Adding and Subtracting Integers

Adding and subtracting positive and negative integers, first on the number line and then by rules, and using them in word problems.

Before you start

The integers are $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$ On the number line, positive numbers are to the right of 0 and negative numbers are to the left.

The numerical value of a number is the number without its sign. It is the number's distance from 0 on the number line: 5 and $-5$ both have numerical value 5.

Adding on the Number Line

Adding is moving along the number line. The first number says where to start. The second number says which way to move and how far.

A negative number that follows an operation sign ( $+$ , $-$ , $\times$ or $\div$ ) is written in brackets, as in $2 + (-3)$ . The brackets keep its sign apart from the operation sign. At the start of an expression the brackets may be left out: $-3 + 7$ means the same as $(-3) + 7$ . In this lesson they are written there too.

Rule

Start at the first number.
To add a positive number, move to the right. To add a negative number, move to the left.
Move as many units as the numerical value of the number you are adding.

The number where you stop is the sum. Adding 0 is no move at all: $(-4) + 0 = -4$ .

Example 1: Adding on the number line

Use a number line to find
(a) $(-3) + 7$
(b) $2 + (-3)$

Solution

(a) Start at $-3$ . The number added, 7, is positive, so move 7 units to the right. You stop at 4.

(b) Start at 2. The number added, $-3$ , is negative, so move 3 units to the left. You stop at $-1$ .

(a) $(-3) + 7 = 4$
(b) $2 + (-3) = -1$

Adding Two Integers with the Same Sign

Two positive integers are added as in primary school: $2 + 3 = 5$ . With two negative integers, you start on the left of 0 and move further left, so the sum is negative.

Example 2: Two negative integers

A fish is 6 m below the surface of a lake. It swims 5 m further down. How far below the surface is the fish now?

Solution

Take the surface as 0 m and positions below it as negative. The fish starts at $-6$ m.

Swimming 5 m further down adds $-5$ , so its new position is $(-6) + (-5)$ .

Start at $-6$ and move 5 units to the left. You stop at $-11$ .

$(-6) + (-5) = -11$ , so the fish is at $-11$ m.

The fish is now 11 m below the surface.

Rule: same sign

To add two integers with the same sign, add their numerical values and give the answer that sign.

$6 + 5 = 11$ , so $(-6) + (-5) = -11$ . In the same way, $23 + 48 = 71$ , so $(-23) + (-48) = -71$ .

Adding Two Integers with Different Signs

When the signs are different, the two numbers pull in opposite directions. In Example 1, $(-3) + 7$ is positive because 7 has the larger numerical value, and $2 + (-3)$ is negative because $-3$ has the larger numerical value.

Rule: different signs

To add two integers with different signs, subtract the smaller numerical value from the larger one. Give the answer the sign of the number with the larger numerical value. If the two numerical values are equal, the sum is 0.

Example 3: Different signs

Evaluate
(a) $(-8) + 5$
(b) $(-12) + 30$
(c) $17 + (-45)$

Solution

(a) The numerical values are 8 and 5, and $8 - 5 = 3$ . The number with the larger numerical value is $-8$ , so the answer is negative.

(b) $30 - 12 = 18$ . The number with the larger numerical value is 30, so the answer is positive.

(a) $(-8) + 5 = -3$
(b) $(-12) + 30 = 18$
(c) $17 + (-45) = -28$

Opposites

Two numbers with the same numerical value and different signs, such as 5 and $-5$ , are opposites: the opposite of 5 is $-5$ , and the opposite of $-5$ is 5. A number and its opposite add to zero: $5 + (-5) = 0$ and $(-5) + 5 = 0$ . The opposite of 0 is 0.

Subtracting Integers

Subtracting undoes adding: $7 - 5 = 2$ because $2 + 5 = 7$ . So on the number line, subtracting a number moves the opposite way to adding it.

