Adding and Subtracting Integers

Learn how to add and subtract positive and negative numbers using number line strategies and rules.

Elementary25 minLesson
Printable worksheet

Definition

Adding integers means combining positive and negative numbers. Subtracting integers means finding the difference, which we can rewrite as adding the opposite.
The absolute value ∣a∣|a| is the distance of aa from zero: ∣−5∣=5|-5| = 5 and ∣5∣=5|5| = 5.

Key Rules

Adding Integers

  • Same signs: Add the absolute values, keep the sign
  • 5+3=85 + 3 = 8 (both positive)
  • (−5)+(−3)=−8(-5) + (-3) = -8 (both negative)
  • Different signs: Subtract the smaller absolute value from the larger, keep the sign of the larger
  • 5+(−3)=25 + (-3) = 2 (positive wins)
  • (−5)+3=−2(-5) + 3 = -2 (negative wins)

Subtracting Integers

  • Subtracting is adding the opposite: a−b=a+(−b)a - b = a + (-b)
  • 5−3=5+(−3)=25 - 3 = 5 + (-3) = 2
  • 5−(−3)=5+3=85 - (-3) = 5 + 3 = 8
  • (−5)−3=(−5)+(−3)=−8(-5) - 3 = (-5) + (-3) = -8

Try it now

What is (−3)+(−5)(-3) + (-5)?

Worked Examples

Calculate (−4)+(−7)(-4) + (-7)
1

Identify the signs

Both numbers are negative → Same signs
2

Add the absolute values

∣−4∣+∣−7∣=4+7=11|-4| + |-7| = 4 + 7 = 11 → Sum is 11
3

Apply the common sign

Both negative, so result is negative → −11-11

Common Mistakes

Thinking 5−(−3)=25 - (-3) = 2 instead of 88

Why it's wrong: Forgetting that subtracting a negative number means adding its opposite: the subtraction sign and the negative sign together become a plus.

Correct: 5−(−3)=5+3=85 - (-3) = 5 + 3 = 8. Subtracting −3-3 is the same as adding 33.

Writing (−4)+(−6)=−2(-4) + (-6) = -2 instead of −10-10

Why it's wrong: Mistakenly subtracting instead of adding when both numbers are negative.

Correct: Same signs means ADD absolute values: 4+6=104 + 6 = 10, then keep the negative: −10-10.

Getting the wrong sign when subtracting like 3−8=53 - 8 = 5

Why it's wrong: Forgetting to check which number has the larger absolute value.

Correct: 3−8=3+(−8)=−53 - 8 = 3 + (-8) = -5 because 8>38 > 3 and we keep the negative sign.

Confusing −5+3-5 + 3 with −5−3-5 - 3

Why it's wrong: Not carefully reading the operation sign.

Correct: −5+3=−2-5 + 3 = -2 (moving right on number line), but −5−3=−8-5 - 3 = -8 (moving left).

Interactive Visual

Click the buttons to jump left (subtract) or right (add) on the number line.

Interactive Sandbox

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Practice Problems

16 problems
Problem 1 of 16
Easy
What is (−3)+(−5)(-3) + (-5)?

Why It Matters

Integer operations are essential in everyday life:
  • Temperature changes: If it's −3°C-3°C and drops 5 more degrees: −3+(−5)=−8°C-3 + (-5) = -8°C
  • Money and debt: You owe 50 dollars and borrow 30 more: (−50)+(−30)=−80(-50) + (-30) = -80, a debt of 80 dollars
  • Elevators: From floor 2, going down 5 floors: 2+(−5)=−32 + (-5) = -3 (basement level 3)
  • Football: A team gains 8 yards then loses 12: 8+(−12)=−48 + (-12) = -4 yards total
Understanding these operations is the foundation for algebra!

Real World Applications

Banking Transactions

Banks use integer operations to track deposits (positive) and withdrawals (negative).

Example:

If your account has 120 dollars and you withdraw 85 dollars, then deposit 40 dollars: 120+(−85)+40=75120 + (-85) + 40 = 75 dollars.

1Try It Yourself

Your bank account shows −45-45 dollars (overdraft). You deposit 80 dollars.

What is your new balance?

Step 1: Write the mathematical expression

Calculate: −45+80-45 + 80

Altitude and Depth

Pilots and divers use integers to track altitude above sea level (positive) or depth below (negative).

Example:

A submarine at −80-80 meters dives 25 meters deeper: −80+(−25)=−105-80 + (-25) = -105 meters.

2Try It Yourself

A submarine is at −120-120 meters. It rises 75 meters toward the surface.

What is the new depth?

Step 1: Write the mathematical expression

Calculate: −120+75-120 + 75

Game Scores

Many games track gains and losses as positive and negative points.

Example:

In a trivia game, you earn 50 points, lose 30, then earn 45: 50+(−30)+45=6550 + (-30) + 45 = 65 points.

3Try It Yourself

Your game score is −15-15 points (in the negative). You answer correctly for 40 points, then incorrectly and lose 10.

What is your final score?

Step 1: Write the mathematical expression

Calculate: −15+40+(−10)-15 + 40 + (-10)

Key Takeaways

  • 1Same signs: Add absolute values and keep the common sign (−4+(−3)=−7-4 + (-3) = -7)
  • 2Different signs: Subtract absolute values and keep the sign of the larger (−4+7=3-4 + 7 = 3)
  • 3Subtracting = adding the opposite: a−b=a+(−b)a - b = a + (-b)
  • 4Double negative: a−(−b)=a+ba - (-b) = a + b
  • 5Use a number line: right is positive, left is negative

Frequently Asked Questions

Think of it as removing a debt. If you owe 5 dollars (−5-5) and someone cancels that debt (subtracts −5-5), you gain 5 dollars. 0−(−5)=0+5=50 - (-5) = 0 + 5 = 5. Subtracting negative = adding positive!
Think of it as removing a debt. If you owe 5 dollars (−5-5) and someone cancels that debt (subtracts −5-5), you gain 5 dollars. 0−(−5)=0+5=50 - (-5) = 0 + 5 = 5. Subtracting negative = adding positive!
For addition with different signs, the answer has the sign of the number with the larger absolute value. −8+5=−3-8 + 5 = -3 because ∣−8∣=8>5=∣5∣|-8| = 8 > 5 = |5|, so the answer is negative.
Yes! Change subtraction to addition and flip the sign of the second number. Then follow the addition rules. Example: 7−(−3)7 - (-3) becomes 7+3=107 + 3 = 10.

Glossary

Integer
A whole number that can be positive, negative, or zero (e.g., −5-5, 00, 1212)
Absolute value
The distance of a number from zero, never negative (e.g., ∣−7∣=7|-7| = 7, ∣0∣=0|0| = 0)
Opposite
A number that is the same distance from zero but on the other side (e.g., opposite of −3-3 is 33)
Additive inverse
Another name for the opposite; a number plus its additive inverse equals zero (5+(−5)=05 + (-5) = 0)
Today's challengeOne problem, three tries

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