Problem 139 - Intermediate Algebra

Question

Explain why the sum of -8 and 2 is negative, but the sum of 8 and -2 is positive.

Step-by-Step Solution

Answer

The magnitude of -8 is greater than 2, resulting in a negative sum. Conversely, the magnitude of 8 is greater than -2, resulting in a positive sum.

Step 1: Understanding Negative and Positive Numbers

First, recall that negative numbers are less than zero and positive numbers are greater than zero. For example, -8 is a negative number, and 2 is a positive number.

Step 2: Calculate the Sum of -8 and 2

Add -8 and 2: -8 + 2 = -6. The result is negative because -8 is much larger in magnitude compared to 2 (i.e., it is further from zero), making the sum negative.

Step 3: Calculate the Sum of 8 and -2

Now, add 8 and -2: 8 + (-2) = 6. The result is positive because 8 is larger in magnitude than -2, making the sum positive.

Step 4: Summarize the Reasoning

The sign of the sum of a positive and negative number depends on the magnitude (absolute value) of each number. The number with the larger magnitude determines the sign of the sum.

Key Concepts

Understanding Positive and Negative Numbers

Positive and negative numbers are fundamental concepts in mathematics. Positive numbers are greater than zero, like 1, 2, 3, and so on. On the other hand, negative numbers are less than zero, like -1, -2, -3, etc. When dealing with both positive and negative numbers, remember that they are essentially opposites.

• Positive numbers indicate an increase or a gain.
• Negative numbers indicate a decrease or a loss.

The arithmetic operations (like addition and subtraction) involving positive and negative numbers require paying attention to their signs. Combining these numbers can lead to different results based on their magnitude and direction.

Understanding Magnitude

Magnitude refers to the size or quantity of a number, without considering its sign. For example, the magnitude of both +5 and -5 is 5. It's how far a number is from zero on a number line.

• Magnitude helps in comparing numbers.
• It's essential for determining the outcome of adding or subtracting numbers with different signs.

When adding -8 and 2, you look at the magnitude of each number. Here, -8 has a larger magnitude than 2, resulting in a negative sum. Conversely, when adding 8 and -2, the positive number (8) has a larger magnitude, resulting in a positive sum.

Understanding Absolute Value

Absolute value measures the distance of a number from zero, regardless of its sign. It is always a non-negative value. For example, the absolute value of -8 is 8, and the absolute value of 2 is 2. Absolute value is denoted by two vertical bars around a number, like this: \(|-8| = 8 and |2| = 2\).

• Absolute value helps in comparing magnitudes without considering the sign.
• It is useful in determining the final sign of a sum or difference.

In the provided exercise, understanding absolute value is crucial. The sum of -8 and 2 is -6 because \(|-8| > |2|\). The sum of 8 and -2 is 6 because \(|8| > |-2|\). Thus, the number with the larger absolute value determines the sign of the result.