Adding and subtracting integers

Adding and subtracting integers is a foundational skill in math that applies to everything from basic arithmetic to algebra, finance, and everyday problem-solving. Integers include positive whole numbers (1, 2, 3…), negative whole numbers (-1, -2, -3…), and zero – and adding/subtracting them follows simple rules based on their signs (positive/negative). This guide breaks down adding and subtracting integers in plain language, with clear rules, step-by-step examples, real-world use cases, and tips to avoid the most common errors.

Key Terms to Know Before Adding and Subtracting Integers

Before diving into adding and subtracting integers, clarify these core terms to avoid confusion:

  • Integer: A whole number (no fractions/decimals) that includes positive numbers (1, 5, 10), negative numbers (-2, -7, -15), and zero (0).
  • Positive Integer: A number greater than 0 (e.g., 3, 8, 22 – represents gain, increase, or above zero).
  • Negative Integer: A number less than 0 (e.g., -4, -9, -17 – represents loss, decrease, or below zero).
  • Sign: The “+” (positive) or “-” (negative) symbol in front of an integer (zero has no sign).
  • Absolute Value: The distance of an integer from zero on the number line (always positive – e.g., |-5| = 5, |3| = 3).

Rules for Adding and Subtracting Integers

The rules for adding and subtracting integers depend on whether the numbers have the same sign (both positive or both negative) or different signs (one positive, one negative).

Rule 1: Adding Integers with the Same Sign

When adding integers with the same sign (both + or both -):

  1. Add the absolute values of the numbers (ignore the signs for this step).
  2. Keep the common sign for the result.

Examples of Adding Same-Sign Integers

  • Positive + Positive: 4 + 7 = 11 (add 4 + 7 = 11; keep + sign → 11)
  • Negative + Negative: -5 + (-8) = -13 (add 5 + 8 = 13; keep – sign → -13)

Rule 2: Adding Integers with Different Signs

When adding integers with different signs (one +, one -):

  1. Subtract the smaller absolute value from the larger one (find the difference).
  2. Take the sign of the integer with the larger absolute value for the result.

Examples of Adding Different-Sign Integers

  • 6 + (-9) = -3 (|6|=6, |-9|=9; 9-6=3; larger absolute value is -9 → -3)
  • -4 + 10 = 6 (|-4|=4, |10|=10; 10-4=6; larger absolute value is 10 → 6)

Rule 3: Subtracting Integers (The “Add the Opposite” Trick)

Subtracting integers is easiest when you rewrite the problem as adding the opposite of the number being subtracted:

  1. Keep the first integer (the “minuend”) the same.
  2. Change the subtraction sign (-) to an addition sign (+).
  3. Change the sign of the second integer (the “subtrahend”) to its opposite (positive ↔ negative).
  4. Follow the addition rules above to solve.

Examples of Subtracting Integers

  • 8 – 5 = 8 + (-5) = 3 (rewrite as adding -5; different signs → 8-5=3, + sign → 3)
  • 7 – (-3) = 7 + 3 = 10 (rewrite as adding +3; same signs → 7+3=10, + sign → 10)
  • -6 – 4 = -6 + (-4) = -10 (rewrite as adding -4; same signs → 6+4=10, – sign → -10)
  • -9 – (-2) = -9 + 2 = -7 (rewrite as adding +2; different signs → 9-2=7, – sign → -7)

Step-by-Step Examples: Adding and Subtracting Integers

Let’s apply the rules to more complex examples of adding and subtracting integers:

Example 1: Adding Multiple Integers

Problem: 3 + (-7) + (-2) + 9Solution:

  1. Group same-sign integers: (3 + 9) + [(-7) + (-2)]
  2. Add same-sign groups: 12 + (-9)
  3. Add different signs: 12 – 9 = 3 (sign of 12 → +3)Result: 3

Example 2: Subtracting Integers with Multiple Steps

Problem: -10 – (-5) + 8 – 4Solution:

  1. Rewrite subtractions as adding opposites: -10 + 5 + 8 + (-4)
  2. Group same-sign integers: (-10 + (-4)) + (5 + 8)
  3. Add same-sign groups: -14 + 13
  4. Add different signs: 14 – 13 = 1 (sign of -14 → -1)Result: -1

Example 3: Using a Number Line (Visual Method)

For visual learners, a number line helps with adding and subtracting integers:

  • Problem: 5 + (-8)
    • Start at 5 on the number line.
    • Move 8 units to the left (for -8).
    • Land at -3 → Result = -3.
  • Problem: -3 – (-6)
    • Start at -3.
    • Rewrite as -3 + 6 → move 6 units to the right.
    • Land at 3 → Result = 3.