Example 4: Subtracting on the number line

Use a number line to find
(a) $(-4) - 5$
(b) $2 - (-6)$

Solution

(a) Start at $-4$ . Adding 5 would move 5 units to the right, so subtracting 5 moves 5 units to the left. You stop at $-9$ .

Adding $-5$ makes the same move, so $(-4) - 5 = (-4) + (-5) = -9$ .

(b) Start at 2. Adding $-6$ would move 6 units to the left, so subtracting $-6$ moves 6 units to the right. You stop at 8.

Adding 6 makes the same move, so $2 - (-6) = 2 + 6 = 8$ . Check: $8 + (-6) = 2$ .

(a) $(-4) - 5 = -9$
(b) $2 - (-6) = 8$

Rule: subtracting

Subtracting a number is the same as adding its opposite. The first number does not change.

$a - b = a + (-b)$ , as in $3 - 9 = 3 + (-9)$

$a - (-b) = a + b$ , as in $3 - (-9) = 3 + 9$

Here $a$ and $b$ stand for any two numbers, and $-b$ stands for the opposite of $b$ .

Example 5: Using the rule

Evaluate
(a) $3 - 9$
(b) $(-3) - 17$
(c) $(-6) - (-8)$
(d) $15 - (-27)$

Solution

(a) The opposite of 9 is $-9$ , so $3 - 9 = 3 + (-9)$ . The signs are different: $9 - 3 = 6$ , and $-9$ has the larger numerical value, so the answer is negative.

(b) $(-3) - 17 = (-3) + (-17)$ . The signs are the same: $3 + 17 = 20$ , and both numbers are negative.

(c) The opposite of $-8$ is 8, so $(-6) - (-8) = (-6) + 8$ . The signs are different: $8 - 6 = 2$ , and 8 has the larger numerical value, so the answer is positive.

(d) $15 - (-27) = 15 + 27$

(a) $3 - 9 = -6$
(b) $(-3) - 17 = -20$
(c) $(-6) - (-8) = 2$
(d) $15 - (-27) = 42$

Several Integers and Brackets

Brackets round a single negative number, as in $(-5)$ , only keep its sign apart from the operation sign. A sum or a difference in brackets, as in $(4 - 10)$ , is worked out first. Then change every subtraction into adding the opposite and work from left to right. Once every operation is an addition, the numbers may be added in any order.

Example 6: More than two integers

Evaluate
(a) $8 - (-3) + (-5) - 4$
(b) $(-9) - (4 - 10) + (-3)$

Solution

(a) Change each subtraction into adding the opposite: $8 - (-3) + (-5) - 4 = 8 + 3 + (-5) + (-4)$

From left to right: $8 + 3 = 11$ , then $11 + (-5) = 6$ , then $6 + (-4) = 2$ .

Or add the positive numbers and the negative numbers separately: $(8 + 3) + [(-5) + (-4)] = 11 + (-9) = 2$ . The square brackets are a second pair of brackets, used so that the pairs are easy to tell apart.

(b) Brackets first: $4 - 10 = 4 + (-10) = -6$

$(-9) - (-6) + (-3) = (-9) + 6 + (-3)$

From left to right: $(-9) + 6 = -3$ , then $(-3) + (-3) = -6$ .

(a) $8 - (-3) + (-5) - 4 = 2$
(b) $(-9) - (4 - 10) + (-3) = -6$

Word Problems

Decide which direction is positive (up, warmer, money you have). A rise is added and a fall is subtracted. The difference between two values is the higher value minus the lower value.

Example 7: Temperature

One winter day in Seoul, the temperature was 4 °C at noon and $-7$ °C at midnight.
(a) By how many degrees did the temperature fall from noon to midnight?
(b) After midnight the temperature rose by 9 °C. What was it then?

Solution

(a) The fall is the higher temperature minus the lower one: $4 - (-7) = 4 + 7 = 11$

Check on the number line. The drawing below shows the fall itself, not the calculation: from 4 down to 0 is 4 units and from 0 down to $-7$ is 7 more, so it takes 11 units to the left to go from 4 to $-7$ .

