Adding and subtracting integers
🎯 In this topic you will
- Add and subtract with positive and negative integers
🧠 Key Words
- integers
- inverse
- inverse operation
- number line
- negative integers
- positive integers
Show Definitions
- integers: Whole numbers that can be positive, negative, or zero.
- inverse: The opposite or reverse of something in mathematics, such as an operation.
- inverse operation: An operation that reverses the effect of another, like subtraction is the inverse of addition.
- number line: A straight line with numbers placed at equal intervals, used to show numerical values and operations.
- negative integers: Whole numbers less than zero, like -1, -2, -3, etc.
- positive integers: Whole numbers greater than zero, like 1, 2, 3, etc.
📚 Integers
When you count objects, you use the positive whole numbers 1, 2, 3, 4, …
Whole numbers are the first numbers that humans invented.
You can use these numbers for more than counting.
For example, to measure temperature it is useful to have the number 0 (zero) and negative whole numbers −1, −2, −3, …
You can put these numbers on a number line.
Integers greater than zero are positive integers: 1, 2, 3, 4, …
Integers less than zero are negative integers: −1, −2, −3, −4, …
Positive and negative whole numbers together with zero are called integers.
💡 Quick Math Tip
Using “...”: The ellipsis (…) shows that a list continues indefinitely.
➖ Inverse operations with integers
Subtraction is the inverse operation of addition.
The inverse of 3 is –3. The inverse of –5 is 5.
To subtract an integer, you add the inverse.
You can draw a number line to help you.
❓ EXERCISES
1. Do these additions.
- a) $-3 + 4$
- b) $3 + (-7)$
- c) $-4 + (-4)$
- d) $9 + (-5)$
👀 Show answer
b) $-4$
c) $-8$
d) $4$
2. Do these subtractions.
- a) $-1 - 5$
- b) $3 - (-5)$
- c) $-3 - 7$
- d) $-4 - (-6)$
👀 Show answer
b) $8$
c) $-10$
d) $2$
3. Work out:
- a) $4 + (-6)$
- b) $4 - (-6)$
- c) $-4 + 6$
- d) $-4 - 6$
👀 Show answer
b) $10$
c) $2$
d) $-10$
4. Work out the missing integers.
- a) $6 + \square = 10$
- b) $6 + \square = 4$
- c) $6 + \square = -4$
- d) $6 + \square = 0$
👀 Show answer
b) $-2$
c) $-10$
d) $-6$
5. Two integers add up to $-4$. One of the integers is $5$. Work out the other integer.
👀 Show answer
6. $-1$ and $7$ is a pair of integers that add up to $6$.
- a) Find four pairs of integers that add up to $1$.
- b) How can you see immediately that two integers add up to $1$?
💡 Quick Math Tip
Pair of integers: This simply means two integers.
👀 Show answer
b) If the two numbers are $x$ and $1 - x$, their sum is always $1$
7. ● and ▲ are two integers.
- a) Show that ● + ▲ and ▲ + ● have the same value.
- b) Do ● − ▲ and ▲ − ● have the same value? Give evidence to justify your answer.
👀 Show answer
b) No, subtraction is not commutative. For example, if ● = 3 and ▲ = 2, then ● − ▲ = 1, but ▲ − ● = −1.
8. Copy and complete this addition table.
| + | $-4$ | $6$ | $-2$ |
|---|---|---|---|
| $3$ | $9$ | ||
| $-5$ |
👀 Show answer
| + | $-4$ | $6$ | $-2$ |
|---|---|---|---|
| $3$ | $-1$ | $9$ | $1$ |
| $-5$ | $-9$ | $1$ | $-7$ |
9. Copy and complete these addition pyramids. The first one has been started for you.
👀 Show answer
b) Middle: $-5$, $2$; Top: $-3$
c) Middle: $-2$, $-2$; Top: $-4$
d) Middle: $-1$, $5$; Top: $4$
e) Middle: $-13$, Top: $-20$
10. Estimate the answers to these questions. Round the numbers to the nearest whole number.
- a) $-3.14 + 8.26$
- b) $-5.93 - 6.37$
- c) $3.2 - (-6.73)$
- d) $-13.29 + (-5.6)$
👀 Show answer
b) $-6 - 6 = -12$
c) $3 + 7 = 10$
d) $-13 + (-6) = -19$
11. Estimate the answers to these questions.
- a) $-67 + 29$
- b) $-82 - 47$
- c) $688 - (-512)$
- d) $-243 + (-514)$
👀 Show answer
b) About $-80 - 50 = -130$
c) About $690 + 510 = 1200$
d) About $-240 + (-510) = -750$
12. Work out:
- i) $-3 + 4 + -5$
- ii) $-5 + 4 + -3$
- iii) $-3 + -5 + 4$
- iv) $4 + -3 + -5$
💡 Quick Math Tip
Adding step by step: For part i, first add −3 and 4. Then add −5 to that result.
What do the answers show? Is this true for any three integers?
👀 Show answer
ii) $-5 + 4 + -3 = -4$
iii) $-3 + -5 + 4 = -4$
iv) $4 + -3 + -5 = -4$
The answer is always $-4$, showing that the sum is the same regardless of grouping or order (commutative and associative properties). Yes, this is true for any set of three integers — their sum is independent of the order in which they are added.
🧠 Think like a Mathematician
Question: What patterns can you observe by completing and analyzing a simple addition table?
Equipment: Pencil, paper
Follow-up Questions:
Copy and complete this addition table:
| + | −5 | 7 |
|---|---|---|
| 4 | ||
| −3 |
👀 Show Answers
| + | −5 | 7 |
|---|---|---|
| 4 | −1 | 11 |
| −3 | −8 | 4 |
- 1:$-1 + 11 + (-8) + 4 = 6$
- 2:$4 + (-3) + (-5) + 7 = 3$
- 3: The sum of the values inside the table is always double the sum of the row and column headers. This happens because each cell is formed by adding a row and column label, so the total sum inside the table equals the sum of all possible row-column combinations — effectively twice the total of the headers.
❓ EXERCISES
14. Three integers are equally spaced on a number line. Two of the integers are $-3$ and $7$. What is the other integer? Is there more than one possible answer?
👀 Show answer
There is only one solution if $-3$ and $7$ are the outer numbers. But if $-3$ and $7$ are the first and second values, another integer like $17$ could also complete the sequence. So multiple answers are possible depending on the interpretation.