Problem 108 - Elementary Algebra

Question

Simplify. $$ |1| $$

Step-by-Step Solution

Answer

|1| = 1.

Step 1: Identify the Absolute Value

The exercise involves the absolute value symbol, which is represented by vertical bars \(|\cdot|\). The objective is to determine the absolute value of the number inside the bars, which is \(1\) in this case.

Step 2: Understand Absolute Value

Absolute value represents the distance a number is from zero on the number line, irrespective of direction. This means the absolute value of a number is always non-negative.

Step 3: Simplify the Absolute Value

Since \(1\) is already a non-negative number, its absolute value is itself. Thus, \(|1| = 1\).

Key Concepts

Number Line

The number line is a visual representation of numbers in a straight line. Imagine a horizontal line where each point corresponds to a number. The center of this line is marked by zero, and numbers extend in both directions.
To the right of zero, you'll find positive numbers, while to the left, you'll encounter negative numbers. This layout helps in understanding properties such as order, magnitude, and specially the absolute value.

• Zero is the central point and acts as the reference for measuring distance.
• Every step to the right increases the number's value, while a step to the left decreases it.

By using a number line, you can easily visualize the distance a number is from zero, which directly relates to understanding the concept of absolute values.

Non-negative Numbers

Non-negative numbers are simple to understand: they include all positive numbers and zero. In mathematical terms, a non-negative number is any number that is zero or greater.
This concept is crucial when dealing with absolute values because the result of any absolute value operation will always be a non-negative number.

• Examples include numbers like 0, 3, 15.7, or even large numbers such as 1000.
• Negative numbers, like -2 or -10, are not considered non-negative.

Remember, when you are calculating or simplifying absolute values, you will always end up with a non-negative number because absolute value measures how far a number is from zero without considering direction.

Distance from Zero

The concept of distance from zero is foundational to understanding absolute values. In essence, the absolute value of a number tells you how far that number is from zero on a number line, without regard to whether it's to the left or right.
For instance, if you have the number -5, its absolute value is 5, which indicates that it is 5 units away from zero, even though it's on the negative side of the line. Similarly, the absolute value of 5 is also 5, as it is the same distance from zero but on the positive side.

• The key point is that absolute value disregards direction; it only considers magnitude.
• Hence, this measure of distance is always non-negative.

Understanding the distance from zero helps simplify the process of working with absolute values, as it emphasizes that you're only focused on how far, not where, a number is from zero.