(b) A rise of 9 °C adds 9: $(-7) + 9 = 2$

(a) The temperature fell by 11 °C.
(b) The temperature was then 2 °C.

Example 8: Floors

A lift in a shopping mall is in the basement car park, 3 floors below the ground floor. It goes up 8 floors and then down 7 floors. Where is the lift now?

Solution

Take the ground floor as 0 and floors above it as positive. The lift starts at $-3$ .

Going up 8 floors adds 8, and going down 7 floors subtracts 7: $(-3) + 8 - 7$

$(-3) + 8 = 5$ , then $5 - 7 = 5 + (-7) = -2$

The lift is now 2 floors below the ground floor.

Common Mistakes

Taking the sign of the larger number. In $(-8) + 5$ the larger number is 5, but the answer is $-3$ . Take the sign of the number with the larger numerical value, which is $-8$ .
Thinking that two minus signs always give a plus. $(-4) + (-7) = -11$ and $(-4) - 7 = -11$ . Only subtracting a negative number becomes adding a positive number: $3 - (-5) = 3 + 5 = 8$ .
Changing the first number. In $(-2) - 4 = (-2) + (-4)$ , only the number being subtracted is changed to its opposite.

Practice

Work each question on paper before you open its solution.

Question 1

Draw a number line from $-6$ to 6 and use it to find
(a) $(-5) + 9$
(b) $3 + (-8)$
(c) $(-2) - 4$
(d) $1 - (-4)$

ShowHide solution

(a) Start at $-5$ and move 9 units to the right. You stop at 4.

(b) Start at 3 and move 8 units to the left. You stop at $-5$ .

(d) Start at 1. Adding $-4$ would move 4 units to the left, so subtracting $-4$ moves 4 units to the right. You stop at 5.

(a) $(-5) + 9 = 4$
(b) $3 + (-8) = -5$
(c) $(-2) - 4 = -6$
(d) $1 - (-4) = 5$

Question 2

Evaluate
(a) $(-14) + (-29)$
(b) $(-36) + 21$
(c) $52 + (-18)$
(d) $(-47) + 47$

ShowHide solution

(a) Same sign: $14 + 29 = 43$ , and both numbers are negative.

(b) Different signs: $36 - 21 = 15$ , and $-36$ has the larger numerical value, so the answer is negative.

(d) $-47$ and 47 are opposites, so they add to zero.

(a) $(-14) + (-29) = -43$
(b) $(-36) + 21 = -15$
(c) $52 + (-18) = 34$
(d) $(-47) + 47 = 0$

Question 3

Write each subtraction as an addition, then evaluate it.
(a) $6 - 15$
(b) $(-8) - 13$
(c) $(-9) - (-4)$
(d) $20 - (-35)$

ShowHide solution

(a) $6 - 15 = 6 + (-15)$ . Different signs: $15 - 6 = 9$ , and $-15$ has the larger numerical value.

(b) $(-8) - 13 = (-8) + (-13)$ . Same sign: $8 + 13 = 21$ , and both numbers are negative.

(d) $20 - (-35) = 20 + 35$

(a) $6 - 15 = -9$
(b) $(-8) - 13 = -21$
(c) $(-9) - (-4) = -5$
(d) $20 - (-35) = 55$

Question 4

Evaluate $(-5) + 8 - (-3) - 6$ .

ShowHide solution

Change each subtraction into adding the opposite: $(-5) + 8 - (-3) - 6 = (-5) + 8 + 3 + (-6)$

From left to right: $(-5) + 8 = 3$ , then $3 + 3 = 6$ , then $6 + (-6) = 0$ .

$(-5) + 8 - (-3) - 6 = 0$

Question 5

Evaluate each of the following. Look for an order that makes the adding easy.
(a) $(-38) + 57 + 38$
(b) $25 - 60 + 75 - 90$

ShowHide solution

(a) Every operation is an addition, so the numbers may be added in any order. $-38$ and 38 are opposites, so they add to zero: $(-38) + 38 = 0$ , and $0 + 57 = 57$ .

