Rational Numbers on the Number Line

Where fractions, mixed numbers and decimals, positive and negative, lie on the number line, how to read them off it, and how to compare and order them with the signs <, >, ≤ and ≥.

Before you start

A rational number is a number that can be written as $\frac{a}{b}$ , where $a$ and $b$ are integers and $b \neq 0$ . Integers, fractions, mixed numbers, and decimals that terminate or recur are all rational.

On the number line, the positive numbers are to the right of 0 and the negative numbers are to the left. A number further to the right is greater, so $-3 > -8$ .

The numerical value of a number is its distance from 0 on the number line: 5 and $-5$ both have numerical value 5. Two numbers like these, with the same numerical value and different signs, are opposites.

A mixed number is a whole number plus a fraction: $2\frac{1}{4} = 2 + \frac{1}{4} = \frac{9}{4}$ . The negative sign of a mixed number belongs to the whole of it: $-2\frac{3}{5}$ means $-\left(2 + \frac{3}{5}\right) = -\frac{13}{5}$ , which is $2\frac{3}{5}$ units to the left of 0. It does not mean $-2 + \frac{3}{5}$ .

The negative sign of a fraction can be written in front, in the numerator or in the denominator: $-\frac{3}{4} = \frac{-3}{4} = \frac{3}{-4}$ .

A fraction is changed to a decimal by dividing its numerator by its denominator: $\frac{3}{8} = 3 \div 8 = 0.375$ . A decimal that ends like this is a terminating decimal. For many fractions the division never stops, and a digit or a group of digits repeats without end. Such a decimal is a recurring decimal. Three dots show that the digits go on: $\frac{1}{3} = 0.333\ldots$ , $\frac{2}{3} = 0.666\ldots$ and $\frac{5}{6} = 0.8333\ldots$ For a negative fraction, divide without the sign, then put the sign back: $4 \div 3 = 1.333\ldots$ , so $-\frac{4}{3} = -1.333\ldots$

The sign $<$ means "is less than", $>$ means "is greater than", $\le$ means "is less than or equal to" and $\ge$ means "is greater than or equal to".

Placing Rational Numbers on the Number Line

Every rational number has its own point on the number line. Numbers that are not integers lie between the integers, the negative ones to the left of 0.

Placing a fraction or a mixed number

Change an improper fraction to a mixed number, so that you can see its whole part: $\frac{9}{5} = 1\frac{4}{5}$ .
Find the two integers it lies between. $\frac{3}{5}$ lies between 0 and 1, $-\frac{4}{5}$ lies between $-1$ and 0, and $-1\frac{2}{5}$ lies between $-2$ and $-1$ .
Divide that unit into as many equal parts as the denominator.
Start at the one of the two integers that is nearer to 0. Count as many parts as the numerator, moving away from 0: to the right for a positive number, to the left for a negative number.

Example 1: Placing fractions and mixed numbers

Represent $\frac{3}{5}$ , $-\frac{4}{5}$ , $-1\frac{2}{5}$ and $\frac{9}{5}$ on a number line.

Solution

All four denominators are 5, so divide every unit into 5 equal parts. Each part is $\frac{1}{5}$ .

$\frac{3}{5}$ lies between 0 and 1. Start at 0 and count 3 parts to the right.

$-\frac{4}{5}$ lies between $-1$ and 0. Start at 0 and count 4 parts to the left.

$-1\frac{2}{5}$ lies between $-2$ and $-1$ . Start at $-1$ , which is nearer to 0 than $-2$ is, and count 2 parts to the left.

$\frac{9}{5} = 1\frac{4}{5}$ , which lies between 1 and 2. Start at 1 and count 4 parts to the right.

The four numbers are marked on the number line. Check: $-1\frac{2}{5} = -\frac{7}{5}$ , and its point is 7 parts to the left of 0.

Placing a decimal

A decimal with one decimal place counts tenths. Divide the unit into 10 equal parts: $1.4$ is 4 parts to the right of 1, and $-0.7$ is 7 parts to the left of 0.

A decimal with two decimal places lies between two of these marks: $0.25$ is halfway between $0.2$ and $0.3$ .

Some decimals are easier to place as fractions: $-1.5 = -1\frac{1}{2}$ is halfway between $-2$ and $-1$ .

Example 2: Placing decimals

Represent $-0.7$ , $0.25$ and $1.4$ on a number line.

Solution

Draw a number line from $-1$ to 2 and divide every unit into 10 equal parts. Each part is $0.1$ . A longer mark at the fifth part of each unit makes the parts easier to count.

$-0.7$ lies between $-1$ and 0. Start at 0 and count 7 parts to the left.

