Problem 13 - Fundamentals of Mathematics

Question

Use the algebraic definition of absolute value to find the following values. $$ -|-52| $$

Step-by-Step Solution

Answer

The value is -52.

Step 1: Understanding Absolute Value

The absolute value of a number is the non-negative value of that number without regard to its sign. The absolute value of any number \( x \) is denoted \( |x| \) and is defined as \( |x| = x \) if \( x \geq 0 \), and \( |x| = -x \) if \( x < 0 \).

Step 2: Calculate Absolute Value of -52

We are given \( |-52| \). Since -52 is less than 0, we use the definition \( |-52| = -(-52) \), which simplifies to \( 52 \).

Step 3: Apply Negative Sign

After calculating the absolute value, we still have to apply the negative sign in front of the absolute value. Therefore, we are looking for \( -|52| \). As calculated, \( |52| = 52 \), hence \( -|52| = -52 \).

Key Concepts

Algebraic Definition

The concept of absolute value is fundamental in mathematics. It essentially measures how far a number is from zero on the number line, ignoring any negative sign. In algebraic terms, the absolute value of a number \( x \) is symbolized by \(|x|\). Here's how it breaks down:

• If \( x \geq 0 \), then \(|x| = x\). This means if \( x \) is zero or a positive number, its absolute value is itself.
• If \( x < 0 \), then \(|x| = -x\). This indicates that for a negative number, we take its opposite value, which is positive.

This definition helps in various mathematical contexts, especially dealing with distance, error estimation, and real-world problems where direction or sign is not significant but magnitude is.

Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign (-). They are crucial in contexts such as finances, temperatures, and altitude. Understanding negative numbers helps make sense of situations where quantities can decrease or go below a starting point.

In the context of our original exercise, the given number is \(-52\). This number is negative, which means it lies to the left of zero on a number line. Despite negative numbers appearing smaller, they often represent large quantities, especially in absolute terms. The existence of negative numbers allows for full range expressions in mathematics, essential for complete and accurate calculations.

Mathematical Operations

Mathematical operations consist of basic manipulations such as addition, subtraction, multiplication, and division. When working with absolute values, it's important to handle these operations correctly.

Let’s break down the steps involved in our exercise:

• First, determine the absolute value: since the number was -52, we find its absolute value by recognizing it is less than zero and calculating \(|-52| = 52\).
• Apply any existing operations: in our case, a negative sign in front of the absolute value. The operation necessary transforms \( |52| \) into \(-|52|\), resulting in -52.

Understanding how these operations interplay ensures correctness in results and fosters a deeper understanding of mathematical concepts.