Problem 14 - Fundamentals of Mathematics

Question

Find each value. \(-3+(-8)\)

Step-by-Step Solution

Answer

The value is \(-11\).

Step 1: Identify the Operation

The given expression is \(-3 + (-8)\). Here, we need to perform an addition operation between two negative numbers, \(-3\) and \(-8\).

Step 2: Convert Addition to Subtraction

When adding a negative number, it is the same as subtracting its absolute value. So, converting \(-3 + (-8)\) is equivalent to \(-3 - 8\).

Step 3: Apply the Rule for Adding Negative Numbers

To add two negative numbers, add their absolute values and attach a negative sign to the result. The absolute values are \(|-3| = 3\) and \(|-8| = 8\). Thus, add them: \(3 + 8 = 11\).

Step 4: Attach the Negative Sign

Since both numbers in the original expression were negative, the result is also negative. Therefore, the sum is \(-11\).

Key Concepts

Operation on Integers

When we talk about operations on integers, we usually refer to basic arithmetic operations like addition, subtraction, multiplication, and division. Integers include positive numbers, negative numbers, and zero. Understanding how to perform these operations is crucial since integers are foundational in mathematics.

In our specific problem, we focus on **addition of negative numbers.** Adding integers, especially negative ones, can be tricky, but there's a straightforward approach. When both numbers have the same sign (like both being negative), you essentially ignore the negative signs to add their absolute values together first, and then put back the common sign. This ensures you're accurately reflecting the nature of the addition operation between the integers involved.

Remember, the key is to focus on the process and manage any signs correctly at the final step.

Absolute Value

The **absolute value** of a number is all about its distance from zero on a number line, irrespective of direction. It's always a non-negative number. For any integer \( x \), its absolute value is denoted by \(|x|\).

For the integers mentioned in our exercise, \(|-3| = 3\) and \(|-8| = 8\). The absolute value just strips away any negative sign. So, it tells us how far a number is from zero, without worrying about its direction on the number line.

Understanding absolute value is pivotal because it allows us to handle operations, particularly addition and subtraction, more effectively. By converting integers to their absolute values during calculations, you simplify the process and minimize errors, especially when negative numbers complicate the operation.

Negative Numbers

**Negative numbers** represent values below zero and are denoted by a minus sign (\(-\)). They can often appear daunting, especially when combined in operations, but the key is understanding their properties and how they interact with each other.

When adding two negative numbers, like in the exercise, it's essential to know that the result stays negative. This is because negative numbers, when visualized on a number line, are always moving to the left from zero. Thus, adding them means going further left.

To ensure accuracy when working with negative numbers:

• Always focus on adding their absolute values first.
• Attach the negative sign to the result since both operands are negative.

This conceptual step not only aids in finding the correct solution but also helps in mentally managing further calculations involving negative numbers.

Subtraction

Subtraction can often be thought of as the inverse of addition, but it plays a unique role, especially when dealing with negative numbers. In the exercise, we converted addition of a negative number to subtraction, which follows the general rule that addition of \(-x\) is equivalent to subtraction of \(|x|\).

This conversion isn't just a mechanical step; it reflects a deeper understanding of how moving along the number line works. When you subtract a value, you move left, meaning you decrease the total. This makes a subtraction operation a powerful tool when dealing with both positive and negative values. It can significantly simplify calculations by turning them into more manageable forms.

In our example,

• Adding \(-8\) was turned into subtracting \(|-8|\) from \(-3\).
• Thus, it became \(-3 - 8\), a procedure that is often simpler to comprehend.

Grasping this concept helps in handling more complex arithmetic involving various integer operations.