Problem 42 - Fundamentals of Mathematics

Question

Determine each value. $$ -|-1| $$

Step-by-Step Solution

Answer

The value is -1.

Step 1: Understand Absolute Value

The absolute value of a number, denoted as \(|x|\), is the non-negative value of \(x\). This means it represents the distance of \(x\) from zero on the number line, without considering direction. Therefore, for any negative number such as \(-1\), the absolute value is simply \(1\). Thus, we have \(|-1| = 1\).

Step 2: Apply the Negative Sign

The expression \(-|x|\) indicates taking the negative or opposite of the absolute value of \(x\). From Step 1, we found \(|-1| = 1\). Therefore, by adding a negative sign to this absolute value, we get \(-|-1| = -1\).

Key Concepts

Non-negative Value

When we talk about the absolute value, we are focusing on the non-negative aspect of a number. Absolute value basically strips away any negative sign and represents just the size of the number.
For example, with \(-1\), its absolute value is \(|-1| = 1\). Here 1 is the non-negative value. Think of it as the numerical part of the number, without worrying about any direction or sign.

• Absolute values are always zero or positive.
• This assists in depicting pure magnitude on the number line.

Number Line

A number line is a straightforward tool that helps visualize how numbers are positioned relative to zero. It’s a simple line with zero in the middle, positive numbers to the right, and negative numbers to the left.
The absolute value of a number can be understood by its distance from zero on this line. If you imagine standing at zero, the absolute value tells you how many steps you need to take to reach the number, ignoring which direction you're going.

• Positive numbers: Right of zero.
• Negative numbers: Left of zero, but absolute value treats them as positive distance.
• Zero itself is at the center, with its own absolute value being zero.

Negative Sign Application

Applying a negative sign to an absolute value might seem a bit confusing at first. Though absolute values are positive, when a negative sign is applied to the absolute value, it simply changes the direction or sign of the result.
Take our example \(-|-1|\) where \(|-1| = 1\). Adding a negative sign before this result changes it to -1. It’s important to keep track of where negative signs are applied to ensure the correct outcome.

• The operation \( -|x| \) implies the opposite of the non-negative absolute value.
• Always apply such negative signs last, after computing the absolute value.
• Make sure to differentiate between negative signs applied to numbers versus absolute values.