Problem 63 - Prealgebra

Question

Find each sum. $$(-2)+5+(-3)$$

Step-by-Step Solution

Answer

The sum is 0.

Step 1: Add the first two numbers

We'll start by adding the first two numbers in the sequence: \((-2) + 5\)When you add a negative number to a positive number, you can think about moving on a number line. Starting from \(-2\), moving 5 units to the right brings us to 3.

Step 2: Add the result to the last number

Now, take the result from Step 1, which is 3, and add the last number in the sequence, \(-3\):\(3 + (-3)\)Adding \(-3\) to 3 means moving 3 steps to the left on the number line starting from 3, which brings us to 0.

Key Concepts

Number Line

A number line is a visual tool that helps in understanding the position and order of numbers within the set of real numbers. It is like a path that stretches infinitely in both directions, with zero in the center. From zero:

• Numbers to the right are positive.
• Numbers to the left are negative.

Each step on a number line represents a unit change. For example, if you start at \(-2\) and move five steps to the right, you end up at 3. This movement is exactly what you do mathematically when handling addition of integers. By using the number line, calculations like adding \((-2) + 5\) become easier to visualize since you can see the movement towards the larger numbers to the right. Similarly, adding \(-3\) means moving three steps to the left starting from 3. Therefore, the number line is not just useful for simple arithmetic but is essential for grasping concepts of integer addition visually.

Positive and Negative Numbers

In mathematics, numbers are classified into positive and negative, based on whether they are greater or smaller than zero. This classification is crucial:

• Positive numbers are greater than zero and are normally represented without a sign, like 5.
• Negative numbers are less than zero and carry a negative sign, such as \(-3\).

Understanding this concept is important because arithmetic operations behave differently with positive and negative numbers. When positive numbers are added to negative numbers, the effect is similar to subtracting values, as you move towards zero. Conversely, adding a negative number equates to subtracting that number's absolute value, effectively moving left from the current position on the number line. The sum \((-2) + 5 + (-3)\) perfectly illustrates this concept, as starting at \(-2\), you move right with the positive number and left again with the negative, ending at zero.

Arithmetic Operations

Arithmetic operations refer to basic mathematical processes including addition, subtraction, multiplication, and division. In the context of integer addition, understanding how these operations manipulate positive and negative numbers is key. With addition:

• Adding a positive number moves attention to the right on the number line.
• Adding a negative number shifts focus to the left.

When performing the addition of several numbers, it's important to consider the sequence. Step-by-step addition helps avoid confusion and makes better sense, especially when negative numbers are involved. In our example, the calculation begins with \((-2) + 5\), resulting in 3 due to moving right on the number line. Then adding \(-3\) changes that 3 to a 0 by moving left. Such operations highlight the importance of order in arithmetic processes and using the number line provides a useful reference to imagine these steps physically.