{4}{6}$−2×23×2​=−46​. Here are both points plotted on the same number line.

 

Now let's compare $-\frac{1}{3}$−13​ and $-\frac{2}{5}$−25​. In this case we need to rewrite both fractions so that they have a common denominator. Let's choose a denominator of $15$15, which is the lowest common denominator of the two fractions. We rewrite $-\frac{1}{3}$−13​ as $-\frac{1\times5}{3\times5}=-\frac{5}{15}$−1×53×5​=−515​ and we rewrite $-\frac{2}{5}$−25​ as $-\frac{2\times3}{5\times3}=-\frac{6}{15}$−2×35×3​=−615​. Here are both fractions plotted on the same number line.

 

Worked example 2

Use a number line to determine the greatest fraction out of $\frac{3}{4}$34​ and $\frac{5}{7}$57​.

Think: To be able to plot both fractions on the same number line we need to rewrite them with a common denominator. The greatest fraction will then be the one furthest in the positive direction on the number line.

Do: The lowest common denominator of the two fractions is $4\times7=28$4×7=28. We can rewrite $\frac{3}{4}$34​ as $\frac{3\times7}{4\times7}=\frac{21}{28}$3×74×7​=2128​, and we can rewrite $\frac{5}{7}$57​ as $\frac{5\times4}{7\times4}=\frac{20}{28}$5×47×4​=2028​. Here are the two fractions plotted on the same number line.

 

We can see that the fraction $\frac{21}{28}$2128​ is further to the right, so $\frac{21}{28}$2128​ is greater than $\frac{20}{28}$2028​. Looking back at our original fractions, we can say that $\frac{3}{4}$34​ is greater than $\frac{5}{7}$57​.

Reflect: Another way to compare two fractions is to use two different number lines. In the image below to top number line has ticks at multiples of $\frac{1}{4}$14​, and the bottom number line has ticks at multiples of $\frac{1}{7}$17​.

 

We can see that $\frac{3}{4}$34​ is further to the right than $\frac{5}{7}$57​, so $\frac{3}{4}$34​ is the greater fraction.

 

Arithmetic with Directed Fractions

Adding and subtracting positive and negative fractions works in the same way as adding and subtracting positive and negative integers. Let's look at an example.

 

Worked example 3

Find the value of $\frac{11}{8}+\left(-\frac{17}{8}\right)$118​+(−178​).

Think: We can simplify the signs using the fact that adding a negative fraction is the same as subtracting a positive fraction.

Do: By combining the adjacent signs we can rewrite the expression as $\frac{11}{8}+\left(-\frac{17}{8}\right)=\frac{11}{8}-\frac{17}{8}$118​+(−178​)=118​−178​. This expression tells us that we start at $\frac{11}{8}$118​ on the number line, and we want to move $\frac{17}{8}$178​ units in the negative direction.

On the number line below each tick represents $\frac{1}{8}$18​ units. We can move $\frac{17}{8}$178​ units to the left by counting down $17$17 ticks.

 

We land at the fraction $-\frac{6}{8}$−68​, which means that $\frac{11}{8}+\left(-\frac{17}{8}\right)=-\frac{6}{8}$118​+(−178​)=−68​.

Reflect: The fraction $\frac{17}{8}$178​ can be written as the mixed number $2\frac{1}{8}$218​. So subtracting $\frac{17}{8}$178​ can be thought of as subtracting $2$2 wholes, then a further $\frac{1}{8}$18​. This is shown on the number line below.

 

Practice questions

Question 1

Question 2

Question 3

 

Number Lines with Decimals

 

When we divide $1$1 unit into $10$10 equal pieces, it is broken into $10$10 tenths, as shown below.

When we divide $1$1 unit into $100$100 equal pieces we end up with hundredths.  We can also get the size of $1$1 hundredth by dividing $1$1 tenth up into $10$10 equal pieces, as shown below. 

 

Practice Questions

Question 4

 

We can also have negative decimals on the number line similar to the one with fractions: 

 

Worked example 4

Plot $-0.2,0.7$−0.2,0.7 and 0.5 on a number line.  

Since all the decimals are integer multiples of $0.1$0.1, we should go up by $0.1$0.1 on our number line. You should choose a minimum lower than the smallest number $-0.2$−0.2, and a maximum higher than the largest number $0.7$0.7. Therefore, the following number line will be suitable: 

Now we can plot the points to get: 

From the number line we can see that $-0.2$−0.2 is the smallest of the three decimals because because it is the most to the left. Also, $0.7$0.7 is the largest of the three decimals because it is most to the right of the number line.

 

 

Opposites

 

Two integers are opposites if they are equal distance from $0$0 on the number line and of opposite sign. e.g. $-2$−2 and $2$2. This is shown on the number line below:
 

This is also true for fractions and decimals.  $\frac{1}{2}$12​ and $-\frac{1}{2}$−12​ are opposites, because they are opposite signs and both are $\frac{1}{2}$12​ a unit away from $0$0:

 

Worked example 5

 

What is the opposite of $1.5$1.5?

 

We need to find the number with a distance of $1.5$1.5 units from $0$0 on the number line and has the opposite sign. Therefore, the opposite number of $1.5$1.5 is: $-1.5$−1.5.

 

Worked example 6

What is the opposite of $-\frac{6}{7}$−67​?

 

We need to find the number with a distance of  $-\frac{6}{7}$−67​ units from $0$0 on the number line and has the opposite sign. Therefore, the opposite number of $-\frac{6}{7}$−67​ is: $\frac{6}{7}$67​.