Practice Maths

Adding & Subtracting Integers

Key Terms

integer
Any whole number, including negatives, zero, and positives: …, −2, −1, 0, 1, 2, …
adding a negative
The same as subtracting the positive; a + (−b) = a − b. Moves left on the number line.
subtracting a negative
The same as adding the positive; a − (−b) = a + b. Moves right on the number line.
opposite
Two integers the same distance from zero on opposite sides; e.g. +3 and −3 are opposites. Their sum is always 0.

Rules at a glance: Adding a negative = subtracting  |  Subtracting a negative = adding  |  Number line: positive = right, negative = left.

Memory trick: Two negatives together always become a positive. Think of it as "two wrongs make a right" in maths!

Worked Examples

Example 1: −8 + 3

Start at −8, move 3 to the right.   Answer: −5

Example 2: 7 − (−4)

Subtracting a negative = adding: 7 + 4 = 11

Example 3: −5 + (−4)

Adding a negative = subtracting: −5 − 4 = −9

What Are Integers and Why Do Negatives Exist?

Integers are the whole numbers ..., −3, −2, −1, 0, 1, 2, 3, ... They extend the counting numbers to include zero and negatives. Negatives have real-world meaning: temperatures below zero, debt (owing money), floors below ground in a building, and depths below sea level. Understanding operations with negatives is essential because algebra, physics, and finance all use them constantly.

The number line is the most reliable tool. Positive moves take you to the right; negative moves take you to the left. From any starting point, addition moves right and subtraction moves left — in the direction of the operation's sign. Start at −4 and add 7: move 7 right, reaching +3. Start at 5 and subtract 8: move 8 left, reaching −3.

Adding a Negative: Same as Subtracting

5 + (−3) means "start at 5 and move 3 to the left" (because −3 is a leftward movement). Result: 2. This is identical to 5 − 3 = 2. So adding a negative is the same as subtracting the positive value. The two symbols +(–) simplify to a single –. This is why we write 5 + (−3) = 5 − 3 = 2.

Another example: −6 + (−4). Start at −6, move 4 more to the left: reach −10. Equivalently: −6 − 4 = −10. When two negatives are "added," the magnitudes (sizes) add up and the result is more negative.

Key Tip: Two signs together always simplify: +(+) = +, +(−) = −, −(+) = −, −(−) = +. The rule you need here is "adding a negative = subtracting" and "subtracting a negative = adding." If you see two signs, combine them first before calculating.

Subtracting a Negative: Same as Adding

7 − (−4): we're subtracting a negative 4. Subtracting a negative reverses the subtraction, making it an addition: 7 + 4 = 11. Intuitively: if a negative temperature change of −4°C is removed (subtracted), the temperature goes up by 4°C. Another intuition: the "negative of a negative" is positive. The two negatives cancel each other, leaving a net positive direction.

Example: −3 − (−8) = −3 + 8 = 5. Check on number line: start at −3, move 8 to the right (because we're now adding), reach +5. Correct!

Key Tip: When subtracting a negative, rewrite it as addition first, then calculate. −5 − (−9) → write as −5 + 9 = 4. Always do this rewriting step first — it prevents the most common integer errors and makes the calculation straightforward.

Applying Integers to Real Situations

Temperature changes: if it's −7°C and the temperature rises by 12°C, the new temperature is −7 + 12 = 5°C. Bank account: you have −$45 (in debt) and deposit $60: −45 + 60 = $15 (now positive). Altitude: a submarine at −120 m rises 35 m: −120 + 35 = −85 m (still below surface). In each case, setting up the integer calculation carefully — paying attention to signs — gives the right answer. Integers are not just abstract; they describe real change.