$1.4$ lies between 1 and 2. Start at 1 and count 4 parts to the right.

$0.25$ lies between $0.2$ and $0.3$ , halfway between them. Start at 0, count 2 parts to the right, then go half a part further.

The three numbers are marked on the number line.

Example 3: Fractions, mixed numbers and decimals on one number line

Represent $-1.5$ , $-\frac{2}{3}$ , $\frac{3}{4}$ and $2\frac{1}{4}$ on a number line.

Solution

$-1.5 = -1\frac{1}{2}$ lies between $-2$ and $-1$ . Divide that unit into 2 equal parts, start at $-1$ and count 1 part to the left. So $-1.5$ is halfway between $-2$ and $-1$ .

$-\frac{2}{3}$ lies between $-1$ and 0. Divide that unit into 3 equal parts, start at 0 and count 2 parts to the left.

$\frac{3}{4}$ lies between 0 and 1. Divide that unit into 4 equal parts, start at 0 and count 3 parts to the right.

$2\frac{1}{4}$ lies between 2 and 3. Divide that unit into 4 equal parts, start at 2 and count 1 part to the right.

The four numbers are marked on the number line. From left to right they are in ascending order: $-1.5 < -\frac{2}{3} < \frac{3}{4} < 2\frac{1}{4}$ .

Reading Rational Numbers off a Number Line

To read a number off a number line, first find the size of one part: count the equal parts between two integers that are next to each other.

Example 4: Reading a number line

On the number line below, the marks are at equal spaces. Write down the numbers that the points A, B, C and D represent.

Solution

There are 3 equal parts between 0 and 1, so each part is $\frac{1}{3}$ .

A lies between $-3$ and $-2$ . It is 2 parts to the left of $-2$ , so A is $-2\frac{2}{3}$ .

B is 1 part to the left of $-1$ , so B is $-1\frac{1}{3}$ .

C is 1 part to the left of 0, and D is 2 parts to the right of 0.

A is $-2\frac{2}{3}$ , B is $-1\frac{1}{3}$ , C is $-\frac{1}{3}$ and D is $\frac{2}{3}$ .

Comparing and Ordering Rational Numbers

As with integers, a number further to the right on the number line is greater. Take care with negative numbers: when two positive numbers are both changed to their opposites, their order is turned round.

$\frac{3}{4}$ is to the right of $\frac{1}{2}$ , so $\frac{3}{4} > \frac{1}{2}$ . But $-\frac{3}{4}$ is to the left of $-\frac{1}{2}$ , so $-\frac{3}{4} < -\frac{1}{2}$ .

Comparing two rational numbers

A negative number is less than 0 and less than every positive number.
To compare two positive numbers, write both as fractions with a common denominator, or write both as decimals.
To compare two negative numbers, compare their numerical values in the same way. The number with the larger numerical value is further from zero, so it is the smaller number.

The signs $<$ , $>$ , $\le$ and $\ge$ are used as they are for integers. A statement with $\le$ or $\ge$ is also true when the two numbers are equal.

Example 5: Comparing two rational numbers

Write $<$ or $>$ between the two numbers of each pair.
(a) $\frac{5}{8}$ and $\frac{7}{12}$
(b) $-\frac{5}{8}$ and $-\frac{7}{12}$
(c) $-2\frac{1}{3}$ and $-2.3$
(d) $-\frac{5}{2}$ and $0.01$

Solution

(a) The LCM of 8 and 12 is 24: $\frac{5}{8} = \frac{15}{24}$ and $\frac{7}{12} = \frac{14}{24}$ . Since $15 > 14$ , $\frac{5}{8}$ is the greater.

(b) Both numbers are negative. From (a), $-\frac{5}{8}$ has the larger numerical value, so it is further from zero and is the smaller number.

(c) As a decimal, $-2\frac{1}{3} = -2.333\ldots$ Both numbers are negative, and $2.333\ldots > 2.3$ . So $-2\frac{1}{3}$ is further from zero and is the smaller number.

(d) $-\frac{5}{2}$ is negative and $0.01$ is positive, and every negative number is less than every positive number.

(a) $\frac{5}{8} > \frac{7}{12}$ (b) $-\frac{5}{8} < -\frac{7}{12}$ (c) $-2\frac{1}{3} < -2.3$ (d) $-\frac{5}{2} < 0.01$

Example 6: Statements with ≤ and ≥

(a) Is $-\frac{1}{2} \ge -0.5$ true or false?
(b) Is $-\frac{2}{3} > -0.6$ true or false?
(c) List all the integers $n$ for which $-\frac{7}{3} < n \le 1.5$ .

