Problem 11 - Fundamentals of Mathematics

Question

Use the algebraic definition of absolute value to find the following values. $$ -|8| $$

Step-by-Step Solution

Answer

The value is \(-8\).

Step 1: Understand the Absolute Value

Absolute value of a number is its distance from zero on the number line, regardless of direction. For any real number \(a\), the absolute value is defined as \(|a| = a\) if \(a \geq 0\) and \(|a| = -a\) if \(a < 0\). For \(8\), which is positive, \(|8| = 8\).

Step 2: Find the Absolute Value

Since \(8\) is a positive number, its absolute value is just the number itself. Therefore, \(|8| = 8\).

Step 3: Apply the Negative Sign

We have \(-|8|\), which means the negative of the absolute value of \(8\). From Step 2, we found that \(|8| = 8\), so we need to take the negative of this value. Thus, \(-|8| = -8\).

Key Concepts

Algebraic Definition of Absolute Value

Absolute value is a fundamental concept in mathematics, especially when dealing with real numbers. The algebraic definition states that the absolute value of a real number \(a\) is expressed as the number's non-negative magnitude.
This can be mathematically defined as:

• \(|a| = a\) if \(a \geq 0\)
• \(|a| = -a\) if \(a < 0\)

In essence, absolute value measures how far a number is from zero on the number line, without regard to its sign.
For example, the absolute value of both 8 and -8 is 8, demonstrating that distance is always positive.

Understanding Real Numbers

Real numbers include all the numbers on the number line. This encompasses whole numbers, fractions, and irrational numbers, such as \(\pi\) and \(\sqrt{2}\).
Real numbers are essential because they provide a complete picture of the number line, where each point corresponds to a unique real number.
They allow us to perform a vast array of calculations and to express quantities with precision. When dealing with absolute values, it's important to remember that the concept applies across all real numbers, ensuring consistent interpretation of distance from zero.

Number Line Distance

The notion of absolute value ties directly into the concept of number line distance. On a number line, the absolute value represents the exact distance a number travels from zero, irrespective of direction.
Think of it like measuring steps taken, where both 8 steps forward and 8 steps backward result in an absolute value of 8 steps.
This characteristic is why the absolute value is always non-negative – distance can't be negative.
For any number \(a\), whether positive or negative, its absolute value \(|a|\) signifies this neutral measurement of distance.

Negative Sign Application

When dealing with expressions like \(-|8|\), you encounter an extra step: applying the negative sign after determining the absolute value.
Once the absolute value is found (in this case, \(|8| = 8\)), the expression \(-|8|\) directs us to take the negative of the absolute value.

• Here, \(-|8| = -8\)

This step is crucial because it affects the final result.
Applying a negative sign to an absolute value effectively reverses the sign of the result, highlighting the importance of careful interpretation of the expression's components.