Multiplying Integers

Multiplying Integers

Core Rules for Integer Multiplication

• Multiplying Integers with the Same Sign:

  • Rule: The product of two integers with the same sign is positive.
  • Examples:
  • $2 \cdot 3 = 6$
  • $-2 \cdot (-3) = 6$

• Multiplying Integers with Different Signs:

  • Rule: The product of two integers with different signs is negative.
  • Examples:
  • $2 \cdot (-3) = -6$
  • $-2 \cdot 3 = -6$

Step-by-Step Multiplication Examples

• Example 1a: Find $-5 \cdot (-6)$

  • Sign Analysis: Both integers, $-5$ and $-6$, are negative. Therefore, the integers have the same sign.
  • Rule Application: The product of two integers with the same sign is positive.
  • Calculation:     $-5 \cdot (-6) = 30$
  • Result: The product is $30$.

• Example 1b: Find $3(-4)$

  • Sign Analysis: The integer $3$ is positive and the integer $-4$ is negative. Therefore, the integers have different signs.
  • Rule Application: The product of two integers with different signs is negative.
  • Calculation:     $3(-4) = -12$
  • Result: The product is $-12$.

Practice Exercises: Integer Multiplication

• Problem 1: Find $5 \cdot 5$
• Problem 2: Find $-1(-9)$
• Problem 3: Find $-7 \cdot (-8)$
• Problem 4: Find $12 \cdot (-2)$
• Problem 5: Find $4(-6)$
• Problem 6: Find $-25(0)$

Dividing Integers

Fundamental Rules and Properties of Integer Division

• Division by Zero: Division by $0$ is undefined.

• Dividing Integers with the Same Sign:

  • Rule: The quotient of two integers with the same sign is positive.
  • Examples:
  • $8 \div 2 = 4$
  • $-8 \div (-2) = 4$

• Dividing Integers with Different Signs:

  • Rule: The quotient of two integers with different signs is negative.
  • Examples:
  • $8 \div (-2) = -4$
  • $-8 \div 2 = -4$

• Equivalence of Negative Rational Expressions:

  • If $a$ and $b$ are integers (where $b \neq 0$), the placement of the negative sign does not alter the overall value of the rational expression:     $-\frac{a}{b} = \frac{-a}{b} = \frac{a}{-b}$
  • Application: The expression $\frac{-54}{6}$ can also be conceptualized as:     $-\frac{54}{6} = -9$

Step-by-Step Division Examples

• Example 1: Find $-18 \div (-6)$

  • Sign Analysis: The dividend $-18$ and the divisor $-6$ are both negative integers, meaning they have the same sign.
  • Rule Application: The quotient of two integers with the same sign is positive.
  • Calculation:     $-18 \div (-6) = 3$
  • Result: The quotient is $3$.

• Example 2a: Find $75 \div (-25)$

  • Sign Analysis: The dividend $75$ is positive and the divisor $-25$ is negative, meaning they have different signs.
  • Rule Application: The quotient of two integers with different signs is negative.
  • Calculation:     $75 \div (-25) = -3$
  • Result: The quotient is $-3$.

• Example 2b: Find $\frac{-54}{6}$

  • Sign Analysis: The numerator $-54$ is negative and the denominator $6$ is positive, meaning they have different signs.
  • Rule Application: The quotient of two integers with different signs is negative.
  • Calculation:     $\frac{-54}{6} = -9$
  • Result: The quotient is $-9$.

Practice Exercises: Integer Division

• Problem 1: Find $14 \div 2$
• Problem 2: Find $-32 \div (-4)$
• Problem 3: Find $-40 \div (-8)$
• Problem 4: Find $0 \div (-6)$
• Problem 5: Find $\frac{-49}{7}$
• Problem 6: Find $\frac{21}{-3}$