Problem 17 - Algebra 1: Concepts and Skills

Question

ADDING REAL NUMBERS Match the exercise with its answer. A. \(-2\) B. 0 C. \(-3\) $$ -2+2 $$

Step-by-Step Solution

Answer

The answer to -2 + 2 is 0. So, the correct match is with 'B. 0'

Step 1: Understanding Negative and Positive Integers

Negative integers like -2 are less than zero, while positive integers like 2 are greater than zero. When adding a negative and a positive integer, we are essentially subtracting the absolute value of the negative number from the positive number. That is how we can visualize -2 + 2 on the number line.

Step 2: Adding -2 and 2

Adding -2 to 2 involves starting at 2 on the number line and moving two steps to the left (because -2 is like moving left), which gets us to the point marked 0 on the number line. So, -2 + 2 is equal to 0.

Key Concepts

Integers

Integers are whole numbers that include both positive and negative numbers, as well as zero. Understanding integers is essential when learning to add real numbers. Let's break down what integers are and why they matter:

• Whole Numbers: Integers are whole numbers, which means they don't have any fractions or decimals.
• Types of Integers: Integers can be positive (like 1, 2, 3), negative (like -1, -2, -3), or zero (0).
• Order of Integers: On the number line, integers are arranged from negative to positive, making it easy to see which are larger or smaller. For example, -3 is less than -2, which is less than 0, 1, or 2.

It's important to understand the concept of integers because when adding them, the direction you move on the number line varies depending on if they are positive or negative: positive moves to the right, and negative moves to the left.

Negative Numbers

Negative numbers are a fundamental part of the set of integers. They are numbers less than zero and have various applications in mathematics and real-life situations. Here's what makes negative numbers unique:

• Representation: Negative numbers are represented with a minus sign (-) before them, such as -1, -2, -3, and so on.
• Addition and Subtraction: Negative numbers play a crucial role when added or subtracted from other numbers. When you add a negative number, it's like subtracting that number's absolute value from the positive number.
• Real-life Applications: Negative numbers are used in many real-life contexts, such as temperatures below zero, elevations below sea level, or financial debts.

When learning about adding real numbers, it helps to visualize adding a negative number as moving to the left on the number line, reducing the overall value of the number.

Number Line

A number line is a visual tool used in mathematics to represent numbers in a straight line. It makes understanding the concept of adding integers, particularly when dealing with negative numbers, more intuitive. Here’s how it helps:

• Visual Representation: The number line shows numbers increasing from left to right with negative numbers on the left and positive numbers on the right.
• Addition on the Number Line: To add two numbers, start at the position of the first number. If you add a positive number, move to the right. If adding a negative number, move to the left.
• Understanding Zero: Zero is the central point on the number line, acting as the boundary between positive and negative numbers.

In the example of adding -2 and 2, you start at 2 on the number line and move two steps left to end at 0. This clearly shows why -2 + 2 equals 0, making the number line a powerful tool for understanding mathematical operations.