Problem 17 - Fundamentals of Mathematics

Question

Find the value of each of the following. Use a calculator to check each result. $$ (-4)(-8) $$

Step-by-Step Solution

Answer

The value of \((-4)\times(-8)\) is 32.

Step 1: Understanding Integer Multiplication

Multiplication of integers involves multiplying their absolute values and assigning the sign according to the rules of multiplication for negative values. A negative number multiplied by another negative number results in a positive product.

Step 2: Calculate Absolute Values

First, calculate the absolute values of the integers involved in the multiplication: Absolute value of -4 is 4, and the absolute value of -8 is 8.

Step 3: Multiply Absolute Values

Multiply the absolute values obtained in the previous step: \[4 \times 8 = 32\]

Step 4: Apply Sign Rules

Since both numbers in the original multiplication problem are negative, their product will be positive. Therefore, the result of \((-4) \times (-8)\) is 32.

Step 5: Verification Using a Calculator

Use a calculator to compute \((-4) \times (-8)\). You should find that it confirms our manual calculation, yielding a result of 32.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers that are less than zero. They are usually used to represent values such as losses, debts, or decreases. Negative numbers have a distinct place on the number line, opposite the positive numbers.

• Positive numbers are greater than zero and lie to the right on a number line.
• Negative numbers are less than zero and lie to the left on a number line.

When you multiply or divide with one negative number, the sign of the result changes. For example:

• Positive × Negative = Negative
• Negative × Positive = Negative
• Negative × Negative = Positive

This understanding is fundamental in operations involving negative numbers.

Exploring Absolute Value

The absolute value of a number is the number's distance from zero on the number line, without considering its direction. It is always a non-negative number.
For example:

• The absolute value of 4 is 4.
• The absolute value of -4 is also 4.
• Symbolically, the absolute value of a number \( x \) is represented as \(|x|\).

Understanding absolute value is important because it allows us to work with numbers without worrying about which side of zero they fall on.When multiplying two numbers, focusing on their absolute values first removes the complexity of signs and helps simplify calculations.

Comprehending Multiplication Rules

The rules of multiplication with integers help in predicting the sign and correctly calculating the product of two numbers. These rules are particularly important when working with both positive and negative numbers.
Here are the basic rules:

• If both numbers are positive, the product is positive.
• If both numbers are negative, the product is positive.
• If one number is negative and the other is positive, the product is negative.

These rules apply regardless of whether the numbers are large or small, whole numbers or decimals. By using absolute values to calculate the magnitude and the multiplication rules to determine the sign, you can handle any integer multiplication with confidence.

Importance of Calculator Verification

Using a calculator for verification is a practical step to ensure accuracy, especially when dealing with complex or large numbers. Although understanding the manual process is important, calculators help in double-checking our results.
Here’s how to effectively use a calculator for verifying multiplication results:

• Enter the numbers carefully, making sure to include the negative signs if needed.
• Check the result given by the calculator against your manual calculation.
• If both answers match, it reinforces the understanding of the multiplication process and rules.

Calculator verification is a good habit, ensuring precision in both academic and real-world mathematical applications. It helps build trust in your manual calculations and highlights any potential mistakes.