Adding and Subtracting Integers

Adding and subtracting integers is where negative numbers start to feel real. You are not just reading them on a thermometer anymore; you are computing with them. The rules may feel unfamiliar at first, but they follow a clear logic. Once you internalize the two cases for addition and the one key insight for subtraction, integer arithmetic becomes second nature.

Adding Integers: Two Cases

When adding two integers, there are two situations to consider.

Case 1: Same Signs

When both numbers have the same sign (both positive or both negative), add their absolute values and keep the common sign.

$a + b = +(|a| + |b|) \quad \text{when both are positive}$

$a + b = -(|a| + |b|) \quad \text{when both are negative}$

Why it works: If you walk 3 steps right and then 4 more steps right, you end up 7 steps to the right. If you walk 3 steps left and then 4 more steps left, you end up 7 steps to the left.

Case 2: Different Signs

When the numbers have different signs (one positive, one negative), subtract the smaller absolute value from the larger absolute value. The result takes the sign of the number with the larger absolute value.

$\text{Subtract absolute values: } \big||a| - |b|\big|$

$\text{Use the sign of whichever has the greater absolute value.}$

Why it works: If you walk 5 steps right and then 3 steps left, you end up 2 steps to the right. The larger movement (right) wins, and the net result is the difference.

Worked Examples: Addition

Example 1: Adding Two Positives

$8 + 5 = 13$

Same sign (both positive), so add the absolute values and keep the positive sign.

Example 2: Adding Two Negatives

$(-6) + (-4) = -10$

Same sign (both negative). Add the absolute values: $6 + 4 = 10$. Keep the negative sign: $-10$.

Example 3: Adding a Positive and a Negative

$9 + (-14) = ?$

Different signs. Absolute values: $|9| = 9$ and $|-14| = 14$.

Subtract: $14 - 9 = 5$.

The number with the larger absolute value is $-14$ (negative), so the result is negative:

$9 + (-14) = -5$

Example 4: Adding a Negative and a Positive

$(-7) + 10 = ?$

Different signs. Absolute values: $|-7| = 7$ and $|10| = 10$.

Subtract: $10 - 7 = 3$.

The number with the larger absolute value is $10$ (positive), so the result is positive:

$(-7) + 10 = 3$

Example 5: Adding Opposites

$(-6) + 6 = 0$

A number and its opposite always sum to zero. This is called the additive inverse property.

Subtracting Integers: One Key Rule

Subtraction of integers can be converted to addition using this essential rule:

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

To subtract an integer, add its opposite.

This single rule turns every subtraction problem into an addition problem, which you already know how to handle.

Worked Examples: Subtraction

Example 6: Subtracting a Positive

$3 - 8 = 3 + (-8) = ?$

Different signs. Absolute values: $3$ and $8$. Subtract: $8 - 3 = 5$. The larger absolute value is $8$ (negative), so:

$3 - 8 = -5$

Example 7: Subtracting a Negative

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

Change “subtract $-3$” to “add $3$.” Now both numbers are positive, so add normally.

Example 8: Subtracting from a Negative

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

Change “subtract $7$” to “add $-7$.” Both numbers are now negative. Add their absolute values ($4 + 7 = 11$) and keep the negative sign.

Example 9: Subtracting a Negative from a Negative

$(-2) - (-9) = (-2) + 9 = 7$

Change “subtract $-9$” to “add $9$.” Now the signs differ. Absolute values: $2$ and $9$. Subtract: $9 - 2 = 7$. The larger absolute value is $9$ (positive), so the result is $7$.

Number Line Model

You can visualize these operations on a number line:

• Adding a positive: move right.
• Adding a negative: move left.
• Subtracting converts to adding the opposite, then follow the rules above.

Example: Compute $(-3) + 7$ on a number line.

1. Start at $-3$.
2. Move 7 units to the right (adding a positive).
3. Land at $4$.

$(-3) + 7 = 4$

Practice Problems

Problem 1: $(-9) + (-6)$

Problem 2: $12 + (-5)$

Problem 3: $(-8) - 4$

Problem 4: $6 - (-11)$

Problem 5: $(-15) + 15$

Key Takeaways

• Same signs: Add the absolute values and keep the shared sign.
• Different signs: Subtract the absolute values and take the sign of the number with the larger absolute value.
• Subtraction becomes addition: $a - b = a + (-b)$. Rewrite every subtraction as adding the opposite, then apply the addition rules.
• A number plus its opposite (additive inverse) always equals zero.
• When in doubt, use a number line: adding a positive moves right, adding a negative moves left.

Return to Arithmetic for more foundational math topics.