Problem 42 - 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

Absolute value measures the distance of a number from zero on the number line. It is always a non-negative number, regardless of whether the original number is positive or negative.

Step 2: Identify the Number

In this exercise, you are given the number -457. You need to find the absolute value of this negative number.

Step 3: Apply Absolute Value Definition

The absolute value of -457 is found by removing the negative sign, essentially finding the distance from zero. Mathematically, \[|-457| = 457\]This is because -457 is 457 units away from zero on the number line.

Key Concepts

Number Line

The number line is a crucial mathematical tool that helps us visualize and understand the position of numbers in a straight horizontal line. This line extends indefinitely in both directions—one side represents positive numbers and the other represents negative numbers. The center point of this line is labeled as zero.

On the number line, each notch or mark corresponds to a real number. Moving to the right from zero, we see positive numbers, and moving to the left, we find negative numbers. Visualizing this can help in understanding mathematical concepts like absolute value, as these numbers can be represented as a position or point on the number line.

• A negative number is to the left of zero.
• A positive number is to the right of zero.
• The absolute value is basically the position of the number, regardless of sign.

The concept of the number line is particularly useful in solving problems involving absolute values, as it provides a clear visual representation of the distance of any given number from zero, which is always non-negative.

Non-Negative Numbers

Non-negative numbers are numbers that are greater than or equal to zero. This includes all positive numbers and zero itself. Negative numbers do not belong to this category because they are less than zero.

In the context of absolute value, every absolute value result is a non-negative number. This is because the absolute value is fundamentally concerned with the distance from zero, and distances cannot be negative.

• Numbers like 0, 1, 10, and 23 are non-negative.
• The absolute value of a negative number is non-negative. For example, \(|-5| = 5\).
• Even zero itself has an absolute value of zero, when represented on the number line, it is its own distance from zero.

Understanding non-negative numbers is essential when dealing with absolute values, as the result of any absolute value calculation must fall into this category, showcasing its nature independent of the original number's sign.

Distance from Zero

The concept of distance from zero is at the heart of understanding absolute values. It is the measure of how far a number is from zero on the number line, not considering the direction (left or right) from which the number originates.

When we talk about distance in everyday life, we understand that distance is inherently positive. This same idea applies to numbers on the line: regardless of whether the number is negative or positive, the distance from zero always results in a non-negative value.

• For a positive number, the absolute value is the number itself. For example, \( |3| = 3 \).
• For a negative number, the absolute value involves changing its sign. For example, \( |-8| = 8 \).
• Zero has no distance from itself, so \( |0| = 0 \).

The property of being a distance from zero makes the absolute value a powerful tool in mathematics, especially in determining the magnitude of numbers, free from the constraints of their directional value on the number line.