Mastery Practice

  1. Adding Integers Fluency

    1. −8 + 3
    2. −5 + (−4)
    3. 6 + (−9)
    4. −12 + 7
    5. −3 + (−8)
    6. 4 + (−4)
    7. −15 + 6
    8. −7 + (−5)

  2. Subtracting Integers Fluency

    1. 7 − (−3)
    2. −4 − 6
    3. 2 − (−8)
    4. −9 − (−4)
    5. 0 − (−5)
    6. −3 − (−3)
    7. 11 − (−6)
    8. −7 − 8

  3. Mixed Addition and Subtraction Fluency

    1. −6 + (−2) − 3
    2. 4 − (−5) + (−7)
    3. −3 − (−8) + (−2)
    4. 10 + (−4) − (−1)
    5. −9 − 3 + (−4)
    6. 0 − (−6) + (−6)
    7. −2 + (−7) − (−5)
    8. 8 − (−3) + (−10)

  4. Temperature and Elevation Contexts Understanding

    1. The temperature in Canberra is −3°C in the morning. By midday it has risen 9°C. What is the midday temperature?
    2. A mountain summit is at 1850 m. A valley floor is 1850 m below the summit. What is the elevation of the valley floor?
    3. The temperature drops from 4°C to −6°C overnight. By how much did it fall?
    4. A diver is at −12 m. She descends a further 8 m. What is her new depth?
    5. The overnight low is −7°C. The daytime high is 11°C. What is the range (difference) in temperature?
    6. A hot-air balloon is at 350 m altitude. It descends 420 m. What is its new position?
    7. Death Valley is at −86 m below sea level. Mount Whitney is 4421 m above sea level. What is the difference in their elevations?
    8. The temperature is −2°C. It falls 5°C, then rises 8°C. What is the final temperature?

  5. Find the Missing Integer Understanding

    Find the value of n:

    1. n + (−3) = 5
    2. −7 + n = −2
    3. n − (−4) = 1
    4. 6 + n = −3
    5. n − 8 = −15
    6. −5 + n = 0
    7. n + (−6) = −10
    8. 3 − n = 9

  6. Multi-Step Real-World Problems Problem Solving

    1. Priya's bank account starts with a balance of $45. She spends $60 on Monday, then receives a $30 transfer on Tuesday, then spends $25 on Wednesday. What is her final balance?
    2. Over five days, the temperature changes recorded are: −3°C, +5°C, −2°C, −4°C, +6°C. If the starting temperature was 2°C, what is the final temperature?
    3. A submarine starts at −30 m. It rises 15 m, then descends 22 m, then rises 8 m. What is its final depth?
    4. A football team's score starts at 0. They gain 7 points, lose 4 points (penalty), gain 12 points, and lose 3 points. What is the final score?
    5. The ground floor of a building is floor 0. A lift starts at floor −3 (carpark). It travels up 5 floors, then down 7 floors, then up 4 floors. What floor does it finish on?
    6. A trader's profit/loss over four weeks is: −$120, +$85, +$200, −$65. What is the total profit or loss?

  7. Number Line Reasoning Understanding

    Use a number line to explain or check your answers:

    1. Start at −4. Move left 6 units. Where do you finish?
    2. Start at 3. Move left 9 units. Where do you finish?
    3. Start at −8. Move right 8 units. Where do you finish?
    4. Start at −5. Move right 12 units. Where do you finish?
    5. Start at 0. Move left 7 units, then right 3 units. Where do you finish?
    6. Start at −2. Move left 4 units, then left 3 units. Where do you finish?

  8. Explain the Rule Understanding

    For each pair of expressions, decide if they are equal. Explain why or why not:

    1. Is −5 + (−3) the same as −5 − 3? Explain.
    2. Is 4 − (−6) the same as 4 + 6? Explain.
    3. Is (−8) − (−8) the same as 0? Explain.
    4. A student says “−3 + (−7) = 4 because two negatives make a positive.” Identify the error and give the correct answer.

  9. Patterns with Integers Understanding

    Find the next three terms in each sequence and state the rule:

    1. 10, 7, 4, 1, …
    2. −20, −15, −10, −5, …
    3. 8, 5, 2, −1, …
    4. −3, −7, −11, −15, …

  10. Extended Problem Solving Problem Solving

    1. The Dead Sea is approximately 430 m below sea level. Mount Everest is approximately 8849 m above sea level. Using integers, write a subtraction to represent the difference in their elevations. What is the difference?
    2. A golfer’s scores for four rounds of golf (relative to par) are: −3, +2, −4, −1. A lower total score is better. What is the golfer’s total score relative to par?
    3. A number line shows three points: A at −9, B at an unknown position, and C at +5. The distance from A to B is 7 units (moving right). The distance from B to C is equal. Is this possible? Find the position of B and verify.
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