Subtracting integers​

Subtracting integers is a foundational skill in math that extends basic subtraction to include positive and negative whole numbers (e.g., 5 – (-3), -8 – 4). While it may seem tricky at first, subtracting integers follows a simple core rule: subtraction = adding the opposite. This guide breaks down subtracting integers in plain language, with clear rules, step-by-step examples, visual number line explanations, real-world uses, and tips to avoid common errors – perfect for students, parents, or anyone refreshing their math skills.

Key Terms to Know Before Subtracting Integers

Before diving into subtracting integers, clarify these essential terms to avoid confusion:

  • Integer: A whole number (no fractions/decimals) including positive numbers (1, 6, 10), negative numbers (-2, -7, -15), and zero (0).
  • Positive Integer: A number greater than 0 (represents gain, increase, or above zero – e.g., 4, 12).
  • Negative Integer: A number less than 0 (represents loss, decrease, or below zero – e.g., -5, -9).
  • Absolute Value: The distance of an integer from zero on the number line (always positive – e.g., |-6| = 6, |8| = 8).
  • Opposite Integer: A number with the same absolute value but opposite sign (e.g., the opposite of -4 is 4; the opposite of 7 is -7).

The Golden Rule for Subtracting Integers: “Add the Opposite”

The single most important rule for subtracting integers is to rewrite the subtraction problem as adding the opposite of the number being subtracted. This turns every integer subtraction problem into an addition problem (which most learners find easier to solve).

Step-by-Step for the “Add the Opposite” Rule

  1. Keep the first integer (minuend) the same: Do not change its sign or value.
  2. Change the subtraction sign (-) to an addition sign (+): This is the critical shift from subtraction to addition.
  3. Change the second integer (subtrahend) to its opposite: Flip its sign (positive ↔ negative).
  4. Solve using integer addition rules:
    • If the numbers have the same sign: Add their absolute values, keep the common sign.
    • If the numbers have different signs: Subtract the smaller absolute value from the larger one, take the sign of the integer with the larger absolute value.

Subtracting Integers: Step-by-Step Examples

Let’s apply the “add the opposite” rule to different scenarios of subtracting integers:

Example 1: Subtracting a Positive Integer from a Positive Integer

Problem: 8 – 5Solution:

  1. Keep 8 the same.
  2. Change – to +.
  3. Change 5 to its opposite (-5).
  4. Solve: 8 + (-5) = 3 (different signs: 8 – 5 = 3, keep positive sign).

Example 2: Subtracting a Negative Integer from a Positive Integer

Problem: 7 – (-3)Solution:

  1. Keep 7 the same.
  2. Change – to +.
  3. Change -3 to its opposite (3).
  4. Solve: 7 + 3 = 10 (same signs: 7 + 3 = 10, keep positive sign).

Example 3: Subtracting a Positive Integer from a Negative Integer

Problem: -6 – 4Solution:

  1. Keep -6 the same.
  2. Change – to +.
  3. Change 4 to its opposite (-4).
  4. Solve: -6 + (-4) = -10 (same signs: 6 + 4 = 10, keep negative sign).

Example 4: Subtracting a Negative Integer from a Negative Integer

Problem: -9 – (-2)Solution:

  1. Keep -9 the same.
  2. Change – to +.
  3. Change -2 to its opposite (2).
  4. Solve: -9 + 2 = -7 (different signs: 9 – 2 = 7, keep negative sign).

Example 5: Subtracting Zero from an Integer

Problem: -12 – 0Solution:

  1. Keep -12 the same.
  2. Change – to +.
  3. Change 0 to its opposite (0 – zero has no opposite).
  4. Solve: -12 + 0 = -12 (adding zero doesn’t change the integer).

Visualizing Subtracting Integers with a Number Line

For visual learners, a number line makes subtracting integers easier to understand – here’s how to use it:

Step-by-Step for Number Line Subtraction

  1. Start at the first integer (minuend) on the number line.
  2. Determine direction:
    • If subtracting a positive integer: Move to the left (negative direction) by the absolute value of the second integer.
    • If subtracting a negative integer: Move to the right (positive direction) by the absolute value of the second integer (since you’re adding its opposite).

