Problem 31 - Algebra 1: Concepts and Skills

Question

RULES OF ADDITION Find the sum. $$8+(-5)$$

Step-by-Step Solution

Answer

The sum is 3.

Step 1: Understand Adding a Negative Number

The fundamental principle is that adding a negative number is the same as subtracting the absolute value of that number. In this case, \(8 + (-5)\) is the same as \(8 - 5\).

Step 2: Perform the Subtraction

Subtract 5 from 8 to obtain the result. \(8 - 5 = 3\) .

Key Concepts

Rules of Addition

The rules of addition are straightforward, but they lay the foundation for all arithmetic operations. Addition involves combining two numbers to obtain a sum. When you add two positive numbers, the sum is simply the total of both amounts. However, addition becomes a bit different when negative numbers come into play.
- When adding a positive number to another positive number, simply combine their values.- When adding negative numbers to positive numbers, think of it as subtracting the absolute value of the negative number from the positive number.- If both numbers are negative, their absolute values add up, but the resulting sum is still negative.
For instance, in the operation \(8 + (-5)\), we interpret \((-5)\) as subtracting 5 from 8 to simplify the calculation.

Negative Numbers

Negative numbers represent values less than zero and have a range of applications, from simple arithmetic to describing real-world losses or debt. When working with negative numbers, it’s important to understand their behavior in mathematical operations containing addition and subtraction.
- A negative number, by definition, is the opposite of a positive one. If you have \(-5\), it’s 5 units below zero on the number line.- In mixing negative with positive numbers through addition, you will either enhance or lessen the magnitude of the resulting number, depending on which is larger in absolute terms.- Think of negative numbers in addition operations as debts—if you're adding a debt to a credit, you'll end up with less credit, or possibly an overall loss.- Navigating calculations, such as \(8 + (-5)\), involves subtracting the absolute value of the negative number from the positive to find the outcome.

Subtraction as Addition

Understanding subtraction as a form of addition might seem confusing at first, but it's based on the idea of adding negative numbers. The subtraction operation \(a - b\) essentially transforms into \(a + (-b)\). This approach aligns addition and subtraction into a unified set of operations, helping simplify our arithmetic understanding.
- Viewing subtraction as adding a negative makes calculations more fluid, especially when dealing with longer mathematical problems. - Using subtraction as addition can help prevent errors, allowing consistent application of addition rules across different operations.- In the example \(8 + (-5)\), the operation is inherently subtracting 5 from 8. Listing it as addition of a negative number serves to illustrate this principle simply and consistently. Looking at subtraction in this way aids not only in algebraic contexts but also enhances problem-solving strategies across various mathematical fields.