3.4 Multiply and Divide Integers

Multiplying and dividing integers builds directly on what you already know about addition and subtraction of integers. The core idea is straightforward: the signs of the numbers determine whether your answer is positive or negative. Once you nail down the sign rules, the actual computation is just regular multiplication and division.

Multiplying and Dividing Integers

Multiplication of positive and negative integers

Two simple rules control the sign of every multiplication problem:

• Same signs → positive result
  • Positive × Positive = Positive: $2 \times 3 = 6$
  • Negative × Negative = Positive: $-4 \times -5 = 20$
• Different signs → negative result
  • Positive × Negative = Negative: $3 \times -7 = -21$
  • Negative × Positive = Negative: $-6 \times 4 = -24$

To find the product, multiply the absolute values of the two numbers, then apply the sign rule above. In other words, $|a \times b| = |a| \times |b|$. For example, $|-6 \times 4| = 6 \times 4 = 24$, and since the signs are different, the answer is $-24$.

Division operations with integers

The sign rules for division are exactly the same as for multiplication:

• Same signs → positive result
  • $12 \div 3 = 4$
  • $-20 \div -5 = 4$
• Different signs → negative result
  • $18 \div -6 = -3$
  • $-24 \div 8 = -3$

Just like with multiplication, divide the absolute values first, then apply the sign rule: $|a \div b| = |a| \div |b|$.

One thing that never changes: division by zero is undefined. You can't divide any integer by 0. For example, $10 \div 0$ has no answer.

Properties of Integer Operations

• Identity property: Multiplying any integer by 1, or dividing it by 1, gives you the same integer. For example, $-8 \times 1 = -8$ and $-8 \div 1 = -8$.
• Multiplication by zero: Any integer multiplied by 0 equals 0. For example, $-5 \times 0 = 0$.
• Opposite integers: Two integers with the same absolute value but different signs, like $5$ and $-5$.
• Additive inverse: The opposite of an integer. Adding an integer and its additive inverse always gives 0: $5 + (-5) = 0$.
• Multiplicative inverse: The reciprocal of a non-zero integer. Multiplying an integer by its reciprocal gives 1: $5 \times \frac{1}{5} = 1$.

Simplification of integer algebraic expressions

When an expression has multiple operations, follow PEMDAS (Parentheses, Exponents, Multiplication/Division left to right, Addition/Subtraction left to right). Apply the integer sign rules at each step.

1. Simplify inside parentheses first: $2 \times (3 - 5) = 2 \times (-2) = -4$

2. Perform multiplication and division from left to right: $12 \div -3 \times 2 = -4 \times 2 = -8$

3. Perform addition and subtraction from left to right: $-6 + 4 - 3 = -2 - 3 = -5$

Here's a longer example that combines several steps:

$-3 \times (4 - 7) \div 2$

1. Parentheses first: $4 - 7 = -3$

2. Multiply left to right: $-3 \times -3 = 9$ (same signs → positive)

3. Divide: $9 \div 2 = 4.5$

Variable expressions with integer values

To evaluate a variable expression, substitute the given integer for the variable, then simplify using PEMDAS and the sign rules.

Example 1: Evaluate $3x - 2$ when $x = -4$

1. Substitute: $3(-4) - 2$

2. Multiply: $-12 - 2$

3. Subtract: $-14$

Example 2: Evaluate $-2y^2 + 5y - 3$ when $y = -1$

1. Substitute: $-2(-1)^2 + 5(-1) - 3$

2. Exponent first: $(-1)^2 = 1$

3. Multiply: $-2(1) + 5(-1) - 3 = -2 + (-5) - 3$

4. Add and subtract left to right: $-2 + (-5) = -7$, then $-7 - 3 = -10$

Word problems to algebraic expressions

Translating a word problem into algebra takes practice. Here's a reliable approach:

1. Identify the unknown and assign it a variable (let $x$ represent the unknown number).

2. Look for keywords that tell you which operation to use:

   • "Sum," "more than," "increased by" → addition (3 more than twice a number: $2x + 3$)
   • "Difference," "less than," "decreased by" → subtraction (5 less than a number: $x - 5$)
   • "Product," "times," "of" → multiplication (3 times a number: $3x$)
   • "Quotient," "divided by," "ratio" → division (a number divided by 4: $\frac{x}{4}$)

3. Substitute the given integer value and simplify.

Watch out for "less than" since it reverses the order. "5 less than a number" is $x - 5$, not $5 - x$.

Example: If $x = -6$, evaluate "3 more than twice a number": $2(-6) + 3 = -12 + 3 = -9$

Applying Integer Operations

Solve real-world problems

Real-world integer problems often involve temperature, money, elevation, or scores. Follow these steps:

1. Read carefully and identify what you know and what you need to find.
2. Assign a variable to the unknown quantity.
3. Write an expression or equation based on the situation.
4. Substitute any given integer values and simplify using the sign rules.
5. Interpret your answer in context.

Example: A city's temperature starts at $5°C$ and decreases by $7°C$. What's the final temperature?

• Expression: $5 - 7$
• Simplify: $5 - 7 = -2$
• The final temperature is $-2°C$, which means 2 degrees below zero.