Problem 111 - Algebra A Combined Function

Question

In your own words, explain how to add two negative numbers.

Step-by-Step Solution

Answer

To add two negative numbers, add their absolute values and make the result negative.

Step 1: Understanding Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign (e.g., -1, -2). They are used to represent a deficit or a position on the left side of zero on a number line.

Step 2: Writing the Problem in Mathematical Form

Let's say we need to add two negative numbers, such as -3 and -5. The mathematical expression is: \(-3 + (-5)\).

Step 3: Adding Negative Numbers

To add two negative numbers, add their absolute values (ignore the negative signs) and then put a negative sign in front of the result. In our example, the absolute values are 3 and 5. So, calculate 3 + 5 = 8, and then apply the negative sign to get -8.

Step 4: Verifying the Result

We can verify by visualizing on a number line. Start at zero, move 3 steps left (for -3), and then 5 more steps left (for -5). You end up 8 steps left of zero, at -8, confirming the solution.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers that are less than zero. They are represented by a minus sign, such as -1 or -5. These numbers are not just theoretical—negative numbers can represent real-life scenarios like a bank overdraft or temperatures below freezing.

When you add negative numbers, you are actually performing an operation similar to subtraction. Why? Because each negative number can be thought of as moving to the left on the number line.

For example:

• If it's -3 degrees outside today and the temperature drops further by 2 degrees, you end up at -5 degrees.

This action of moving left on the number line helps us visualize how negative numbers work in the context of addition.

Exploring Absolute Value

Absolute value is a concept that helps simplify the addition of negative numbers. The absolute value of a number is its distance from zero on the number line, without considering direction. Simply put, it is the number without its negative sign.

For instance, the absolute value of -3 is 3, same goes for the absolute value of -5 which is 5. Absolute values are important because when adding negative numbers, we first add their absolute values.

Here's a little equation to visualize this:

• The absolute value of -3 + the absolute value of -5 equals 3 + 5, which equals 8.

After getting this result, you attach a negative sign to it because both original numbers were negative, resulting in -8. Knowing how to use absolute values can make adding negative numbers much simpler and straightforward.

Visualizing with a Number Line

A number line is a straight line with numbers placed at equal intervals or segments along its length. It is a wonderful visual tool for adding negative numbers. Numbers to the right of zero are positive, while those to the left are negative.

When using a number line to add two negative numbers, you start at zero. Move left the number of spaces corresponding to the first negative number, and then move left again for the second negative number.

• For example, begin at zero, move 3 spaces left to reach -3, then shift 5 more spaces left to land on -8.

The number line perfectly illustrates how adding negative numbers essentially means increasing their distance from zero in the leftward direction. This process provides a clear and intuitive understanding of how negative numbers interact to collectively "pull" further into the negative side.