Problem 61 - Elementary and Intermediate Algebra

Question

Find each absolute value. $$ |83| $$

Step-by-Step Solution

Answer

The absolute value of 83 is 83.

Step 1: Understand Absolute Value

The absolute value of a number is its distance from 0 on the number line, disregarding any direction. It is always a non-negative number. In this case, we need to find the absolute value of 83.

Step 2: Apply Absolute Value Definition

Since 83 is a positive number, its absolute value is the number itself. According to the definition of absolute value, \(|a| = a\) if \(a\) is positive.

Step 3: Finalize the Answer

Based on the definition from Step 2, the absolute value of 83 is simply 83.

Key Concepts

Understanding the Number Line

Imagine a straight line that stretches infinitely in both directions. This is the number line. It acts like a map for numbers, helping us to visualize where they fall in relation to each other and to zero, which is the central point of the number line. When we talk about a number's position on the number line, especially for absolute value calculations, we're interested in how far it is from zero. Think of it as counting the 'steps' from zero to that number.

• Numbers to the right of zero are positive numbers.
• Numbers to the left of zero are negative numbers.

Thus, the number line is not just useful for understanding positive and negative numbers, but it's also essential for grasping how absolute values work.

Positive Numbers Explained

Positive numbers are quite straightforward. They are any numbers greater than zero. On the number line, positive numbers are placed to the right of zero. They include numbers like 1, 2, 3, 4, and so on. When we consider the absolute value of a positive number, it remains unchanged. That's because the absolute value simply signifies its distance from zero. A positive number's distance to zero is the number itself, which is why its absolute value doesn't change. For example, if you see the number 83, its absolute value is 83 since it's already positive.

Mathematical Definition of Absolute Value

The absolute value of a number is defined as how far that number is from zero on the number line. Mathematically, it is represented as \[|a|\]The formula changes slightly depending on whether the number is positive or negative:

• For any positive number or zero, the absolute value of \(a\) is \(a\) itself: \( |a| = a \).
• For any negative number, the absolute value is its positive counterpart, which means \( |a| = -a \).

This mathematical definition is universal and helps us apply the concept of absolute value across all real numbers, ensuring that the value is never negative, always reflecting a non-negative distance from zero.