Negative Numbers and the Number Line

What negative numbers and integers are, and how to place, read, compare and order them on the number line.

Before you start

The whole numbers are 0, 1, 2, 3 and so on.

On a number line they are marked at equal spaces, and a number further to the right is greater: 7 is to the right of 4, so 7 is greater than 4.

Negative Numbers

Some quantities can go below zero. We say what 0 stands for, and write a minus sign in front of an amount below it.

Setting	What 0 stands for	A number below zero
Temperature	0 °C, at which water freezes	$-18$ °C is 18 °C below 0 °C (a freezer)
Floors of a building	the ground floor	$-2$ is 2 floors below the ground floor (Basement 2)
Money	no profit and no loss	−$350 is a loss of $350
Height	sea level	$-15$ m is 15 m below sea level (a diver)

Definitions

A number less than zero is a negative number. It is written with a minus sign in front: $-5$ is read "negative five".

A number greater than zero is a positive number. It is written with a plus sign or with no sign: $+5$ and 5 are the same number.

Zero is neither positive nor negative.

Example 1: Writing positive and negative numbers

Write each quantity as a positive or a negative number.
(a) A temperature 4 °C below 0 °C.
(b) The profit of a company in a year when it made a loss of $50 500.
(c) The height of Bukit Timah Hill, which is 164 m above sea level to the nearest metre.

Solution

(a) Below 0 °C, so the number is negative.

(b) A loss is a profit below zero, so the number is negative.

(a) $-4$ °C (b) −$50 500 (c) $+164$ m, or 164 m

Integers

Negative numbers, like positive numbers, can be decimals or fractions: $-12.5$ and $-\frac{3}{2}$ are negative numbers. Those with no fraction or decimal part join the whole numbers to make the integers.

Definition

The integers are the numbers $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$ The numbers $1, 2, 3, \ldots$ are the positive integers and $-1, -2, -3, \ldots$ are the negative integers. Zero is an integer that is neither positive nor negative.

Example 2: Picking out the integers

Which of these numbers are integers?
$-45$ , $+\frac{1}{2}$ , $+0.59$ , 21, $-\frac{100}{9}$ , 0

Solution

$-45$ , 21 and 0 are in the list $\ldots, -2, -1, 0, 1, 2, \ldots$ , so they are integers.

$+\frac{1}{2}$ and $+0.59$ lie between 0 and 1, so they are not integers.

$100 \div 9 = 11$ remainder 1, so $-\frac{100}{9}$ has a fraction part. It is not an integer.

The integers are $-45$ (a negative integer), 21 (a positive integer) and 0.

The Number Line

Every number has its own place on a number line.

Drawing a number line

Draw a horizontal line and mark a point on it as 0.
Mark $1, 2, 3, \ldots$ at equal spaces to the right of 0.
Mark $-1, -2, -3, \ldots$ at the same spacing to the left of 0.

So 3 is 3 units to the right of 0, and $-3$ is 3 units to the left of 0.

The arrows at the two ends show that the line goes on without end in both directions. A number line can also be drawn upright, like a thermometer. Then the positive numbers are above 0 and the negative numbers are below it.

Example 3: Placing numbers on a number line

In a town with cold winters, the temperature at noon was $-14$ °C on a day in January, $-7$ °C on a day in February, 5 °C on a day in April and 20 °C on a day in June. Show the four temperatures on a number line.

Solution

Draw a number line from $-20$ to 20 with a small space for each unit, and label every 10.

$-14$ is 14 units to the left of 0: go to $-10$ , then 4 more spaces to the left.

$-7$ is 7 units to the left of 0. 5 and 20 are 5 units and 20 units to the right of 0.

From left to right, the points are $-14$ °C, $-7$ °C, 5 °C and 20 °C.

Example 4: Reading a number line

Write down the integers marked A, B, C and D on this number line.

Solution

First find the size of one space. There are 5 spaces from 0 to 5, so each space is $5 \div 5 = 1$ unit.

A is 2 spaces to the left of $-5$ . Going left from $-5$ , the numbers are $-6$ and then $-7$ .

B is 2 spaces to the left of 0. C is 3 spaces to the right of 0. D is 1 space to the right of 5.

A is $-7$ , B is $-2$ , C is 3 and D is 6.

Example 5: When one space is not 1 unit

Write down the integers marked K, L, M and N on this number line.

Solution

Find the size of one space from two labelled marks. There are 4 spaces from 0 to 20, so each space is $20 \div 4 = 5$ units.

K is 1 space to the left of $-20$ , and 5 units to the left of $-20$ is $-25$ .

L is 1 space to the left of 0. M is 3 spaces to the right of 0: 5, 10, then 15.

N is 2 spaces to the right of 20: 25, then 30.

K is $-25$ , L is $-5$ , M is 15 and N is 30.

Numerical Value

On the number line, 3 and $-3$ are on opposite sides of 0, but they are the same distance from it.

Definition

The distance of a number from 0 is its numerical value: 3 and $-3$ are both 3 units from 0, so both have numerical value 3.

