Calculate The Sum Of 24 + (-6) A Step-by-Step Guide

Understanding Integers and the Number Line

Before diving into the specifics of the problem, it's crucial to grasp the concept of integers. Integers are whole numbers, meaning they don't have any fractional or decimal parts. They include:

• Positive integers: These are whole numbers greater than zero (e.g., 1, 2, 3, ...).
• Negative integers: These are whole numbers less than zero (e.g., -1, -2, -3, ...).
• Zero: Zero is an integer that is neither positive nor negative.

The number line provides a visual representation of integers. It's a straight line with zero at the center, positive integers extending to the right, and negative integers extending to the left. Each integer is represented by a point on the line.

Understanding the number line is essential for visualizing the addition of integers. When adding a positive integer, we move to the right on the number line, and when adding a negative integer, we move to the left. This concept forms the basis for solving problems like $24 + (-6)$.

The Rules of Adding Integers

Adding integers involves specific rules that govern the outcome based on the signs of the numbers being added. Here's a breakdown of the key rules:

1. Adding integers with the same sign:
   • If both integers are positive, add their absolute values and keep the positive sign (e.g., $3 + 5 = 8$).
   • If both integers are negative, add their absolute values and keep the negative sign (e.g., $-3 + (-5) = -8$).
2. Adding integers with different signs:
   • Find the absolute value of each integer.
   • Subtract the smaller absolute value from the larger absolute value.
   • The result takes the sign of the integer with the larger absolute value (e.g., $-7 + 4 = -3$ because $|-7| > |4|$ and -7 is negative).

These rules are fundamental to accurately adding integers and will be applied in solving the problem at hand.

Solving $24 + (-6)$ Step-by-Step

Now, let's apply these rules to solve the specific problem: $24 + (-6)$.

1. Identify the signs: We are adding a positive integer (24) and a negative integer (-6).
2. Find the absolute values:
   • The absolute value of 24 is $|24| = 24$.
   • The absolute value of -6 is $|-6| = 6$.
3. Subtract the smaller absolute value from the larger absolute value: $24 - 6 = 18$.
4. Determine the sign of the result: Since the absolute value of 24 is greater than the absolute value of -6, and 24 is positive, the result will be positive.

Therefore, $24 + (-6) = 18$.

Visualizing on the Number Line

To further illustrate the solution, we can use the number line. Start at 24 on the number line. Adding -6 means moving 6 units to the left. This movement will land you at 18, confirming our calculated answer.

Alternative Methods and Strategies

While the step-by-step method is effective, there are alternative ways to think about adding integers. One such method involves thinking of negative numbers as debts or deductions.

In the context of $24 + (-6)$, you can think of it as having $24 and owing $6. After paying off the $6 debt, you would have $18 remaining. This analogy provides a practical understanding of the operation.

Another strategy is to rewrite the addition of a negative number as subtraction. Adding a negative number is the same as subtracting its absolute value. So, $24 + (-6)$ can be rewritten as $24 - 6$, which directly leads to the answer of 18.

Common Mistakes to Avoid

When adding integers, it's easy to make mistakes if the rules are not carefully followed. Here are some common pitfalls to avoid:

• Incorrectly applying the sign rules: A frequent error is forgetting to consider the signs of the integers when adding them. For example, incorrectly adding $-5 + 3$ as 8 instead of -2.
• Confusion with multiplication rules: The rules for multiplying integers are different from those for addition. Confusing these rules can lead to errors. For instance, incorrectly applying the rule "a negative times a negative is a positive" to addition.
• Neglecting absolute values: When adding integers with different signs, it's crucial to use absolute values to determine the magnitude of the result. Ignoring this step can lead to incorrect answers.

Real-World Applications of Integer Addition

The addition of integers is not just a theoretical concept; it has numerous real-world applications. Here are a few examples:

• Finance: Integers are used to represent financial transactions, with positive numbers representing income and negative numbers representing expenses. Adding integers helps in calculating net balances.
• Temperature: Temperatures below zero are represented using negative integers. Adding integers can be used to calculate temperature changes.
• Altitude: Sea level is considered zero altitude. Altitudes above sea level are positive integers, and altitudes below sea level are negative integers. Adding integers can determine the difference in altitude between two points.
• Sports: In some sports, points can be both positive and negative. Adding integers helps in calculating the final score or point difference.

Practice Problems

To solidify your understanding of adding integers, try solving these practice problems:

1. $-15 + 7 =$
2. $32 + (-12) =$
3. $-8 + (-11) =$
4. $19 + (-19) =$
5. $-25 + 10 =$

Conclusion

Finding the sum of integers is a fundamental skill in mathematics with wide-ranging applications. By understanding the rules of integer addition, visualizing the process on the number line, and practicing regularly, you can master this essential concept. In this guide, we have thoroughly explored the solution to $24 + (-6)$, along with the underlying principles and practical applications. Continue to practice and apply these concepts to enhance your mathematical proficiency.