Problem 2 - Fundamentals of Mathematics

Question

Determine each a bsolute value. $$ |-3| $$

Step-by-Step Solution

Answer

3

Step 1: Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. This means that the absolute value of both positive and negative numbers is positive.

Step 2: Applying the Rule to -3

The number given is -3. Using the rule for absolute value, we take the distance of -3 from zero, which is 3, as distance is always positive.

Step 3: Writing the Solution

Since the absolute value of -3 is 3, we write the solution as \(|-3| = 3\).

Key Concepts

Understanding the Number Line

The number line is a simple yet powerful tool to visualize numbers and their relationships. It is a horizontal line where each point represents a unique number, flowing from negative to positive. The left side holds negative numbers, zero is at the center, and positive numbers stretch towards the right.

• Negative numbers: to the left of zero
• Zero: the center point
• Positive numbers: to the right of zero

Every number on this line has a specific 'address' or position based on its value. The number line helps in understanding basic operations like addition, subtraction, and evaluating the absolute value. When calculating absolute values, our focus is on measuring how far a number is from zero, not the direction.

Exploring Positive Numbers

Positive numbers are all the numbers that you find to the right of zero on the number line. They are whole numbers such as 1, 2, 3, and so on, but also fractions and decimals like 0.5 or 1.75.

• Characteristics: Greater than zero
• Found to the right of zero, increasing in value

Positive numbers have a natural order and are generally what people refer to when they talk about counting. They reflect values above zero, which means anytime you're dealing with positive numbers, you are dealing with units of increase or gain.

Calculating Distance from Zero

The concept of 'distance from zero' is central to understanding absolute value. When we talk about distance on the number line, it is always a non-negative measurement. Imagine standing at zero and counting steps to a number, whether it's positive like 4 or negative like -3. The distance is simply how many steps you take, ignoring direction.

For example:

• Distance from zero to 3 is 3 steps, so ext{equation} |3| = 3
• Distance from zero to -3 is also 3 steps, thus ext{equation} |-3| = 3

The absolute value of a number, noted as ext{equation} | | (such as ext{equation} |-3| ), reflects this concept of distance, illustrating that regardless of a number's position left or right of zero, its absolute value is always a positive measure.