The numerical value of a number is the number without its sign. The numerical value of 0 is 0.

Example 6: Finding numerical values

(a) Write down the numerical value of $-6$ and of 4.
(b) Which of $-6$ and 4 is further from 0?
(c) Write down the two integers that have numerical value 2.

Solution

(a) $-6$ is 6 units to the left of 0, and 4 is 4 units to the right of 0.

(b) 6 units is more than 4 units, so $-6$ has the larger numerical value.

(a) 6 and 4 (b) $-6$ is further from 0. (c) 2 and $-2$

Greater Than, Less Than and Ordering

Rule

On a number line, the numbers get greater from left to right. A number is greater than every number to its left and less than every number to its right. On an upright number line, the numbers get greater from the bottom to the top: a number is greater than every number below it and less than every number above it.

"Greater than" and "less than" say the same thing from the two ends: 7 is greater than 4, and 4 is less than 7. Every negative number is less than 0 and less than every positive number. Of two negative numbers, the one further from zero, which is the one with the larger numerical value, is the smaller.

Example 7: Comparing two numbers

Which is greater?
(a) $-8$ or $-3$
(b) 0 or $-6$

Solution

(a) $-3$ is to the right of $-8$ , so $-3$ is greater than $-8$ . Said the other way round, $-8$ is less than $-3$ .

(b) 0 is to the right of $-6$ , so 0 is greater than $-6$ .

(a) $-3$ is greater than $-8$ . (b) 0 is greater than $-6$ .

Example 8: Ascending and descending order

Arrange 3, $-2$ , 0, $-5$ , 7 and $-1$
(a) in ascending order (from the least to the greatest),
(b) in descending order (from the greatest to the least).

Solution

Mark the six numbers on a number line.

(a) Ascending order is their order from left to right.

(b) Descending order is their order from right to left.

(a) $-5$ , $-2$ , $-1$ , 0, 3, 7 (b) 7, 3, 0, $-1$ , $-2$ , $-5$

Example 9: Ordering without a number line

Without drawing a number line, arrange $-15$ , 8, $-40$ , 0 and $-7$ in ascending order.

Solution

Sort the numbers into negative numbers, zero and positive numbers. $-15$ , $-40$ and $-7$ are negative, and 8 is positive.

Every negative number is less than 0, so the negative numbers come first. Of these, the one furthest from zero is the least: $-40$ , then $-15$ , then $-7$ .

Next comes 0, and last the positive number, 8.

In ascending order, the numbers are $-40$ , $-15$ , $-7$ , 0, 8.

Common Mistakes

Saying $-8$ is greater than $-3$ because 8 is greater than 3. $-8$ is to the left of $-3$ on the number line, so $-8$ is less than $-3$ .
Calling zero positive. Zero is neither positive nor negative.
Counting the wrong way to the left of zero. One step to the left of $-5$ is $-6$ , not $-4$ .
Taking every space on a number line as 1 unit. Look at the labels first: if there are 5 spaces from 0 to 10, each space is 2 units.

Practice

Work each question on paper before you open its solution.

Question 1

Write each quantity as a positive or a negative number.
(a) The position of a submarine 150 m below sea level.
(b) The change in Mr Tan's bank balance when his salary of $7000 is paid in.
(c) The change in his bank balance when he then takes out $500.

ShowHide solution

(a) Below sea level, so the number is negative.

(b) Money paid in makes the balance go up, so the change is positive.

(a) $-150$ m (b) +$7000 (c) −$500

Question 2

Say in words what each number tells you.
(a) A lift is at floor $-3$ , where the ground floor is floor 0.
(b) The profit of a stall for one week is −$120.
(c) The change in the water level of a reservoir is $+40$ cm in one week and $-15$ cm in the next week.

ShowHide solution

(a) The number is negative, so the lift is below the ground floor.

(b) A profit below zero is a loss.

(a) The lift is 3 floors below the ground floor (Basement 3). (b) The stall made a loss of $120. (c) The water level rose by 40 cm in the first week and fell by 15 cm in the next week.

Question 3

From the numbers $-9$ , $\frac{4}{5}$ , 0, 17, $-2.5$ and $+6$ , write down
(a) the positive integers,
(b) the negative integers,
(c) the number that is neither positive nor negative,
(d) all the negative numbers.

ShowHide solution

$\frac{4}{5}$ lies between 0 and 1, and $-2.5$ has a decimal part, so they are not integers.

$+6$ is the same number as 6.

Part (d) asks for negative numbers, not only negative integers, so $-2.5$ is included.

(a) 17 and 6 (b) $-9$ (c) 0 (d) $-9$ and $-2.5$

Question 4

Write down the integers marked P, Q, R and S on this number line.

ShowHide solution

There are 5 spaces from 0 to 10, so each space is $10 \div 5 = 2$ units.

Counting in twos to the left of 0, the marks are $-2$ , $-4$ , $-6$ , $-8$ , $-10$ and $-12$ . Q is on the second of these and P is on the sixth.

