Problem 63 - Prealgebra

Question

Find the sum of \(-8,-10,\) and \(-3\).

Step-by-Step Solution

Answer

The sum is -21.

Step 1: Understand the Numbers

We need to find the sum of three negative numbers: -8, -10, and -3. It's important to remember that when you add negative numbers, you're essentially adding their absolute values and then applying the negative sign to the result.

Step 2: Calculate the Absolute Sum

Find the absolute values of the numbers: |-8| = 8, |-10| = 10, and |-3| = 3. Next, add these absolute values together: \[ 8 + 10 + 3 = 21 \]

Step 3: Apply the Negative Sign

Since all original numbers are negative, we apply the negative sign to the sum of their absolute values calculated in Step 2. Therefore, the result is \[ -21 \]

Key Concepts

Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. Simply put, it's the number itself without any positive or negative sign. Absolute value is always non-negative because it solely represents magnitude without considering direction. For example, the absolute value of  and 6 are both 6, written as 6, because they are both 6 units away from zero on the number line. Understanding absolute values is crucial when working with negative numbers because it helps clarify their magnitude. In our example exercise, the absolute values of -8, -10, and -3 are 8, 10, and 3, respectively. This step is vital because it allows us to easily add their magnitudes without worrying about their signs at first. After calculating the total magnitude, we can then consider the direction by re-applying the negative sign, as all original numbers were negative.

Integers and Their Properties

Integers include all whole numbers, both positive and negative, as well as zero. Examples of integers are -5, 0, 3, and 11. Unlike decimals or fractions, integers do not have parts between whole numbers. They are only whole units that can be positive, negative, or zero. When dealing with addition or subtraction, the sign of the integers plays a significant role. When you add two negative integers, you can think of it as combining debts, which always increases the total negative value. For instance, adding -8 and -3 adds the obligation, resulting in -11. Recognizing the properties of integers, such as their ordering on the number line and their relationships with each other, is helpful in understanding arithmetic operations like addition, which often require sign awareness to determine the correct final result.

Summing Up Numbers: A Step-by-Step Approach

When adding any set of numbers, including negative ones, it is important to handle them systematically. Let's look at the exercise of adding -8, -10, and -3.

• First, find the absolute values: |-8| = 8, |-10| = 10, |-3| = 3.
• Next, sum these absolute values: 8 + 10 + 3 = 21. This step allows you to focus simply on the size of the numbers, letting you add them like straightforward positive integers.
• Finally, since the original numbers were negative, apply a negative sign to reach the final sum: -21. This step reflects the negative nature of all original integers.

By following the structured process above, you ensure accuracy and clarity, treating each part of addition carefully, whether the numbers are positive or negative. Consistently using this approach not only simplifies calculations but also deepens your understanding of how integers interact with each other when added.