Adding and Subtracting Integers — Rules and Examples

Grade: 6-7 | Topic: Arithmetic

What You Will Learn

By the end of this page you will be able to add any two integers -- whether both are positive, both are negative, or one of each -- using clear sign rules. You will also master subtracting integers by converting every subtraction into an addition problem, and you will feel confident applying these skills to multi-step calculations.

Theory

Why integer addition needs special rules

When you first learned addition, every number was positive. Adding $3 + 5$ just meant "move 5 steps to the right on a number line." Negative numbers add a new direction -- moving to the left. That is why we need rules that account for the signs of the numbers.

Rule 1 -- Adding integers with the same sign

When both numbers have the same sign, add their absolute values and keep the common sign.

$\text{Same signs: } (+a) + (+b) = +(a + b)$

$\text{Same signs: } (-a) + (-b) = -(a + b)$

Example: $(-7) + (-5) = -(7 + 5) = -12$

Think of it as combining two debts: if you owe $7 and then owe $5 more, you now owe $12 total.

Rule 2 -- Adding integers with different signs

When the signs are different, subtract the smaller absolute value from the larger one, then take the sign of the number whose absolute value is larger.

$\text{Different signs: } (-a) + b = \pm(|{-a}| - |b|)$

Example: $(-9) + 4 = -(9 - 4) = -5$ (since $|-9| > |4|$, the result is negative)

Example: $(-3) + 10 = +(10 - 3) = 7$ (since $|10| > |-3|$, the result is positive)

Rule 3 -- Subtracting integers (add the opposite)

Every subtraction problem can be rewritten as an addition problem. Change the subtraction sign to addition and flip the sign of the number being subtracted:

$a - b = a + (-b)$

$a - (-b) = a + (+b) = a + b$

This is the single most useful rule in integer arithmetic. Once you rewrite the problem as addition, apply Rule 1 or Rule 2 above.

Quick reference table

Expression type	Rewrite as	Then apply
$a + b$ (same signs)	--	Rule 1: add absolute values, keep sign
$a + b$ (different signs)	--	Rule 2: subtract absolute values, keep sign of larger
$a - b$	$a + (-b)$	Rule 1 or 2
$a - (-b)$	$a + b$	Rule 1 or 2

Number line approach

You can always verify on a number line:

• Adding a positive number: move right.
• Adding a negative number: move left.
• Subtracting is the same as adding the opposite direction.

$\longleftarrow \;-5 \;\; -4 \;\; -3 \;\; -2 \;\; -1 \;\;\; 0 \;\;\; 1 \;\;\; 2 \;\;\; 3 \;\;\; 4 \;\;\; 5\; \longrightarrow$

Starting at $-2$ and adding $+5$ means moving 5 steps right, landing on $3$: $(-2) + 5 = 3$.

Worked Examples

Example 1: Adding two negative integers (easy)

Problem: Calculate $(-12) + (-8)$.

Step 1: Both signs are negative (same sign), so add the absolute values. $12 + 8 = 20$

Step 2: Keep the common sign (negative).

Answer: $(-12) + (-8) = \mathbf{-20}$

Example 2: Adding integers with different signs (easy)

Problem: Calculate $(-15) + 9$.

Step 1: The signs are different, so subtract the smaller absolute value from the larger one. $15 - 9 = 6$

Step 2: The larger absolute value belongs to $-15$ (negative), so the result is negative.

Answer: $(-15) + 9 = \mathbf{-6}$

Example 3: Subtracting a positive integer (medium)

Problem: Calculate $4 - 13$.

Step 1: Rewrite as addition. $4 - 13 = 4 + (-13)$

Step 2: Different signs -- subtract absolute values. $13 - 4 = 9$

Step 3: The larger absolute value belongs to $-13$ (negative), so the result is negative.

Answer: $4 - 13 = \mathbf{-9}$

Example 4: Subtracting a negative integer (medium)

Problem: Calculate $(-6) - (-14)$.

Step 1: Rewrite as addition by flipping the sign of the second number. $(-6) - (-14) = (-6) + 14$

Step 2: Different signs -- subtract absolute values. $14 - 6 = 8$

Step 3: The larger absolute value belongs to $14$ (positive), so the result is positive.

Answer: $(-6) - (-14) = \mathbf{8}$

Example 5: Multi-step calculation (challenging)

Problem: Simplify $(-3) + 7 - (-4) - 9$.

Step 1: Rewrite every subtraction as addition. $(-3) + 7 + 4 + (-9)$

Step 2: Group positives and negatives. $\text{Positives: } 7 + 4 = 11$ $\text{Negatives: } (-3) + (-9) = -12$

Step 3: Combine the totals (different signs). $11 + (-12) = -(12 - 11) = -1$

Answer: $(-3) + 7 - (-4) - 9 = \mathbf{-1}$

Common Mistakes

Mistake 1: Forgetting to flip the sign when subtracting a negative

❌ $8 - (-5) = 8 - 5 = 3$

✅ $8 - (-5) = 8 + 5 = 13$

Why this matters: The double negative becomes a positive. Missing this step gives an answer that is off by twice the subtracted value. Always rewrite subtraction as addition first.

Mistake 2: Applying multiplication sign rules to addition

❌ $(-4) + (-6) = +10$ (thinking "two negatives make a positive")

✅ $(-4) + (-6) = -10$

Why this matters: The "two negatives make a positive" rule applies only to multiplication and division. When adding two negatives, the result is always more negative.

Mistake 3: Losing track of signs in multi-step problems

❌ Solving $5 - 8 + 3 - (-2)$ left-to-right without rewriting: $5 - 8 = -3$, $-3 + 3 = 0$, $0 - (-2)$ ... confusion

✅ Rewrite all subtractions first: $5 + (-8) + 3 + 2$. Group: positives $= 10$, negatives $= -8$. Total $= 2$.

Why this matters: Rewriting everything as addition before computing prevents sign errors in longer expressions.

Practice Problems

Try these on your own before checking the answers:

1. $(-9) + (-7)$
2. $(-11) + 18$
3. $15 - 23$
4. $(-4) - (-12)$
5. $(-6) + 10 - (-3) - 8$

Click to see answers

1. $(-9) + (-7) = -(9 + 7) = \mathbf{-16}$ (same signs: add absolute values, keep negative)
2. $(-11) + 18 = +(18 - 11) = \mathbf{7}$ (different signs: 18 has the larger absolute value, so positive)
3. $15 - 23 = 15 + (-23) = -(23 - 15) = \mathbf{-8}$ (rewrite, then different signs)
4. $(-4) - (-12) = (-4) + 12 = +(12 - 4) = \mathbf{8}$ (flip the sign, then different signs)
5. Rewrite: $(-6) + 10 + 3 + (-8)$. Positives: $10 + 3 = 13$. Negatives: $(-6) + (-8) = -14$. Total: $13 + (-14) = \mathbf{-1}$.

Summary

• Same signs: add absolute values, keep the common sign.
• Different signs: subtract absolute values, take the sign of the number with the larger absolute value.
• Subtraction rule: always rewrite $a - b$ as $a + (-b)$ before computing.
• Double negative: $a - (-b) = a + b$ -- subtracting a negative is the same as adding.
• For multi-step problems, rewrite all subtractions as additions first, then group positives and negatives separately.

Related Topics

• Integers — Operations, Number Line, and Word Problems
• Multiplying and Dividing Integers — Sign Rules Explained
• Integer Word Problems with Step-by-Step Solutions

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