Problem 7 - Prealgebra

Question

Draw a number line from 10 to 10 and use it to add the following numbers. $$-4+(-2)$$

Step-by-Step Solution

Answer

The sum of \(-4 + (-2)\) is -6.

Step 1: Understanding the Problem

We need to add two negative numbers, \(-4 + (-2)\), using a number line that ranges from -10 to 10.

Step 2: Identifying the Start Point

We start at 0 on the number line to represent the starting point before we perform any addition or subtraction.

Step 3: Adding the First Number

Since we have \(-4\), move 4 units to the left from 0 on the number line. This brings us to -4.

Step 4: Adding the Second Number

Next, add \(-2\) by moving another 2 units to the left from -4. This brings us to -6 on the number line.

Step 5: Verifying the Calculation

Initially at 0, moving 4 units left takes us to -4, and moving an additional 2 units further left brings us to -6. Thus, \(-4 + (-2) = -6\).

Key Concepts

Adding Negative Numbers

When you add negative numbers, it's like moving left on a number line. Think of this process as the opposite of adding positive numbers, which would move you to the right.

• Visualize the Movement: Imagine standing at a point on a number line. Negative signs tell you to step to the left. So, if you are starting at 0 and add \(-4\), you move 4 steps to the left, stopping at -4.
• Sequence Matters: Adding another negative number means you continue in the same direction. After reaching -4, adding \(-2\) means you move 2 more steps left, landing at -6.
• This is why subtraction of numbers works similarly in some cases: that's because subtracting a positive number involves moving left as well, just like adding a negative one.

Integers on Number Line

A number line is a powerful tool for visualizing numbers and understanding operations involving them.

• Structure: The number line stretches infinitely in both directions but is often marked within a specific range, in our case, from -10 to 10.
• Position and Movement: Each point on the number line corresponds to an integer. Moving left indicates a decrease, while moving right indicates an increase. Therefore, understanding this can help with both negative and positive operations.
• Usage in Calculations: When performing operations like \(-4 + (-2)\), the number line allows us to precisely track our starting and ending points by sight.

Mastering the use of number lines significantly aids in understanding more complex mathematical concepts.

Mathematical Operations

Mathematics involves various operations that can be visualized in distinct ways, with addition and subtraction being fundamental.

• Addition: This operation involves combining numbers. When dealing with integers, adding positive numbers moves you right on the number line, and adding negative numbers moves you left.
• Subtraction: Often seen as a form of addition of negative numbers. For instance, subtracting \(2\) is the same as adding \(-2\).
• Checking Your Work: It’s always a good idea to verify calculations using the number line. Ensuring movements and final position align with expectations helps spot mistakes.

By understanding these operations deeply, especially with visual aids, you lay a solid foundation for tackling more challenging mathematical problems.