Counting in twos to the right of 0, the marks are 2, 4, 6 and 8. R is on the first of these and S is on the fourth.

P is $-12$ , Q is $-4$ , R is 2 and S is 8.

Question 5

This upright scale shows heights in metres, with sea level marked 0. A seagull is at G, a diver is at D and a shipwreck is at W.
(a) How many metres does one space on the scale stand for?
(b) Write down the heights of G, D and W.
(c) Which of the three is furthest from sea level?

ShowHide solution

(a) There are 4 spaces from 0 to 100, so each space is $100 \div 4 = 25$ m.

(b) On an upright number line, the positive numbers are above 0 and the negative numbers are below it. G is 2 spaces above 0: 25, then 50.

D is 1 space below 0. W is 3 spaces below 0: $-25$ , $-50$ , then $-75$ .

(a) 25 m (b) G is at 50 m, D is at $-25$ m and W is at $-75$ m. (c) The shipwreck, which is 75 m below sea level.

Question 6

(a) Write down the numerical value of $-12$ , of 5 and of 0.
(b) Write down the two integers that have numerical value 7.
(c) Which of $-9$ and 6 has the larger numerical value? Which of them is the greater number?

ShowHide solution

(a) $-12$ is 12 units to the left of 0, and 5 is 5 units to the right of 0. 0 is at 0 itself, so its distance from 0 is 0.

(b) The numbers 7 units from 0 are 7, which is 7 units to the right of 0, and $-7$ , which is 7 units to the left of 0.

(c) $-9$ is 9 units from 0 and 6 is 6 units from 0, so $-9$ has the larger numerical value. But $-9$ is to the left of 6 on the number line, so 6 is the greater number.

(a) 12, 5 and 0 (b) 7 and $-7$ (c) $-9$ has the larger numerical value. 6 is the greater number.

Question 7

(a) Which is greater, $-12$ or $-9$ ?
(b) Which is greater, 0 or $-4$ ?
(c) Which is less, 5 or $-20$ ?
(d) Which is less, $-31$ or $-13$ ?

ShowHide solution

(a) Both numbers are negative, and $-12$ is further from zero, so it is the smaller.

(b) 0 is to the right of every negative number.

(d) Both numbers are negative. $-31$ is 31 units from zero and $-13$ is 13 units from zero, so $-31$ is further to the left.

(a) $-9$ (b) 0 (c) $-20$ (d) $-31$

Question 8

One morning in December the temperature was $-8$ °C in Seoul, 2 °C in Tokyo, $-15$ °C in Harbin and $-3$ °C in Beijing.
(a) Show the four temperatures on a number line.
(b) Which city was the coldest?
(c) Arrange the four temperatures in descending order.
(d) Arrange the four temperatures in ascending order.

ShowHide solution

(a) The number line runs from $-16$ to 4 and is labelled every 2. $-15$ is halfway between $-16$ and $-14$ , and $-3$ is halfway between $-4$ and $-2$ .

(b) The coldest city has the lowest temperature, the number furthest to the left.

(d) Ascending order starts with the least number, so read the marked numbers from left to right.

Check: of the negative numbers, $-3$ is the nearest to zero, so it is the greatest of them.

(b) Harbin was the coldest, at $-15$ °C. (c) 2 °C, $-3$ °C, $-8$ °C, $-15$ °C (d) $-15$ °C, $-8$ °C, $-3$ °C, 2 °C

Question 9

In a quiz, a right answer gains points and a wrong answer loses points, so a score can be below zero. After five rounds the scores are: Aisha 15, Ben $-10$ , Chandra $-25$ , Devi 0 and Ethan $-5$ .
(a) Who has the lowest score?
(b) Arrange the five scores in descending order.
(c) Which pupils have a score that is less than zero but greater than Ben's?
(d) Farah's score is an integer that is greater than Ben's and less than Ethan's. Write down all the scores she could have.

ShowHide solution

(a) The lowest score is the number furthest to the left on the number line. Of the three negative scores, $-25$ is the furthest from zero.

(b) Start with the positive score, then zero, then the negative scores, nearest to zero first.

(d) Her score lies between $-10$ and $-5$ on the number line, and is neither of them.

(a) Chandra, with $-25$ (b) 15, 0, $-5$ , $-10$ , $-25$ (c) Only Ethan, with $-5$ (d) $-9$ , $-8$ , $-7$ or $-6$

Key Points

A negative number is less than zero and is written with a minus sign. A positive number is greater than zero. Zero is neither positive nor negative.
The integers are $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$
On a number line, positive numbers are to the right of 0 and negative numbers are to the left of 0, at equal spaces. To read a number line, first find the size of one space.
The numerical value of a number is its distance from 0: 3 and $-3$ both have numerical value 3.
A number further to the right is greater: $-3$ is greater than $-8$ . Of two negative numbers, the one further from zero is the smaller.
Ascending order goes from the least to the greatest. Descending order goes from the greatest to the least.