Problem 84 - Prealgebra

Question

The problems below review addition, subtraction, multiplication, and division of positive and negative numbers, as covered in this chapter. Perform the indicated operations. $$-8-4$$

Step-by-Step Solution

Answer

The result of \(-8 - 4\) is \(-12\).

Step 1: Understand the Operation

The given problem is an expression that combines two numbers with subtraction: \(-8 - 4\). This operation involves subtracting 4 from -8, which requires careful handling of negative numbers.

Step 2: Subtract Numbers

When subtracting a positive number from a negative number, you actually move further down the negative scale. Therefore, subtracting 4 from -8 means you subtract 4 additional units from -8, resulting in a decrease by 4.

Step 3: Calculate the Result

Calculate \(-8 - 4\) by realizing that you are moving 4 units further into the negatives, which results in \(-12\). Thus, \(-8 - 4 = -12\).

Key Concepts

Addition and Subtraction of Integers

When working with integers, two basic operations are addition and subtraction. These operations often involve positive and negative numbers, which can sometimes be tricky. The key is to think of integers as points on a number line.

• **Addition**: When adding two integers, if they have the same sign (both positive or both negative), you simply add their absolute values and keep the common sign. For instance, \(-3 + (-7) = -10\).
• **Subtraction**: Subtraction of integers can be thought of as adding the opposite. For example, \(a - b\) becomes \(a + (-b)\). This is particularly helpful when dealing with negative numbers, as we'll see next.

Understanding these concepts helps in smoothly executing both operations in various scenarios, especially when the numbers include negatives.

Negative Numbers

Negative numbers represent values less than zero and are typically indicated by a minus sign. They can be found on the left side of a number line from zero. The more negative the number, the further left on the number line it is.

• **Behavior**: Just like positive numbers have their addition and subtraction rules, negative numbers adhere to specific rules. For instance, adding a negative number essentially decreases the value, which is equivalent to subtraction.
• **Subtraction**: When subtracting a positive number from a negative number, you move deeper into negative territory, as you are effectively adding the negative equivalent of the positive number.

Understanding negative numbers and their operations helps avoid common mistakes, especially in contexts like \(-8 - 4\), where care must be taken with the signs involved.

Basic Arithmetic Operations

Basic arithmetic operations include addition, subtraction, multiplication, and division. These foundational operations form the basis of mathematics and cleverly interplay even when working with diverse numbers, such as negative and positive integers.

• **Importance of Order**: In problems involving multiple operations, the correct order of operations is crucial. Usually, calculations are performed from left to right, adhering to the priority of operations (PEMDAS/BODMAS).
• **Handling Negatives**: Special care is needed with negatives, especially in subtraction. Understanding how to correctly process such expressions ensures accuracy, such as when interpreting \(-8 - 4\), where you move further into negatives after subtracting a positive number from a negative one.

Grasping these basic arithmetic principles can simplify complex problems, making them more approachable and easier to solve.