Problem 35 - Fundamentals of Mathematics

Question

For the following 10 problems, on the number line, how many units are there between the given pair of numbers? -3 and 3

Step-by-Step Solution

Answer

6 units

Step 1: Understanding the Problem

The task is to find the distance between the numbers -3 and 3 on a number line. The distance between two numbers on a number line can be found by calculating the absolute difference between them.

Step 2: Identifying the Numbers

Identify the two numbers we need to calculate the distance between. In this case, the two numbers are -3 and 3.

Step 3: Using the Formula for Distance

The distance between two numbers \(a\) and \(b\) on the number line is given by the formula \(|a - b|\), where \(|x|\) denotes the absolute value of \(x\).

Step 4: Substituting the Values

Substitute the values -3 and 3 into the formula: \(|-3 - 3|\).

Step 5: Calculating the Difference

First, calculate the difference: \(-3 - 3 = -6\).

Step 6: Finding the Absolute Value

Calculate the absolute value of the result from the previous step: \(|-6| = 6\).

Step 7: Conclusion

The distance between -3 and 3 on the number line is 6 units.

Key Concepts

Number Line

A number line is a visual representation of numbers laid out in a straight line. This line helps us understand numbers' positions and their relationships with one another. Each point on a number line corresponds to a real number, and the numbers are evenly spaced.
For instance:

• Zero is typically at the center of this line.
• Positive numbers extend to the right.
• Negative numbers extend to the left.

This setup allows us to visually and easily determine the order of numbers, see how close or far apart numbers are, and compare their sizes. Number lines are widely used in math due to their simplicity and effectiveness in illustrating concepts like addition, subtraction, and particularly, understanding absolute values and distances.

Distance Between Numbers

The distance between any two numbers on a number line is simply how far apart the numbers are without considering their direction. For this, we use absolute value. The concept is straightforward and extremely practical.
Here's a quick guide:

• Identify the two numbers, say, \(a\) and \(b\).
• The distance is given by the formula: \(|a - b|\).
• This means you calculate the difference between the numbers, and then take the absolute value of this difference.

Absolute value is used because distance is always a positive quantity. Whether moving from \(a\) to \(b\) or from \(b\) to \(a\), the distance remains the same. This is why, in the example of finding the distance between -3 and 3, we obtain \(6\) since the absolute difference \(|-3 - 3| = 6|\) is calculated.

Mathematical Problem-Solving

When solving a mathematical problem such as finding the distance between numbers on a number line, follow a structured approach to ensure accuracy. Break down the problem into manageable steps and apply logical reasoning.
Here's a simple process to follow:

• Carefully read and understand the problem statement.
• Identify all relevant information, such as the specific numbers.
• Apply the appropriate mathematical formula or rule.
• Perform calculations step-by-step, ensuring the correct order of operations.
• Double-check results for reasoning coherence and correctness.

Using this step-by-step approach not only helps in reaching the correct solution but also boosts your math problem-solving skills by reinforcing understanding and procedural knowledge.