Problem 89 - Beginning and Intermediate Algebra

Question

In your own words, explain how to find the absolute value of a number.

Step-by-Step Solution

Answer

The absolute value is the non-negative distance from zero on a number line.

Step 1: Understand Absolute Value

The absolute value of a number is the distance from that number to zero on the number line, without considering which direction you are going. It is always a non-negative value, regardless of whether the original number is positive or negative.

Step 2: Determine the Type of Number

Check if the number is positive, negative, or zero. The absolute value of zero is zero itself because it's already at zero on the number line.

Step 3: Calculate Absolute Value for Non-Negative Numbers

If the number is positive, the absolute value is the number itself. For example, if the number is 5, then the absolute value is also 5.

Step 4: Calculate Absolute Value for Negative Numbers

If the number is negative, then the absolute value is the number without the negative sign. For instance, if the number is -7, then the absolute value is 7.

Step 5: Conclusion

Therefore, the absolute value of a number is simply its distance from zero on the number line, represented by a non-negative number.

Key Concepts

Non-Negative Numbers

Understanding the absolute value starts with the concept of non-negative numbers. Non-negative numbers include all positive numbers and zero. These numbers are significant because the absolute value of any number is always non-negative.

When you look at non-negative numbers, they act as a foundation for determining the absolute value. If you have a positive number or zero, its absolute value is the number itself. No changes occur to the number because non-negative numbers are already at or above zero.

• A non-negative number can be zero or any positive number.
• Absolute values turn any number, whether it's positive or negative, into a non-negative number.

This principle helps in defining absolute values distinctly and ensures clarity when dealing with both positive and negative numbers.

Number Line

The number line is an essential tool for visualizing absolute values. It's a straight line that shows numbers in an increasing sequence from left to right, with zero as the central point.

On the number line, positive numbers appear to the right of zero, while negative numbers are found to the left. Absolute value is about measuring how far a number is from zero, no matter which side of zero it’s on.

• Visualizing numbers on a line helps understanding their positions relative to zero.
• The number line provides a graphic way to see the distance from zero, which is what absolute value measures.

By using the number line, you can easily determine the absolute value by counting the spaces between the number and zero.

Distance from Zero

The concept of absolute value revolves around the idea of 'distance from zero' on the number line. Whether positive or negative, every number has a 'distance' that can be measured without considering direction.

This distance is simple to conceive:

• Any number's absolute value is just how far it is from zero.
• The absolute value disregards if the number is left or right of zero.
• It's represented as a positive or non-negative number.

For example, the distance of -3 from zero is 3, and similarly, the distance of 3 is also 3. This highlights the absolute value as the number's magnitude without focusing on its sign.