Solution

(a) $-\frac{1}{2} = -0.5$ . The two numbers are equal, and $\ge$ allows "equal to".

(b) $-\frac{2}{3} = -0.666\ldots$ , which is further from zero than $-0.6$ . So $-\frac{2}{3}$ is less than $-0.6$ , not greater.

(c) $-\frac{7}{3} = -2\frac{1}{3}$ , which lies between $-3$ and $-2$ . So the integers greater than $-\frac{7}{3}$ start at $-2$ . $1.5$ lies between 1 and 2, so the integers less than or equal to $1.5$ end at 1.

(a) True (b) False (c) $-2$ , $-1$ , 0 and 1

To order a list that mixes fractions, decimals and integers, write every number as a decimal, with enough decimal places to tell the numbers apart. Order the decimals, then give the answer with each number written as it was in the question.

Example 7: Ordering fractions, decimals and integers

Arrange $-\frac{4}{3}$ , $0.9$ , $-1.4$ , $\frac{5}{6}$ and $-2$ in ascending order.

Solution

Numbers that are close together are easier to compare as decimals. By division, $-\frac{4}{3} = -1.333\ldots$ and $\frac{5}{6} = 0.8333\ldots$ , as in "Before you start".

Of two negative numbers, the one further from 0 is smaller: $-2 < -1.4 < -1.333\ldots$

For the positive numbers, $0.8333\ldots < 0.9$ .

Every negative number is less than every positive number, so the negative numbers come first.

In ascending order: $-2$ , $-1.4$ , $-\frac{4}{3}$ , $\frac{5}{6}$ , $0.9$ .

Common Mistakes

Counting towards 0 for a negative mixed number. $-1\frac{2}{5}$ is 2 parts to the left of $-1$ , between $-2$ and $-1$ . It is not 2 parts to the right of $-1$ .
Ordering negative fractions like positive ones. $\frac{3}{4} > \frac{1}{2}$ , but $-\frac{3}{4} < -\frac{1}{2}$ , because $-\frac{3}{4}$ is further to the left on the number line.
Cutting a recurring decimal short before comparing. $-\frac{2}{3} = -0.666\ldots$ , which is not $-0.6$ . It is further from zero than $-0.6$ , so $-\frac{2}{3} < -0.6$ .
Forgetting that $\le$ and $\ge$ allow "equal to". $-\frac{1}{2} \ge -0.5$ is true, because the two numbers are equal.

Practice

Work each question on paper before you open its solution.

Question 1

On the number line below, the marks are at equal spaces.
(a) Write down the numbers that the points P, Q and R represent.
(b) Is $-1\frac{1}{3}$ to the left or to the right of P?

ShowHide solution

(a) There are 4 equal parts between 0 and 1, so each part is $\frac{1}{4}$ . P is 1 part to the left of $-1$ , Q is 2 parts to the left of 0, and R is 3 parts to the right of 0.

(b) $-1\frac{1}{3} = -1.333\ldots$ and $-1\frac{1}{4} = -1.25$ . Of two negative numbers, the one further from 0 is smaller, so $-1\frac{1}{3} < -1\frac{1}{4}$ .

(a) P is $-1\frac{1}{4}$ , Q is $-\frac{1}{2}$ and R is $\frac{3}{4}$ . (b) $-1\frac{1}{3}$ is to the left of P.

Question 2

Draw a number line from $-3$ to 2 and divide each unit into 3 equal parts. Represent $-\frac{7}{3}$ , $-1\frac{2}{3}$ , $\frac{2}{3}$ and $\frac{4}{3}$ on it.

ShowHide solution

Each part is $\frac{1}{3}$ .

7 divided by 3 is 2 remainder 1, so $-\frac{7}{3} = -2\frac{1}{3}$ , which lies between $-3$ and $-2$ . Start at $-2$ and count 1 part to the left.

$-1\frac{2}{3}$ lies between $-2$ and $-1$ . Start at $-1$ and count 2 parts to the left.

$\frac{2}{3}$ lies between 0 and 1. Start at 0 and count 2 parts to the right.

4 divided by 3 is 1 remainder 1, so $\frac{4}{3} = 1\frac{1}{3}$ , which lies between 1 and 2. Start at 1 and count 1 part to the right.

The four numbers are marked on the number line. Check: the point for $-\frac{7}{3}$ is 7 parts to the left of 0, and the point for $\frac{4}{3}$ is 4 parts to the right of 0.

Question 3

Draw a number line from $-2$ to 1 and divide each unit into 10 equal parts. Represent $-1.3$ , $-0.45$ and $0.8$ on it.

