Problem 40 - Fundamentals of Mathematics

Question

Determine each value. $$ |-8| $$

Step-by-Step Solution

Answer

The absolute value of -8 is 8.

Step 1: Understand Absolute Value

The absolute value of a number is the distance of that number from zero on a number line, regardless of direction. Therefore, it is always non-negative.

Step 2: Apply Absolute Value Definition

Take the given number, -8, and determine its distance from zero. The number -8 is 8 units away from 0, so its absolute value is 8.

Step 3: Write the Result

Based on the absolute value definition, we find that \(|-8|\) equals 8.

Key Concepts

Distance from Zero

When you think about the absolute value, envision it as a measure of how far a number is from zero. It does not matter whether the number is positive or negative when considering its absolute value; what matters is the distance itself. For example, the number \(-8\) on a number line is 8 units away from zero. Therefore, the absolute value of \(-8\) is 8. This is because only the distance is considered, not the direction.

• The focus is solely on how many steps you need to reach zero.
• Both \(+8\) and \(-8\) have the same absolute value of 8.
• Absolute value means ignoring the sign and looking at the value itself.

This idea makes absolute values very useful in various mathematical applications where the size of a number is important but its sign is not.

Non-negative Values

Absolute value is fundamentally linked to the concept of non-negative numbers. This means when you calculate the absolute value, the result will always be zero or positive. This principle assures that you're always dealing with a non-negative distance from zero. No matter whether you're dealing with the positive side, like \(+8\), or the negative, like \(-8\), the absolute value turns it non-negative. Let's remember:

• Absolute values are always zero or greater.
• Negative inputs, such as \(-8\), become positive outputs, such as 8.
• This process helps in calculations where size matters but negativity doesn't.

So, whenever you face a number and are asked to find its absolute value, you're essentially stripping away any negative sign to consider only its magnitude.

Number Line Concept

The number line is a simple yet powerful tool that helps us understand absolute value better. Imagine a straight line with zero in the center, positive numbers spaced evenly to the right, and negative numbers to the left. This visualization helps demonstrate why absolute values are non-negative by showing them as distances. On the number line:

• Zero is the starting point, which means the absence of distance.
• Numbers on the right are positive, and numbers on the left are negative.
• Both directions fit into the idea of measuring how far away they are from zero.

To picture \(-8\), you locate it on the number line, count how many steps it is to zero, and see that you need 8 units regardless of direction. This upholds the concept that absolute value represents the distance from zero, showcasing its neutrality and non-negative nature on the number line.