Example: Number Line for -4 – (-6)

  1. Start at -4 on the number line.
  2. Subtracting -6 means moving 6 units to the right (opposite of left).
  3. Land at 2 → Result = 2.

Example: Number Line for 6 – 9

  1. Start at 6 on the number line.
  2. Subtracting 9 means moving 9 units to the left.
  3. Land at -3 → Result = -3.

Real-World Examples of Subtracting Integers

Subtracting integers isn’t just for math class – it’s used daily in real-life scenarios where positive/negative values matter:

Example 1: Temperature Changes

Problem: The temperature is 5°F in the morning. It drops 8°F by noon (subtract 8) – what’s the noon temperature?Solution: 5 – 8 = 5 + (-8) = -3°F.

Example 2: Finance (Bank Accounts)

Problem: You have $45 in your account. You withdraw $60 (subtract 60) – what’s your account balance?Solution: 45 – 60 = 45 + (-60) = -$15 (overdraft of $15).

Example 3: Elevation (Hiking)

Problem: A hiker is at 1,000 feet above sea level (+1000). They hike down 1,200 feet (subtract 1200) – what’s their final elevation?Solution: 1000 – 1200 = 1000 + (-1200) = -200 feet (200 feet below sea level).

Example 4: Sports (Golf)

Problem: A golfer is at -2 strokes (2 under par) after 9 holes. They score +3 strokes (3 over par) on the next hole (subtract -3) – what’s their new score?Solution: -2 – 3 = -2 + (-3) = -5 strokes (5 under par).

Common Mistakes to Avoid When Subtracting Integers

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

  1. Forgetting to “Add the Opposite”: Never subtract a negative integer directly (e.g., 8 – (-2) ≠ 6 – it’s 8 + 2 = 10).
  2. Mixing Up Direction on the Number Line: Subtracting a negative integer means moving right (not left) on the number line.
  3. Ignoring Absolute Value: Always use the positive distance from zero when calculating (e.g., |-7| = 7, not -7).
  4. Miscalculating the Opposite: The opposite of -9 is 9 (not -9); the opposite of 11 is -11 (not 11).
  5. Rushing Multi-Step Problems: Break down problems like -5 – (-4) + 7 step-by-step (first: -5 + 4 = -1; then: -1 + 7 = 6).

Frequently Asked Questions (FAQs) About Subtracting Integers

Q1: What is the result of 0 – (-8)?

A1: 0 – (-8) = 0 + 8 = 8 (adding the opposite of -8 is 8; zero plus any integer equals that integer).

Q2: How do I subtract large integers (e.g., 200 – (-50))?

A2: Follow the same “add the opposite” rule: 200 – (-50) = 200 + 50 = 250 – the size of the integer doesn’t change the rule.

Q3: Can I use a calculator for subtracting integers?

A3: Yes, but learn the rules first – calculators are tools, but understanding the math helps you catch errors (e.g., if a calculator says 7 – (-3) = 4, you know it’s wrong).

Q4: Why is subtracting a negative number the same as adding a positive?

A4: Subtraction is the inverse of addition – removing a loss (negative) is the same as gaining a positive (e.g., “taking away a $5 debt” = “gaining $5”).

Q5: How does subtracting integers apply to algebra?

A5: In algebra, variables often use integers (e.g., x – (-4) = x + 4; 3y – 7y = -4y) – mastering integer subtraction is critical for simplifying expressions.

Conclusion

Subtracting integers boils down to one simple rule: rewrite subtraction as adding the opposite. Whether you use the “add the opposite” method, a number line, or solve real-world problems, this rule applies to every integer subtraction scenario. With practice, subtracting positive and negative integers becomes second nature – and it’s a skill that’s essential for algebra, finance, and everyday problem-solving.

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