ShowHide solution

Each part is $0.1$ .

$-1.3$ lies between $-2$ and $-1$ . Start at $-1$ and count 3 parts to the left.

$0.8$ lies between 0 and 1. Start at 0 and count 8 parts to the right.

$-0.45$ lies between $-0.5$ and $-0.4$ , halfway between them. Start at 0, count 4 parts to the left, then go half a part further.

The three numbers are marked on the number line. From left to right they are $-1.3$ , $-0.45$ and $0.8$ .

Question 4

Draw a number line from $-3$ to 2. Represent $\frac{1}{3}$ , $-2.5$ , $1\frac{3}{4}$ and $-\frac{3}{4}$ on it. Then write the four numbers in ascending order.

ShowHide solution

$\frac{1}{3}$ lies between 0 and 1. Divide that unit into 3 equal parts, start at 0 and count 1 part to the right.

$-2.5 = -2\frac{1}{2}$ lies between $-3$ and $-2$ . Divide that unit into 2 equal parts, start at $-2$ and count 1 part to the left.

$1\frac{3}{4}$ lies between 1 and 2. Divide that unit into 4 equal parts, start at 1 and count 3 parts to the right.

$-\frac{3}{4}$ lies between $-1$ and 0. Divide that unit into 4 equal parts, start at 0 and count 3 parts to the left.

Ascending order is the order of the points from left to right.

In ascending order: $-2.5$ , $-\frac{3}{4}$ , $\frac{1}{3}$ , $1\frac{3}{4}$ .

Question 5

Write $<$ or $>$ between the two numbers of each pair.
(a) $-\frac{5}{6}$ and $-\frac{7}{9}$
(b) $-0.45$ and $-\frac{4}{9}$
(c) $-1\frac{2}{3}$ and $-1.6$
(d) $\frac{3}{-8}$ and $0.05$

ShowHide solution

(a) The LCM of 6 and 9 is 18: $\frac{5}{6} = \frac{15}{18}$ and $\frac{7}{9} = \frac{14}{18}$ . So $-\frac{5}{6}$ has the larger numerical value. It is further from zero, so it is the smaller number.

(b) 40 divided by 9 is 4 remainder 4 every time, so $\frac{4}{9} = 0.444\ldots$ Since $0.45 > 0.444\ldots$ , $-0.45$ is further from zero and is the smaller number.

(c) $-1\frac{2}{3} = -1.666\ldots$ Since $1.666\ldots > 1.6$ , $-1\frac{2}{3}$ is further from zero and is the smaller number.

(d) $\frac{3}{-8} = -\frac{3}{8}$ , which is negative, and $0.05$ is positive.

(a) $-\frac{5}{6} < -\frac{7}{9}$ (b) $-0.45 < -\frac{4}{9}$ (c) $-1\frac{2}{3} < -1.6$ (d) $\frac{3}{-8} < 0.05$

Question 6

(a) Is $-\frac{13}{4} \ge -3\frac{1}{4}$ true or false?
(b) Is $-0.3 \le -\frac{1}{3}$ true or false?
(c) List all the integers $n$ for which $-\frac{9}{4} \le n < \frac{5}{3}$ .

ShowHide solution

(a) 13 divided by 4 is 3 remainder 1, so $-\frac{13}{4} = -3\frac{1}{4}$ . The two numbers are equal, and $\ge$ allows "equal to".

(b) $-\frac{1}{3} = -0.333\ldots$ , which is further from zero than $-0.3$ . So $-0.3$ is greater than $-\frac{1}{3}$ . It is neither less than $-\frac{1}{3}$ nor equal to it.

(c) $-\frac{9}{4} = -2\frac{1}{4}$ , which lies between $-3$ and $-2$ , so the integers start at $-2$ . $\frac{5}{3} = 1\frac{2}{3}$ , which lies between 1 and 2, so the integers end at 1.

(a) True (b) False (c) $-2$ , $-1$ , 0 and 1

Question 7

Arrange $-\frac{3}{5}$ , $0.6$ , $-1$ , $-0.65$ , $\frac{2}{3}$ and $-\frac{2}{3}$ in descending order.

ShowHide solution

As decimals: $-\frac{3}{5} = -0.6$ , $\frac{2}{3} = 0.666\ldots$ and $-\frac{2}{3} = -0.666\ldots$

The positive numbers are the greatest: $0.666\ldots > 0.6$ .

Of the negative numbers, the one closest to 0 is the greatest: $-0.6 > -0.65 > -0.666\ldots > -1$ .

