Problem 10 - Prealgebra

Question

Find each of the following products. (Multiply.) $$5(-2)$$

Step-by-Step Solution

Answer

The product of \(5\) and \(-2\) is \(-10\).

Step 1: Recognizing the Multipliers

Identify the numbers involved in the multiplication, which are \(5\) and \(-2\). Here, \(5\) is a positive number, and \(-2\) is a negative number.

Step 2: Determine the Sign of the Product

To determine the sign of the product, recall the rule: the product of a positive number and a negative number is negative.

Step 3: Multiply the Absolute Values

Calculate the product of the absolute values of the numbers. The absolute value of \(5\) is \(5\), and the absolute value of \(-2\) is \(2\). Multiply these values: \(5 \times 2 = 10\).

Step 4: Apply the Sign

Since the sign of the product (from Step 2) is negative, apply this sign to the result obtained in Step 3. Therefore, the result of \(5\) times \(-2\) is \(-10\).

Key Concepts

Understanding Negative Numbers

Negative numbers are those that are less than zero. They are usually represented by a minus sign in front of them, like \(-2\). In mathematics, negative numbers can have interesting properties, especially when involved in operations like addition, subtraction, and crucially, multiplication.

• Negative numbers are found on the number line to the left of zero.
• The further you go to the left, the smaller the negative numbers become.

Understanding negative numbers is important in interpreting financial losses in business, measuring below-zero temperatures, and many other real-world applications. When multiplying, negative numbers introduce unique behavior that differs from multiplying with only positives.

Grasping Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering its direction. For instance, both \(3\) and \(-3\) have an absolute value of \(3\), because they are each three units from zero. Absolute value essentially removes the sign of a number, focusing strictly on its magnitude.

• The absolute value is always a non-negative number.
• Written with vertical bars, such as \(|-3| = 3\).

When you multiply numbers, finding the product of their absolute values first can help simplify the process before considering their signs. This is useful in managing calculations involving negative numbers.

Applying Multiplication Rules

Multiplication rules dictate how to handle the signs of numbers to find their product. Here are key rules to remember when multiplying integers:

• When you multiply a positive number by a positive number, the product is positive.
• When you multiply a positive number by a negative number, like in the exercise \(5(-2)\), the product is negative.
• Similarly, when you multiply a negative number by another negative number, the product is positive. This might sound surprising, but it happens because the two negative signs cancel each other out.

Remembering these basic rules helps you predict whether the product of multiplication will be positive or negative, and it's essential for solving problems with various integer combinations.