Operations with Integers

Adding, subtracting, multiplying, and dividing integers

Operations with Integers

What are Integers?

Integers are whole numbers and their opposites: ...,−3,−2,−1,0,1,2,3,......, -3, -2, -1, 0, 1, 2, 3, ...

Positive integers: 1,2,3,...1, 2, 3, ... Negative integers: −1,−2,−3,...-1, -2, -3, ... Zero: Neither positive nor negative

Adding Integers

Same signs: Add the absolute values, keep the sign

  • 5+3=85 + 3 = 8
  • −5+(−3)=−8-5 + (-3) = -8

Different signs: Subtract the absolute values, use the sign of the larger

  • 5+(−3)=25 + (-3) = 2
  • −5+3=−2-5 + 3 = -2

Subtracting Integers

Key rule: Subtracting is the same as adding the opposite

a−b=a+(−b)a - b = a + (-b)

Examples:

  • 7−3=7+(−3)=47 - 3 = 7 + (-3) = 4
  • 7−(−3)=7+3=107 - (-3) = 7 + 3 = 10
  • −7−3=−7+(−3)=−10-7 - 3 = -7 + (-3) = -10

Multiplying Integers

Sign rules:

  • Same signs → positive result
  • Different signs → negative result

Examples:

  • 5×3=155 \times 3 = 15 (both positive)
  • −5×(−3)=15-5 \times (-3) = 15 (both negative)
  • 5×(−3)=−155 \times (-3) = -15 (different signs)
  • −5×3=−15-5 \times 3 = -15 (different signs)

Dividing Integers

Same sign rules as multiplication:

  • Same signs → positive result
  • Different signs → negative result

Examples:

  • 12÷3=412 \div 3 = 4
  • −12÷(−3)=4-12 \div (-3) = 4
  • 12÷(−3)=−412 \div (-3) = -4
  • −12÷3=−4-12 \div 3 = -4

📚 Practice Problems

1Problem 1easy

❓ Question:

Calculate: −8+5-8 + 5

💡 Show Solution

Different signs: subtract and use sign of larger absolute value.

∣−8∣=8,∣5∣=5|-8| = 8, \quad |5| = 5

Since 8 > 5: 8−5=38 - 5 = 3

Use negative sign (from -8): −8+5=−3-8 + 5 = -3

Answer: −3-3

2Problem 2medium

❓ Question:

Calculate: −6−(−9)-6 - (-9)

💡 Show Solution

Change subtraction to adding the opposite: −6−(−9)=−6+9-6 - (-9) = -6 + 9

Different signs: subtract and use sign of larger. 9−6=39 - 6 = 3

Use positive sign (from 9): −6+9=3-6 + 9 = 3

Answer: 33

3Problem 3hard

❓ Question:

Calculate: (−4)×3×(−2)(-4) \times 3 \times (-2)

💡 Show Solution

Work left to right:

Step 1: (−4)×3=−12(-4) \times 3 = -12 (different signs → negative)

Step 2: −12×(−2)=24-12 \times (-2) = 24 (same signs → positive)

Answer: 2424

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