Problem 55 - Algebra 1: Concepts and Skills

Question

Find the product. $$ -3(-12) $$

Step-by-Step Solution

Answer

The product of -3 and -12 is 36.

Step 1: Understand the Rule for Multiplying Negative Numbers

When multiplying two negative numbers, the product is positive. This can be thought of as the negation operation (-) being applied twice, once for each negative number being multiplied. Doing anything twice is essentially undoing it, so the two negations cancel out, resulting in a positive product.

Step 2: Apply the Rule

Applying the rule to our numbers -3 and -12 gives us a positive product. So we simply multiply 3 and 12 together.

Step 3: Find the Product

The product of 3 and 12 is 36. Since we know from Step 2 that our product is positive, the final answer is 36.

Key Concepts

Negative Numbers

In mathematics, negative numbers are simply numbers that are less than zero. They often depict a deficit or below zero value, such as temperatures below freezing, losses in finance, or debts one may owe. When visualized on a number line, negative numbers are always to the left of zero.
Understanding how negative numbers interact is fundamental in math. When two negative numbers interact in operations like addition or multiplication, their properties help define the result. For example, the negative sign in front of a number indicates a direction opposite to that of positive numbers. This concept comes into play significantly when multiplying negative numbers, as shown in the exercise above.

Positive Product

When multiplying a negative number by another negative number, the product becomes positive. This might seem confusing initially, but an easy way to remember this is by thinking about negation.

Understanding Negation

In math, negation can be thought of as reversing or doing the opposite. The operation of negation applied twice essentially undoes itself, resulting in the original value. When dealing with two negative numbers, each number "reverses" the other, turning the sign positive.
Thus,

• Negative times negative equals positive.

Consider the example given:

• -3 times -12.

Both numbers are negative, so multiply them normally (3 times 12 equals 36) and remember the positive product rule: The multiplication of these two negative numbers yields a positive 36.

Mathematical Rules for Multiplication

Multiplication follows a set of fundamental mathematical rules, and understanding these rules helps in dealing with negative numbers effectively. The rules are logical and consistent, offering a structure for calculations.

Key Rules of Multiplication

The key multiplication rules everyone learns are:

• Positive times positive equals positive.
• Positive times negative equals negative.
• Negative times positive equals negative.
• Negative times negative equals positive.

These rules assist you every time you perform multiplication. In our example with -3 * -12, the rule **negative times negative equals positive** guided us to correctly find the positive product of 36. Such rules ensure consistency and accuracy in mathematical operations, removing any guesswork from finding products of negative numbers.