Problem 17 - Fundamentals of Mathematics

Question

Determine each of the values. $$ |6| $$

Step-by-Step Solution

Answer

|6| = 6

Step 1: Understand Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering the direction. It is always non-negative.

Step 2: Apply the Absolute Value Definition

Because absolute value represents the distance from zero, for any positive number like 6, the absolute value is the number itself. Thus, \[|6| = 6\]

Key Concepts

Number Line

A number line is a straight horizontal line marked with numbers placed at equal intervals. It is a tool that allows us to visually represent and understand numbers and their relationships. On a standard number line:

• Positive numbers are located to the right of zero.
• Negative numbers are located to the left of zero.
• Zero is the central point dividing the line between positive and negative numbers.

Using a number line, we can easily identify the position of any given number, including its distance from zero - which is crucial for understanding absolute value. Moreover, the number line provides an intuitive way to see how numbers increase or decrease as you move right or left. This visual tool helps clarify the concept of direction, which is vital for distinguishing between negative and positive numbers. When we're dealing with absolute values, the number line's most relevant feature is the non-negative distance of any number from zero. This concept helps underscore that absolute values are concerned only with how far away a number is from zero, not the direction.

Distance from Zero

Distance from zero on a number line is a way to measure how far a number is from the central zero point, always considering this distance as non-negative. Here's how to interpret this concept:

• If a number is positive, like 6, its distance from zero is simply the number itself.
• If a number is negative, like -6, its distance from zero is still 6, discarding the negative sign.

This concept is also known as the absolute value, denoted by vertical bars around a number, such as \(|x|\).

The idea of distance from zero helps simplify our understanding of numerical differences, focusing purely on separation and not direction. For instance, both 6 and -6 are equally distant from zero, despite being on opposite sides. This is why when we compute the absolute value, \(|6|\), we consider directions irrelevant and conclude that it equals 6. Therefore, the value is always non-negative, embodying the 'distance' without directional bias.

Non-negative Numbers

Non-negative numbers include all positive numbers and zero. They do not contain any negative signs and are fundamental to understanding absolute values. Let's break down what this means:

• A positive number like 6 is non-negative because it is greater than zero.
• Zero itself is non-negative because it is neither positive nor negative.
• Negative numbers do not fall under non-negative numbers because they lie on the opposite side of zero.

When dealing with absolute values, all results are non-negative. For instance, the absolute value of both 6 and -6 is 6. This means adjusting negative numbers to their non-negative counterparts by removing the negative sign. Hence, understanding non-negative numbers is crucial in calculations where absolute values are involved. If you are finding the absolute value, your answer will never be a negative number, emphasizing that you're measuring a non-negative distance from zero. This principle ensures consistency in mathematical expressions and problem-solving involving absolute values.