Problem 50 - Prealgebra

Question

Find each of the following absolute values. $$|457|$$

Step-by-Step Solution

Answer

The absolute value of 457 is 457.

Step 1: Understanding Absolute Value

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

Step 2: Identify the Number

We are given the number 457. To find the absolute value, we need to consider its distance from zero on the number line.

Step 3: Evaluating Absolute Value for Positive Numbers

Since 457 is a positive number, its absolute value is simply the number itself. Therefore, \(|457| = 457\).

Key Concepts

Number Line

A number line is a simple yet vital tool in mathematics. Imagine a straight line with numbers placed at equal intervals or distances along its length. The center of this line is marked as zero. To the right, the numbers increase positively, while to the left, they decrease negatively.

Understanding the number line is crucial because it visually represents numbers in a way that helps explain concepts such as the absolute value. On this line, any number's position is its key feature. This position is used to measure another important value: how far that number is from zero.

Here's why the number line is particularly useful:

• It clearly shows the order of numbers, helping understand greater and lesser values.
• It visually demonstrates the concept of absolute value since you can "see" the distance a number is from zero.
• It is an essential tool for basic arithmetic operations, such as addition and subtraction.

Distance from Zero

The concept of 'distance from zero' is central to understanding absolute value. It refers to how far away a number is from zero on the number line, no matter which direction. Think of it as measuring the gap between two points without worrying about their order.

For example:

• The distance from zero to 5 on the number line is 5 steps.
• Similarly, the distance from zero to -5 is also 5 steps.

The important takeaway is that absolute value considers only this distance, not the direction. Hence, absolute values are always non-negative.

By understanding 'distance from zero,' solving absolute value problems becomes easier and helps in reasoning about numbers in different contexts.

Positive Numbers

Positive numbers are all the numbers greater than zero. They appear to the right of zero on a standard number line. In the realm of absolute values, calculations involving positive numbers are straightforward. This is because their value is already their distance from zero.

When we talk about positive numbers:

• They never involve negative signs.
• Their absolute value is always the number itself.
• Examples of positive numbers include 1, 42, and 457.

In the example provided, where we find the absolute value of 457, it is already positive. Therefore, according to the rules of absolute value, |457| equals 457 itself. This is why understanding positive numbers helps simplify many math problems. Knowing these properties makes recognizing their role in equations easier.