Maths4U
« Back to Subchapter

Integers – Adding and Subtracting Integers

📘 Notes

Year 7 — Adding and Subtracting Integers (Positive & Negative)

Integers are whole numbers and their negatives … …, -3, -2, -1, 0, 1, 2, 3, …. Use a number line: right = bigger (add), left = smaller (subtract).

1) Key Ideas

  • Add a positive: move right. Example: \(3 + 4 = 7\).
  • Add a negative: move left. Example: \(3 + (-4) = -1\).
  • Subtract a positive: move left. Example: \(5 - 7 = -2\).
  • Subtract a negative: move right (two negatives make a plus). Example: \( -2 - (-5) = -2 + 5 = 3\).

Tip: If you see “\(-\)(negative)”, change it to “\(+\)(positive)”.

2) Sign Rules (Quick Reference)

OperationRuleExample
Add a positive Go right on the number line \( -3 + 6 = 3 \)
Add a negative Go left on the number line \( 4 + (-7) = -3 \)
Subtract a positive Go left \( 2 - 5 = -3 \)
Subtract a negative Change to add a positive \( -6 - (-4) = -6 + 4 = -2 \)

3) Method A — Number Line

  1. Start at the first number.
  2. For “\(+\) \(k\)”, jump \(k\) steps to the right. For “\(-\) \(k\)”, jump \(k\) steps to the left.
Example 1: \( -4 + 7 \)

Start at \(-4\). Add \(+7\) → move right 7 steps → \(\boxed{3}\).

Example 2: \( 5 - 9 \)

Start at \(5\). Subtract \(9\) → move left 9 steps → \(\boxed{-4}\).

Example 3: \( -2 - (-6) \)

“Subtract a negative” → change to add: \( -2 + 6 = \boxed{4}\).

4) Method B — Same Sign vs Different Sign (for Addition)

Adding integers (including negatives) can be done by size comparison:

  • Same sign: add the sizes, keep the sign.
    \( (-3) + (-8) = -(3+8) = \boxed{-11} \)
  • Different signs: subtract the smaller size from the bigger size, keep the sign of the bigger size.
    \( (+9) + (-4) = 9 - 4 = \boxed{+5} \)
Example 4: \( -12 + 5 \)

Different signs → \(12-5=7\). Bigger size is 12 (negative), so answer \(\boxed{-7}\).

Example 5: \( -6 + (-7) \)

Same sign (both negative) → \(6+7=13\), keep negative → \(\boxed{-13}\).

5) Subtracting Integers = Add the Opposite

Change every subtraction into addition by flipping the sign of the next number:

\[ a - b \;=\; a + (-b). \]

Example 6: \( 7 - (-3) \)

Change: \(7 - (-3) = 7 + 3 = \boxed{10}\).

Example 7: \( -9 - 4 \)

Change: \(-9 + (-4) = \boxed{-13}\).

Example 8: \( -3 - (-10) \)

Change: \(-3 + 10 = \boxed{7}\).

6) Column (Vertical) Layout (to keep signs clear)

Write the numbers under each other with signs clearly shown. Then combine.

Example 9 (Addition): \( -15 + 9 \) 
  -15
+  9
-----
   -6

\(-15 + 9 = -6\) (more negatives than positives).

Example 10 (Subtraction): \( 8 - (-11) \) 
   8
-(-11)
------
   8 + 11 = 19

Subtracting a negative becomes addition → \(\boxed{19}\).

7) Word Problems (real-life)

Example 11: Temperature change

Morning: \(-3^\circ\text{C}\). It warms by \(+7^\circ\text{C}\). New temperature = \(-3 + 7 = \boxed{4^\circ\text{C}}\).

Example 12: Bank balance

Balance \(-\$12\). You deposit \$20. New balance = \(-12 + 20 = \boxed{\$8}\).

Example 13: Lift (elevator) floors

Start at floor \(-2\). Go up 5 floors → \( -2 + 5 = \boxed{3}\) (3rd floor).

8) Quick Practice (Show Answers)

1) \( 6 + (-9) = \;?\)

\(6 - 9 = \boxed{-3}\).

2) \( -8 + (-7) = \;?\)

Same sign → \(8+7=15\), keep negative → \(\boxed{-15}\).

3) \( -4 - 11 = \;?\)

\(-4 + (-11) = \boxed{-15}\).

4) \( 13 - (-5) = \;?\)

\(13 + 5 = \boxed{18}\).

5) \( -10 - (-3) = \;?\)

\(-10 + 3 = \boxed{-7}\).

9) Common Mistakes to Avoid

  • Forgetting that subtracting a negative is the same as adding.
  • Ignoring signs in word problems (e.g., “a loss of 5” means \(-5\)).
  • Mixing up “move right/left” on the number line.