Real-World Examples of Adding and Subtracting Integers

Adding and subtracting integers isn’t just for math class – it’s used daily in real-life scenarios:

Example 1: Finance (Money Management)

Problem: You have $25 in your wallet. You spend $18 (subtract -18) and then get a $10 cash gift (add +10). How much money do you have? Solution: 25 – 18 + 10 = 25 + (-18) + 10 = 7 + 10 = $17.

Example 2: Temperature

Problem: The temperature is 4°F in the morning. It drops 9°F by noon, then rises 5°F by afternoon. What’s the afternoon temperature?Solution: 4 – 9 + 5 = 4 + (-9) + 5 = -5 + 5 = 0°F.

Example 3: Elevation

Problem: A hiker starts at 1,200 feet above sea level (+1200). They hike down 800 feet (-800) and then up 300 feet (+300). What’s their final elevation?Solution: 1200 – 800 + 300 = 400 + 300 = 700 feet above sea level.

Example 4: Sports (Golf)

Problem: Golf scores are relative to par (0). A golfer scores -2 (birdie), +1 (bogey), and 0 (par) on three holes. What’s their total score vs. par?Solution: -2 + 1 + 0 = -1 (1 stroke under par).

Common Mistakes to Avoid When Adding and Subtracting Integers

These are the most frequent errors when adding and subtracting integers – watch for them!

  1. Forgetting the “Add the Opposite” Rule for Subtraction: Never subtract a negative number directly (e.g., 7 – (-3) ≠ 4 – it’s 7 + 3 = 10).
  2. Mixing Up Signs in Addition: When adding different signs, take the sign of the larger absolute value (e.g., -6 + 4 ≠ -10 – it’s -2).
  3. Ignoring Absolute Value: Always use absolute value (positive distance) when adding/subtracting (e.g., |-8| = 8, not -8).
  4. Miscalculating Multiple Steps: Group same-sign integers first to simplify (e.g., -5 + 8 – 3 = (-5 -3) + 8 = -8 + 8 = 0).
  5. Confusing Zero: Adding/subtracting zero doesn’t change the integer (e.g., 9 + 0 = 9, -7 – 0 = -7).

Frequently Asked Questions (FAQs) About Adding and Subtracting Integers

Q1: What is 0 + (-5) and 0 – (-5)?

A1: 0 + (-5) = -5 (adding a negative to zero keeps the negative sign); 0 – (-5) = 0 + 5 = 5 (add the opposite).

Q2: How do I add/subtract integers with large numbers (e.g., 100 – (-50))?

A2: Follow the same rules: 100 – (-50) = 100 + 50 = 150 (size of the number doesn’t change the sign rules).

Q3: Can I use a calculator for adding/subtracting integers?

A3: Yes, but learn the rules first – calculators are tools, but understanding the math helps catch errors.

Q4: What’s the difference between -(-4) and -(4)?

A4: -(-4) = 4 (the opposite of -4 is 4); -(4) = -4 (the opposite of 4 is -4).

Q5: How do adding/subtracting integers apply to algebra?

A5: In algebra, variables often use integers (e.g., x – (-3) = x + 3; 2y + (-5y) = -3y) – mastering integer rules is critical for simplifying expressions.

Conclusion

Adding and subtracting integers boils down to understanding sign rules: add same signs, subtract different signs (and take the larger absolute value’s sign), and rewrite subtraction as “adding the opposite.” With practice, these rules become second nature – and they’re essential for everything from balancing a budget to solving algebra problems. Whether you’re a student learning the basics or an adult refreshing your skills, adding and subtracting integers is a fundamental math skill that applies to everyday life.

If you have questions about adding/subtracting specific integers, or need help with a tricky problem, leave a comment below!