Problem 34 - Prealgebra

Question

Combine the following by using the rule for addition of positive and negative numbers. $$-765+213$$

Step-by-Step Solution

Answer

-552

Step 1: Understanding the Problem

We need to solve the expression \(-765 + 213\) by combining these numbers. The goal is to find the sum of the two numbers, taking into account the rules for adding negative and positive numbers.

Step 2: Applying the Rule of Addition

When combining a negative number and a positive number, you subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value. For \(-765 + 213\), the absolute values are 765 and 213. Since 765 > 213, we'll subtract 213 from 765.

Step 3: Subtracting the Absolute Values

Subtract the smaller absolute value from the larger one: \(765 - 213\). Perform the subtraction to find the result: \(765 - 213 = 552\).

Step 4: Determining the Sign of the Result

Since the number with the larger absolute value is negative (-765), the result of the addition will also be negative. Therefore, the result is \(-552\).

Key Concepts

Negative Numbers

Negative numbers are a fundamental part of everyday math and essentially represent values less than zero. They are denoted by a minus sign (-) before the number. Understanding negative numbers is crucial because they are used in various contexts like temperature changes, financial transactions, and depth measurements.

When dealing with negative numbers, it's important to recognize their direction on the number line. Unlike positive numbers, which increase as you move to the right, negative numbers become smaller (or more negative) as you move to the left. For example:

• -1 is greater than -5 because it is closer to zero.
• Moving in the positive direction from -3, you'll reach -2, -1, and eventually 0.

These concepts help in understanding calculations involving negative numbers, particularly when they are combined through addition or subtraction.

Absolute Value

The absolute value of a number refers to its distance from zero on the number line, without considering direction. In simpler terms, the absolute value of a number is always non-negative.

To find the absolute value, consider the following:

• The absolute value of a positive number is the number itself. For example, \( |5| = 5 \).
• The absolute value of a negative number is its positive counterpart. For instance, \( |-8| = 8 \).

When doing arithmetic with negative and positive numbers, absolute value helps in determining the magnitude of the numbers to perform operations like addition or subtraction effectively. In the expression \( -765 + 213 \), we take the absolute values 765 and 213 to decide the calculation method.

Arithmetic Rules

Arithmetic rules are guidelines that govern mathematical operations to ensure consistency across calculations. Particularly when adding numbers, understanding these rules simplifies solutions.

Rules for Adding Integers:

• When adding two positive numbers, simply add their values.
• When adding two negative numbers, add their absolute values and make the result negative.
• When adding a negative number to a positive number, subtract the smaller absolute value from the larger one and keep the sign of the number with the larger absolute value.

In \( -765 + 213 \), the rule for a positive and a negative number is applied, where absolute values assist in determining the operation type, and the sign of the larger absolute value dictates the sign of the answer.

Integer Operations

Integer operations encompass all arithmetic activities involving whole numbers, both positive and negative, including zero. They follow straightforward rules but require attention to the signs of the numbers involved. These operations form the base of many mathematical concepts.

Key Integer Operations:

• Addition: Carefully add numbers with regard to their signs as per arithmetic rules.
• Subtraction: Convert the subtraction problem into an addition problem by changing the sign of the second number.
• Multiplication and Division: The result will be positive if the signs of the numbers are the same, and negative if they are different.

In everyday scenarios, adding or subtracting integers helps in balancing budgets, tracking profit and loss, and measuring physical quantities. Understanding how to manipulate and combine integers expands one's ability to solve real-world problems efficiently.