(b) Change each subtraction into adding the opposite: $25 - 60 + 75 - 90 = 25 + (-60) + 75 + (-90)$

Add the positive numbers and the negative numbers separately: $25 + 75 = 100$ and $(-60) + (-90) = -150$ . Then $100 + (-150) = -50$ .

(a) $(-38) + 57 + 38 = 57$
(b) $25 - 60 + 75 - 90 = -50$

Question 6

Evaluate
(a) $18 - (7 - 15) + (-30)$
(b) $(-12) - (-5) - (9 - 20)$

ShowHide solution

(a) Brackets first: $7 - 15 = 7 + (-15) = -8$

$18 - (-8) + (-30) = 18 + 8 + (-30) = 26 + (-30) = -4$

(b) Brackets first: $9 - 20 = 9 + (-20) = -11$

$(-12) - (-5) - (-11) = (-12) + 5 + 11 = (-7) + 11 = 4$

(a) $18 - (7 - 15) + (-30) = -4$
(b) $(-12) - (-5) - (9 - 20) = 4$

Question 7

On a mountain, the temperature was $-7$ °C in the morning. It rose by 12 °C by the afternoon and then fell by 5 °C by night. What was the temperature at night?

ShowHide solution

A rise is added and a fall is subtracted: $(-7) + 12 - 5$

$(-7) + 12 = 5$ , then $5 - 5 = 0$

The temperature at night was 0 °C.

Question 8

Wei Ling owes her mother $40. She pays back $15 and then borrows another $30. How much does she owe her mother now?

ShowHide solution

Money owed is a balance below zero, so Wei Ling starts at −$40.

Paying back $15 adds 15, and borrowing $30 subtracts 30: $(-40) + 15 - 30$

$(-40) + 15 = -25$ , then $(-25) - 30 = (-25) + (-30) = -55$

Wei Ling now owes her mother $55.

Question 9

A drone hovers 35 m above sea level. Straight below it, a diver is 18 m below sea level.
(a) How far apart are the drone and the diver?
(b) The diver then swims 11 m straight up. How far below sea level is the diver now?

ShowHide solution

Take sea level as 0 m and heights above it as positive. The drone is at 35 m and the diver is at $-18$ m.

(a) The distance apart is the higher position minus the lower one: $35 - (-18) = 35 + 18 = 53$

(b) Swimming up 11 m adds 11: $(-18) + 11 = -7$

(a) The drone and the diver are 53 m apart.
(b) The diver is now 7 m below sea level.

Question 10

In a computer game a score can fall below zero. After three rounds Arif has $-35$ points and Devi has $-12$ points.
(a) Who has the higher score, and by how many points?
(b) In the fourth round Arif gains 50 points and Devi loses 20 points. Who has the higher score now, and by how many points?

ShowHide solution

(a) $-12$ is to the right of $-35$ on the number line, so Devi has the higher score. The difference is the higher score minus the lower one: $(-12) - (-35) = (-12) + 35 = 23$

(b) A gain is added: Arif now has $(-35) + 50 = 15$ points. A loss is subtracted: Devi now has $(-12) - 20 = (-12) + (-20) = -32$ points.

15 is greater than $-32$ , so Arif has the higher score. The difference is $15 - (-32) = 15 + 32 = 47$

(a) Devi has the higher score, by 23 points.
(b) Arif has the higher score, by 47 points.

Key Points

On the number line, start at the first number. Adding a positive number moves right; adding a negative number moves left.
Same sign: add the numerical values and keep the sign.
Different signs: subtract the smaller numerical value from the larger, and take the sign of the number with the larger numerical value.
A number and its opposite add to zero: $5 + (-5) = 0$ .
Subtracting a number is the same as adding its opposite: $a - b = a + (-b)$ and $a - (-b) = a + b$ .
With several integers, work out brackets first, change each subtraction into adding the opposite, then add.