Problem 23 - Prealgebra

Question

Combine the following by using the rule for addition of positive and negative numbers. $$-1+(-2)$$

Step-by-Step Solution

Answer

-1 + (-2) = -3.

Step 1: Understanding the Problem

We need to find the sum of the two numbers: -1 and -2. Both numbers are negative, which means we're dealing with the addition of negative numbers.

Step 2: Identify the Rule

The rule for adding two negative numbers is to add their absolute values together and then apply a negative sign to the result.

Step 3: Calculate Absolute Values

Find the absolute value of -1, which is 1, and the absolute value of -2, which is 2.

Step 4: Add Absolute Values

Add the absolute values together: 1 + 2 = 3.

Step 5: Apply the Negative Sign

Since both original numbers were negative, the result of their addition will also be negative. Thus, -1 + (-2) = -3.

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They have a minus sign in front of them. Negative numbers are used to represent losses, debts, or temperatures below zero. Anything below ground level, or under the surface, can also be seen as a negative measurement.
Negative numbers are not only used in math but also in everyday life. Some examples include:

• Temperatures below zero, like -5°C, represent degrees below freezing.
• In finances, a bank overdraft or loss can be represented as a negative number.
• In sports, a team’s score can be negative if they lose points, like in golf or penalty kicks.

Understanding how to work with negative numbers is crucial for solving various mathematical problems, such as the addition of integers. When you add negative numbers, you move further in the negative direction on the number line.

Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering the direction. Simply put, it's the magnitude of the number, whether it is positive or negative.
For instance, the absolute value of -4 is 4, written as \(|-4| = 4\). Similarly, the absolute value of 4 is 4, written as \(|4| = 4\). As you can see, absolute value ignores the sign of a number.
The absolute value helps to simplify the addition and subtraction of integers by converting everything into a positive format. This is particularly useful when combining negative numbers. The rule involves:

• Finding the absolute values of each negative number.
• Add the absolute values together.
• Apply the sign of the original numbers to find the final result.

Sum of Negative Numbers

When adding negative numbers, such as -1 and -2, we're essentially calculating how much we "move left" on the number line, since negative numbers represent values below zero.
The rule for adding negative numbers is to:

• Calculate the absolute values of the numbers. For -1, it's 1, and for -2, it's 2.
• Add these absolute values. Here, it would be 1 + 2 = 3.
• Finally, apply the negative sign to this sum, which results from the original negative signs of the numbers. So, -1 + (-2) = -3.

This method ensures you maintain the correct "direction" on the number line, moving further into the negative as you add them together. The outcome is a larger absolute negative value, showing a more considerable "debt" or "loss," akin to moving deeper underwater or further south below sea level.