In descending order: $\frac{2}{3}$ , $0.6$ , $-\frac{3}{5}$ , $-0.65$ , $-\frac{2}{3}$ , $-1$ .

Question 8

Five friends each guess the mass of a watermelon. For each friend, the guess minus the true mass, in kilograms, is: Aisha $-0.35$ , Ben $-\frac{3}{8}$ , Chitra $\frac{1}{5}$ , Daniel $-0.4$ and Ethan $0.15$ . A negative number means that the guess was too low.
(a) Arrange the five numbers in ascending order.
(b) Who made the highest guess?
(c) Whose guess was nearest to the true mass?

ShowHide solution

(a) As decimals, $-\frac{3}{8} = -0.375$ and $\frac{1}{5} = 0.2$ . Of the negative numbers, the one further from zero is smaller: $-0.4 < -0.375 < -0.35$ . For the positive numbers, $0.15 < 0.2$ .

(b) The highest guess has the greatest number.

(c) The nearest guess has the number nearest to 0, which is the one with the smallest numerical value. The numerical values are $0.35$ , $0.375$ , $0.2$ , $0.4$ and $0.15$ .

(a) $-0.4$ , $-\frac{3}{8}$ , $-0.35$ , $0.15$ , $\frac{1}{5}$ (b) Chitra made the highest guess. (c) Ethan's guess was the nearest: it was 0.15 kg too high.

Question 9

Divers measure heights from sea level, in metres. A negative height is below sea level. The top of a buoy is at $0.4$ m, a reef is at $-2\frac{3}{4}$ m, a wreck is at $-2.8$ m and the mouth of a cave is at $-2\frac{2}{3}$ m.
(a) Arrange the four heights in ascending order.
(b) Which of the four is the deepest?
(c) A fish swims at $-2.7$ m. Is it above or below the reef? Is it above or below the mouth of the cave?

ShowHide solution

(a) As decimals, $-2\frac{3}{4} = -2.75$ and $-2\frac{2}{3} = -2.666\ldots$ Of the negative numbers, the one further from zero is smaller: $-2.8 < -2.75 < -2.666\ldots$ The only positive number, $0.4$ , is the greatest.

(b) The deepest has the lowest height, which is the smallest number.

(c) $-2.7$ is closer to zero than $-2.75$ , so $-2.7 > -2\frac{3}{4}$ . And $-2.7$ is further from zero than $-2.666\ldots$ , so $-2.7 < -2\frac{2}{3}$ .

(a) $-2.8$ , $-2\frac{3}{4}$ , $-2\frac{2}{3}$ , $0.4$ (b) The wreck is the deepest, at $-2.8$ m. (c) The fish is above the reef and below the mouth of the cave: $-2\frac{3}{4} < -2.7 < -2\frac{2}{3}$ .

Question 10

The temperature inside an ice-cream freezer, $t$ °C, must be at most $-18$ °C. It is read four times in one day: $-18.5$ °C, $-17\frac{3}{4}$ °C, $-18$ °C and $-18\frac{1}{4}$ °C.
(a) Write "at most $-18$ " as a statement about $t$ .
(b) Which of the four readings are allowed?
(c) Which reading is the lowest?

ShowHide solution

(a) "At most $-18$ " means $-18$ or anything less.

(b) $-18.5$ is to the left of $-18$ on the number line, so it is less than $-18$ . $-17\frac{3}{4}$ lies between $-18$ and $-17$ , to the right of $-18$ , so it is greater than $-18$ . $-18$ is equal to $-18$ . $-18\frac{1}{4}$ lies between $-19$ and $-18$ , so it is less than $-18$ .

(a) $t \le -18$ (b) $-18.5$ °C, $-18$ °C and $-18\frac{1}{4}$ °C are allowed. $-17\frac{3}{4}$ °C is too warm. (c) The lowest reading is $-18.5$ °C.

Key Points

Every rational number has its own point on the number line. Positive numbers are to the right of 0 and negative numbers are to the left.
To place a fraction on the number line, divide the unit it lies in into as many equal parts as its denominator. Start at the integer nearer to 0 and count as many parts as the numerator, away from 0.
To place a decimal with one decimal place, divide the unit into 10 equal parts: $-0.7$ is 7 parts to the left of 0.
To read a number off a number line, count the equal parts between two integers that are next to each other. That gives the size of one part.
A number further to the right is greater. Of two negative numbers, the one further from zero is the smaller: $-\frac{3}{4} < -\frac{1}{2}$ .
To compare two rational numbers, write both as fractions with a common denominator or write both as decimals. A statement with $\le$ or $\ge$ is also true when the two numbers are equal.
To order fractions, decimals and integers together, write them all as decimals.