Operations and Integers

Adding Integers

  • Basic Concept: When adding integers, determine if the signs are the same or different.

    • Different Signs: Subtract the absolute values and take the sign of the larger absolute value.

    • Same Signs: Add absolute values and keep the common sign.

Example 1: 12 + (-7)

  • Signs: Positive 12 and negative 7 (different signs)

  • Absolute Values: |12| = 12, |−7| = 7

  • Calculation: 12 - 7 = 5 (greater absolute value is positive 12)

  • Final Answer: Positive 5

    • Mental Approach: Start at 12 and decrease by 7 gives 5.

Example 2: -8 + (-10)

  • Signs: Both negative (same signs)

  • Absolute Values: |−8| = 8, |−10| = 10

  • Calculation: 8 + 10 = 18 (use negative sign)

  • Final Answer: Negative 18

    • Mental Approach: Start at -8, decrease by 10 gives -18.

Subtracting Integers

  • Basic Concept: The subtraction of integers can be turned into an addition problem by adding the opposite.

Example 1: 5 - (-9)

  • Transformation: 5 + 9 (opposite of -9 is +9)

  • Calculation: 5 + 9 = 14

  • Final Answer: Positive 14

    • Explanation: Subtracting a negative is like adding that value.

Example 2: -3 - 20

  • Transformation: -3 + (-20)

  • Calculation: -3 + -20 = -23

  • Final Answer: Negative 23

    • Explanation: Start at -3, decreasing by 20 results in -23.

Multiplying Integers

  • Basic Concept: The product of integers is determined by the signs of the integers.

    • Different Signs: Product is negative.

    • Same Signs: Product is positive.

Example 1: -7 x 4

  • Signs: Negative x Positive (different signs)

  • Calculation: 7 x 4 = 28, thus product is negative.

  • Final Answer: Negative 28.

Example 2: -10 x -6

  • Signs: Negative x Negative (same signs)

  • Calculation: 10 x 6 = 60, thus product is positive.

  • Final Answer: Positive 60.

Dividing Integers

  • Basic Concept: The quotient is also determined by the signs in the same way as multiplication.

    • Same Signs: Quotient is positive.

    • Different Signs: Quotient is negative.

Example 1: -48 ÷ -8

  • Calculation Base: 48 ÷ 8 = 6

  • Sign Determination: Negative ÷ Negative gives a positive.

  • Final Answer: Positive 6.

Example 2: 36 ÷ -4

  • Calculation Base: 36 ÷ 4 = 9

  • Sign Determination: Positive ÷ Negative gives a negative.

  • Final Answer: Negative 9.

Final Recap

  • Addition: Same signs—add, keep sign; Different signs—subtract, take the sign of the greater absolute value.

  • Subtraction: Add opposite.

  • Multiplication/Division: Different signs yield negative results; same signs yield positive results.